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1714 questions

Concept of a set · FoundationCount the distinct elements.Original teaching material · Worked example Concept of a set · StandardCount the integers satisfying the condition.Original teaching material · Worked example Concept of a set · TransferDoes “interesting books” specify a well-defined set? Repair the description.Original teaching material · Worked example Concept of a set · FoundationCount the distinct elements.Original teaching material · Answer check Concept of a set · FoundationCount the integers satisfying the condition.Original teaching material · Answer check Concept of a set · FoundationWrite the real solution set.Original teaching material · Reasoning checklist Concept of a set · FoundationAre the two sets equal? Explain.Original teaching material · Reasoning checklist Concept of a set · StandardHow many elements are in this set?Original teaching material · Answer check Concept of a set · StandardGive a set-builder description.Original teaching material · Reasoning checklist Concept of a set · StandardHow many elements are in this real solution set?Original teaching material · Answer check Concept of a set · StandardCount the distinct elements.Original teaching material · Answer check Concept of a set · TransferCount the integers satisfying the condition.Original teaching material · Answer check Concept of a set · TransferWrite the real solution set.Original teaching material · Reasoning checklist Basic relations between sets · FoundationHow many subsets does A have?Original teaching material · Worked example Basic relations between sets · StandardHow many proper subsets does A have?Original teaching material · Worked example Basic relations between sets · TransferCount the nonempty subsets that omit one specified element.Original teaching material · Worked example Basic relations between sets · FoundationHow many subsets does A have?Original teaching material · Answer check Basic relations between sets · FoundationHow many proper subsets does A have?Original teaching material · Answer check Basic relations between sets · FoundationHow many subsets must contain one specified element?Original teaching material · Answer check Basic relations between sets · FoundationExplain why these statements are equivalent.Original teaching material · Reasoning checklist Basic relations between sets · StandardProve that the empty set is a subset of every set.Original teaching material · Reasoning checklist Basic relations between sets · StandardProve transitivity of inclusion.Original teaching material · Reasoning checklist Basic relations between sets · StandardCount the nonempty subsets that omit one specified element.Original teaching material · Answer check Basic relations between sets · StandardHow many subsets does A have?Original teaching material · Answer check Basic relations between sets · TransferHow many proper subsets does A have?Original teaching material · Answer check Basic relations between sets · TransferHow many subsets must contain one specified element?Original teaching material · Answer check Basic operations on sets · FoundationFind the cardinality of the intersection.Original teaching material · Worked example Basic operations on sets · StandardFind the cardinality of the union.Original teaching material · Worked example Basic operations on sets · TransferHow many elements belong to exactly one of A and B?Original teaching material · Worked example Basic operations on sets · FoundationFind the cardinality of the intersection.Original teaching material · Answer check Basic operations on sets · FoundationFind the cardinality of the union.Original teaching material · Answer check Basic operations on sets · FoundationFind the cardinality of the complement.Original teaching material · Answer check Basic operations on sets · FoundationProve the complement-of-union rule.Original teaching material · Reasoning checklist Basic operations on sets · StandardA group has 22 people; 9 take music, 10 sport, and 3 both. How many take neither?Original teaching material · Answer check Basic operations on sets · StandardFind the interval intersection.Original teaching material · Reasoning checklist Basic operations on sets · StandardHow many elements belong to exactly one of A and B?Original teaching material · Answer check Basic operations on sets · StandardFind the cardinality of the intersection.Original teaching material · Answer check Basic operations on sets · TransferFind the cardinality of the union.Original teaching material · Answer check Basic operations on sets · TransferFind the cardinality of the complement.Original teaching material · Answer check Sufficient and necessary conditions · FoundationClassify p as a condition for q.Original teaching material · Worked example Sufficient and necessary conditions · StandardClassify p as a condition for q over the reals.Original teaching material · Worked example Sufficient and necessary conditions · TransferDisprove: a positive product forces both factors to be positive.Original teaching material · Worked example Sufficient and necessary conditions · FoundationClassify p as a condition for q.Original teaching material · Reasoning checklist Sufficient and necessary conditions · FoundationClassify p as a condition for q over the reals.Original teaching material · Reasoning checklist Sufficient and necessary conditions · FoundationGive a necessary and sufficient condition for the product to vanish.Original teaching material · Reasoning checklist Sufficient and necessary conditions · FoundationRepair this proposed implication.Original teaching material · Reasoning checklist Sufficient and necessary conditions · StandardUsing “at least two equal sides” for isosceles, classify equilateral as a condition for isosceles.Original teaching material · Reasoning checklist Sufficient and necessary conditions · StandardClassify the relation between p and q.Original teaching material · Reasoning checklist Sufficient and necessary conditions · StandardFor x>0, classify p as a condition for q.Original teaching material · Reasoning checklist Sufficient and necessary conditions · StandardClassify p as a condition for q.Original teaching material · Reasoning checklist Sufficient and necessary conditions · TransferClassify p as a condition for q over the reals.Original teaching material · Reasoning checklist Sufficient and necessary conditions · TransferClassify being an integer as a condition for being rational.Original teaching material · Reasoning checklist Universal and existential quantifiers · FoundationNegate this statement.Original teaching material · Worked example Universal and existential quantifiers · StandardNegate this existential claim.Original teaching material · Worked example Universal and existential quantifiers · TransferDecide whether each domain supplies a witness.Original teaching material · Worked example Universal and existential quantifiers · FoundationNegate this statement.Original teaching material · Reasoning checklist Universal and existential quantifiers · FoundationNegate this existential claim.Original teaching material · Reasoning checklist Universal and existential quantifiers · FoundationProve the quantified statement.Original teaching material · Reasoning checklist Universal and existential quantifiers · FoundationGive an integer counterexample.Original teaching material · Reasoning checklist Universal and existential quantifiers · StandardCount the possible integer witnesses.Original teaching material · Answer check Universal and existential quantifiers · StandardExplain the effect of exchanging the quantifier order.Original teaching material · Reasoning checklist Universal and existential quantifiers · StandardDetermine whether x²=3 has a witness in the integers and in the reals.Original teaching material · Reasoning checklist Universal and existential quantifiers · StandardNegate this statement.Original teaching material · Reasoning checklist Universal and existential quantifiers · TransferNegate this existential claim.Original teaching material · Reasoning checklist Universal and existential quantifiers · TransferProve this universal statement by cases.Original teaching material · Reasoning checklist Properties of equalities and inequalities · FoundationSolve and justify the sign direction.Original teaching material · Worked example Properties of equalities and inequalities · StandardCompare the reciprocals for 0<a<b.Original teaching material · Worked example Properties of equalities and inequalities · TransferExplain how the sign of a affects multiplying x<b by a.Original teaching material · Worked example Properties of equalities and inequalities · FoundationSolve and justify the sign direction.Original teaching material · Reasoning checklist Properties of equalities and inequalities · FoundationGiven −2<x<5, find the interval containing −3x.Original teaching material · Reasoning checklist Properties of equalities and inequalities · FoundationFind the least integer greater than every admissible sum a+b.Original teaching material · Answer check Properties of equalities and inequalities · FoundationGive a counterexample to squaring as a general order-preserving rule.Original teaching material · Reasoning checklist Properties of equalities and inequalities · StandardConvert to a double inequality.Original teaching material · Reasoning checklist Properties of equalities and inequalities · StandardDoes a<b and c<b guarantee a<c? Explain.Original teaching material · Reasoning checklist Properties of equalities and inequalities · StandardFrom a<b and c<d, may one conclude a−c<b−d? Give a counterexample.Original teaching material · Reasoning checklist Properties of equalities and inequalities · StandardSolve and justify the sign direction.Original teaching material · Reasoning checklist Properties of equalities and inequalities · TransferIf a<0<b, compare 1/a and 1/b.Original teaching material · Reasoning checklist Properties of equalities and inequalities · TransferFind the least integer greater than every admissible sum a+b.Original teaching material · Answer check Basic inequality: AM–GM · FoundationFind the minimum for x>0.Original teaching material · Worked example Basic inequality: AM–GM · StandardPositive a,b have sum 4; maximize their product.Original teaching material · Worked example Basic inequality: AM–GM · TransferMinimize the sum of two positive numbers with this product.Original teaching material · Worked example Basic inequality: AM–GM · FoundationFind the minimum for x>0.Original teaching material · Answer check Basic inequality: AM–GM · FoundationPositive a,b have sum 6; maximize their product.Original teaching material · Answer check Basic inequality: AM–GM · FoundationA rectangle has perimeter 12. Find its largest area.Original teaching material · Answer check Basic inequality: AM–GM · FoundationMinimize the expression over positive x.Original teaching material · Answer check Basic inequality: AM–GM · StandardDoes the function attain a minimum on this domain?Original teaching material · Reasoning checklist Basic inequality: AM–GM · StandardProve AM–GM for nonnegative a,b.Original teaching material · Reasoning checklist Basic inequality: AM–GM · StandardMinimize the sum of two positive numbers with this product.Original teaching material · Answer check Basic inequality: AM–GM · StandardFind the minimum for x>0.Original teaching material · Answer check Basic inequality: AM–GM · TransferPositive a,b have sum 8; maximize their product.Original teaching material · Answer check Basic inequality: AM–GM · TransferA rectangle has perimeter 16. Find its largest area.Original teaching material · Answer check Quadratic functions, equations and inequalities · FoundationSolve the quadratic inequality.Original teaching material · Worked example Quadratic functions, equations and inequalities · StandardCount the integer solutions.Original teaching material · Worked example Quadratic functions, equations and inequalities · TransferSolve the case a=0 separately.Original teaching material · Worked example Quadratic functions, equations and inequalities · FoundationSolve the quadratic inequality.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities · FoundationCount the integer solutions.Original teaching material · Answer check Quadratic functions, equations and inequalities · FoundationCalculate the discriminant and interpret its sign.Original teaching material · Answer check Quadratic functions, equations and inequalities · FoundationSolve the inequality with a negative leading coefficient.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities · StandardFor which m is this positive for every real x?Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities · StandardFind the length of the x-interval where the line is above or on the parabola.Original teaching material · Answer check Quadratic functions, equations and inequalities · StandardSolve the case a=0 separately.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities · StandardSolve the quadratic inequality.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities · TransferCount the integer solutions.Original teaching material · Answer check Quadratic functions, equations and inequalities · TransferCalculate the discriminant and interpret its sign.Original teaching material · Answer check Function concept and representations · FoundationEvaluate the function at the stated input.Original teaching material · Worked example Function concept and representations · StandardFind the real domain.Original teaching material · Worked example Function concept and representations · TransferRecover f(x) from the shifted-input identity.Original teaching material · Worked example Function concept and representations · FoundationEvaluate the function at the stated input.Original teaching material · Answer check Function concept and representations · FoundationFind the real domain.Original teaching material · Reasoning checklist Function concept and representations · FoundationFind the greatest value on the stated closed interval.Original teaching material · Answer check Function concept and representations · FoundationEvaluate the piecewise rule at the breakpoint.Original teaching material · Answer check Function concept and representations · StandardAre these the same function with their natural real domains? Explain.Original teaching material · Reasoning checklist Function concept and representations · StandardDoes the relation define y as a function of x? Explain.Original teaching material · Reasoning checklist Function concept and representations · StandardRecover f(x) from the shifted-input identity.Original teaching material · Reasoning checklist Function concept and representations · StandardEvaluate the function at the stated input.Original teaching material · Answer check Function concept and representations · TransferFind the real domain.Original teaching material · Reasoning checklist Function concept and representations · TransferFind the greatest value on the stated closed interval.Original teaching material · Answer check Basic properties of functions · FoundationState the monotonicity on the real line and justify it.Original teaching material · Worked example Basic properties of functions · StandardClassify parity on the real line.Original teaching material · Worked example Basic properties of functions · TransferIs the function even on its stated domain? Explain.Original teaching material · Worked example Basic properties of functions · FoundationState the monotonicity on the real line and justify it.Original teaching material · Reasoning checklist Basic properties of functions · FoundationClassify parity on the real line.Original teaching material · Reasoning checklist Basic properties of functions · FoundationAn even function has the stated value. Find the reflected value.Original teaching material · Answer check Basic properties of functions · FoundationFind the minimum on the stated interval.Original teaching material · Answer check Basic properties of functions · StandardHow does adding a constant affect monotonicity? Prove your answer.Original teaching material · Reasoning checklist Basic properties of functions · StandardCan a nonzero constant function be odd? Explain.Original teaching material · Reasoning checklist Basic properties of functions · StandardIs f(x)=1/x odd on its natural domain? Justify the domain condition.Original teaching material · Reasoning checklist Basic properties of functions · StandardState the monotonicity on the real line and justify it.Original teaching material · Reasoning checklist Basic properties of functions · TransferClassify parity on the real line.Original teaching material · Reasoning checklist Basic properties of functions · TransferAn even function has the stated value. Find the reflected value.Original teaching material · Answer check Power functions · FoundationEvaluate the power.Original teaching material · Worked example Power functions · StandardGive the real domain and range.Original teaching material · Worked example Power functions · TransferSolve the power comparison for positive x.Original teaching material · Worked example Power functions · FoundationEvaluate the power.Original teaching material · Answer check Power functions · FoundationFind the real domain and range of the square-root power function.Original teaching material · Reasoning checklist Power functions · FoundationEvaluate the real cube root.Original teaching material · Answer check Power functions · FoundationFor 0<x<1, compare x² and x³ and prove the comparison.Original teaching material · Reasoning checklist Power functions · StandardClassify the parity of this power function on the reals.Original teaching material · Reasoning checklist Power functions · StandardFind m so that this is a power function with coefficient one.Original teaching material · Answer check Power functions · StandardSolve the power comparison for positive x.Original teaching material · Reasoning checklist Power functions · StandardEvaluate the power.Original teaching material · Answer check Power functions · TransferInterpret x^(2/3) as the square of the real cube root. Find its domain, range and parity.Original teaching material · Reasoning checklist Power functions · TransferEvaluate the real cube root.Original teaching material · Answer check Applications of functions I · FoundationA taxi charges 2 units initially and 3 per kilometre. Find the charge for 4 km.Original teaching material · Worked example Applications of functions I · StandardFind the maximum profit under the continuous model.Original teaching material · Worked example Applications of functions I · TransferA learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim.Original teaching material · Worked example Applications of functions I · FoundationA taxi charges 3 units initially and 3 per kilometre. Find the charge for 5 km.Original teaching material · Answer check Applications of functions I · FoundationFind the maximum profit under the continuous model.Original teaching material · Answer check Applications of functions I · FoundationA journey covers equal distances at speeds 6 and 18. Find its average speed.Original teaching material · Answer check Applications of functions I · FoundationA service costs C=3+4n for n whole items. A bill is 23. Find n.Original teaching material · Answer check Applications of functions I · StandardA budget is 17 and each item costs 5. Find the greatest number of whole items affordable.Original teaching material · Answer check Applications of functions I · StandardWhy should the input domain of a ticket-revenue model be discrete and bounded?Original teaching material · Reasoning checklist Applications of functions I · StandardA fare is 8 for 0≤d≤2, then 8+3(d−2) for d>2. Find the fare at d=5 and explain why the branches agree at d=2.Original teaching material · Answer check Applications of functions I · StandardA taxi charges 4 units initially and 3 per kilometre. Find the charge for 6 km.Original teaching material · Answer check Applications of functions I · TransferFind the maximum profit under the continuous model.Original teaching material · Answer check Applications of functions I · TransferA journey covers equal distances at speeds 8 and 24. Find its average speed.Original teaching material · Answer check Exponents · FoundationSimplify the product.Original teaching material · Worked example Exponents · StandardEvaluate the negative power.Original teaching material · Worked example Exponents · TransferCompare the powers without decimal approximations.Original teaching material · Worked example Exponents · FoundationSimplify the product.Original teaching material · Answer check Exponents · FoundationEvaluate the negative power.Original teaching material · Answer check Exponents · FoundationEvaluate the rational power.Original teaching material · Answer check Exponents · FoundationSimplify for a>0.Original teaching material · Reasoning checklist Exponents · StandardExplain why the rule a⁰=1 must not be used to assign 0⁰ here.Original teaching material · Reasoning checklist Exponents · StandardSimplify for real x, including negative x.Original teaching material · Reasoning checklist Exponents · StandardCompare the powers without decimal approximations.Original teaching material · Reasoning checklist Exponents · StandardSimplify the product.Original teaching material · Answer check Exponents · TransferEvaluate the negative power.Original teaching material · Answer check Exponents · TransferEvaluate the rational power.Original teaching material · Answer check Exponential functions · FoundationFind the positive exponential base.Original teaching material · Worked example Exponential functions · StandardSolve the exponential equation.Original teaching material · Worked example Exponential functions · TransferExplain why an exponential model with base one is excluded from the usual increasing/decreasing classification.Original teaching material · Worked example Exponential functions · FoundationFind the positive exponential base.Original teaching material · Answer check Exponential functions · FoundationSolve the exponential equation.Original teaching material · Answer check Exponential functions · FoundationCompare the two values for x<y.Original teaching material · Reasoning checklist Exponential functions · FoundationState the range and horizontal asymptote.Original teaching material · Reasoning checklist Exponential functions · StandardFind the y-intercept.Original teaching material · Answer check Exponential functions · StandardA quantity starts at 3 and doubles each hour. Find it after 3 hours.Original teaching material · Answer check Exponential functions · StandardProve that the difference of these two exponentials is odd.Original teaching material · Reasoning checklist Exponential functions · StandardFind the positive exponential base.Original teaching material · Answer check Exponential functions · TransferSolve the exponential equation.Original teaching material · Answer check Exponential functions · TransferCompare the two values for x<y.Original teaching material · Reasoning checklist Logarithms · FoundationEvaluate the logarithm.Original teaching material · Worked example Logarithms · StandardSimplify using the product rule.Original teaching material · Worked example Logarithms · TransferSolve the logarithmic equation.Original teaching material · Worked example Logarithms · FoundationEvaluate the logarithm.Original teaching material · Answer check Logarithms · FoundationSimplify using the product rule.Original teaching material · Answer check Logarithms · FoundationEvaluate by change of base.Original teaching material · Answer check Logarithms · FoundationFind the real domain.Original teaching material · Reasoning checklist Logarithms · StandardDisprove the claimed addition rule using positive numbers.Original teaching material · Reasoning checklist Logarithms · StandardEvaluate the logarithm with a base below one.Original teaching material · Answer check Logarithms · StandardSolve the logarithmic equation.Original teaching material · Answer check Logarithms · StandardEvaluate the logarithm.Original teaching material · Answer check Logarithms · TransferSimplify using the product rule.Original teaching material · Answer check Logarithms · TransferEvaluate by change of base.Original teaching material · Answer check Logarithmic functions · FoundationCompare the values when 0<x<y.Original teaching material · Worked example Logarithmic functions · StandardFind the domain and vertical asymptote.Original teaching material · Worked example Logarithmic functions · TransferExplain why the graphs y=log₂x and y=2ˣ are reflections in y=x.Original teaching material · Worked example Logarithmic functions · FoundationCompare the values when 0<x<y.Original teaching material · Reasoning checklist Logarithmic functions · FoundationFind the domain and vertical asymptote.Original teaching material · Reasoning checklist Logarithmic functions · FoundationFind the inverse function and its domain.Original teaching material · Reasoning checklist Logarithmic functions · FoundationSolve the inequality.Original teaching material · Reasoning checklist Logarithmic functions · StandardFind the x-intercept.Original teaching material · Answer check Logarithmic functions · StandardFind all m for which the expression is defined for every x in [0,1].Original teaching material · Reasoning checklist Logarithmic functions · StandardState the conditions under which these logarithms are reciprocals, then prove it.Original teaching material · Reasoning checklist Logarithmic functions · StandardCompare the values when 0<x<y.Original teaching material · Reasoning checklist Logarithmic functions · TransferFind the domain and vertical asymptote.Original teaching material · Reasoning checklist Logarithmic functions · TransferFind the inverse function and its domain.Original teaching material · Reasoning checklist Applications of functions II · FoundationA population is 200 and increases by 10% each year. Find it after two years.Original teaching material · Worked example Applications of functions II · StandardA sample starts at 160 mg and halves every 3 days. Find its mass after 9 days.Original teaching material · Worked example Applications of functions II · TransferA log-transformed data plot looks straight. Does this prove the original process is exponential at all times? Explain.Original teaching material · Worked example Applications of functions II · FoundationA population is 300 and increases by 10% each year. Find it after two years.Original teaching material · Answer check Applications of functions II · FoundationA sample starts at 240 mg and halves every 3 days. Find its mass after 9 days.Original teaching material · Answer check Applications of functions II · FoundationAn exponential amount grows from 3 to 24 with doubling every 2 hours. How many hours elapsed?Original teaching material · Answer check Applications of functions II · FoundationFind the base-10 logarithmic increase when a positive quantity is multiplied by 1000.Original teaching material · Answer check Applications of functions II · StandardShow that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique.Original teaching material · Reasoning checklist Applications of functions II · StandardCompare the changes over equal unit intervals for L(t)=3t and E(t)=3·2ᵗ.Original teaching material · Reasoning checklist Applications of functions II · StandardA continuous population model predicts 121.5 individuals. Explain how to report this result responsibly.Original teaching material · Reasoning checklist Applications of functions II · StandardA population is 400 and increases by 10% each year. Find it after two years.Original teaching material · Answer check Applications of functions II · TransferA sample starts at 320 mg and halves every 3 days. Find its mass after 9 days.Original teaching material · Answer check Applications of functions II · TransferAn exponential amount grows from 4 to 32 with doubling every 2 hours. How many hours elapsed?Original teaching material · Answer check General angles and radian measure · FoundationWrite the angle as cπ radians; find c.Original teaching material · Worked example General angles and radian measure · StandardConvert to degrees.Original teaching material · Worked example General angles and radian measure · TransferHow many coterminal representatives of 30° lie in this closed interval?Original teaching material · Worked example General angles and radian measure · FoundationWrite the angle as cπ radians; find c.Original teaching material · Answer check General angles and radian measure · FoundationConvert to degrees.Original teaching material · Answer check General angles and radian measure · FoundationA circle has radius r and angle θ radians. Find the arc length for these values.Original teaching material · Answer check General angles and radian measure · FoundationFind the sector area for the stated radian angle.Original teaching material · Answer check General angles and radian measure · StandardGive the coterminal angle in [0°,360°).Original teaching material · Answer check General angles and radian measure · StandardWhich quadrant contains the terminal ray? Explain.Original teaching material · Reasoning checklist General angles and radian measure · StandardHow many coterminal representatives of 30° lie in this closed interval?Original teaching material · Answer check General angles and radian measure · StandardWrite the angle as cπ radians; find c.Original teaching material · Answer check General angles and radian measure · TransferConvert to degrees.Original teaching material · Answer check General angles and radian measure · TransferA circle has radius r and angle θ radians. Find the arc length for these values.Original teaching material · Answer check Concept of trigonometric functions · FoundationFind sin θ from the terminal-ray point.Original teaching material · Worked example Concept of trigonometric functions · StandardFind cos θ when θ is in quadrant II.Original teaching material · Worked example Concept of trigonometric functions · TransferFind the radius of this terminal-ray point.Original teaching material · Worked example Concept of trigonometric functions · FoundationFind sin θ from the terminal-ray point.Original teaching material · Answer check Concept of trigonometric functions · FoundationFind cos θ when θ is in quadrant III.Original teaching material · Answer check Concept of trigonometric functions · FoundationFind tan θ from the terminal-ray point.Original teaching material · Answer check Concept of trigonometric functions · FoundationExplain why tangent is undefined at 90°.Original teaching material · Reasoning checklist Concept of trigonometric functions · StandardDetermine whether the two claimed trigonometric values can occur together.Original teaching material · Reasoning checklist Concept of trigonometric functions · StandardIn which quadrant is the angle, given both signs?Original teaching material · Reasoning checklist Concept of trigonometric functions · StandardFind the radius of this terminal-ray point.Original teaching material · Answer check Concept of trigonometric functions · StandardFind sin θ from the terminal-ray point.Original teaching material · Answer check Concept of trigonometric functions · TransferCan an angle in quadrant IV have sine 4/5? Explain.Original teaching material · Reasoning checklist Concept of trigonometric functions · TransferFind tan θ from the terminal-ray point.Original teaching material · Answer check Reduction formulas · FoundationFind the transformed sine value.Original teaching material · Worked example Reduction formulas · StandardFind the reflected cosine value.Original teaching material · Worked example Reduction formulas · TransferDisprove the claim that cos(π−α)=cos α for every α.Original teaching material · Worked example Reduction formulas · FoundationEvaluate cos 420° without a calculator.Original teaching material · Answer check Reduction formulas · FoundationEvaluate sin 225° exactly.Original teaching material · Reasoning checklist Reduction formulas · FoundationFind tangent after a half turn.Original teaching material · Answer check Reduction formulas · FoundationEvaluate sine at the opposite angle.Original teaching material · Answer check Reduction formulas · StandardSimplify the complementary-angle expression.Original teaching material · Reasoning checklist Reduction formulas · StandardEvaluate without a calculator.Original teaching material · Answer check Reduction formulas · StandardDerive the cotangent reduction identity and state a domain restriction.Original teaching material · Reasoning checklist Reduction formulas · StandardEvaluate tan(−π/4).Original teaching material · Answer check Reduction formulas · TransferSimplify sine after a three-quarter-turn shift.Original teaching material · Reasoning checklist Reduction formulas · TransferFind tangent after a half turn.Original teaching material · Answer check Graphs and properties of trigonometric functions · FoundationThe least positive period is cπ. Find c.Original teaching material · Worked example Graphs and properties of trigonometric functions · StandardState the full range.Original teaching material · Worked example Graphs and properties of trigonometric functions · TransferIs sine increasing on the whole real line? Give a counterexample.Original teaching material · Worked example Graphs and properties of trigonometric functions · FoundationThe least positive period is cπ. Find c.Original teaching material · Answer check Graphs and properties of trigonometric functions · FoundationState the full range.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · FoundationState the excluded inputs for tangent.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · FoundationCount the zeros on the closed interval.Original teaching material · Answer check Graphs and properties of trigonometric functions · StandardClassify the parity of tangent on its natural domain.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · StandardGive every x where sine reaches its maximum.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · StandardFind the minimum of cosine on [0,π] and where it occurs.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · StandardThe least positive period is cπ. Find c.Original teaching material · Answer check Graphs and properties of trigonometric functions · TransferState the full range.Original teaching material · Reasoning checklist Graphs and properties of trigonometric functions · TransferFind the least positive period of sin²x, expressed as cπ; enter c and justify why a half-sized candidate fails.Original teaching material · Answer check Trigonometric identities and transformations · FoundationGiven an acute angle, find sin 2α.Original teaching material · Worked example Trigonometric identities and transformations · StandardFind cos 2α using the given sine.Original teaching material · Worked example Trigonometric identities and transformations · TransferProve the identity and state its domain restriction.Original teaching material · Worked example Trigonometric identities and transformations · FoundationFor an acute α with sin α=5/13, find sin 2α.Original teaching material · Answer check Trigonometric identities and transformations · FoundationGiven cos α=3/5, find cos 2α.Original teaching material · Answer check Trigonometric identities and transformations · FoundationEvaluate sin 75° exactly.Original teaching material · Reasoning checklist Trigonometric identities and transformations · FoundationFind tan(α+β), assuming both tangents have the stated values.Original teaching material · Answer check Trigonometric identities and transformations · StandardFor π<α<3π/2 and cos α=−3/5, determine the sign and value of sin(α/2).Original teaching material · Reasoning checklist Trigonometric identities and transformations · StandardFind the amplitude of the combined sinusoid.Original teaching material · Answer check Trigonometric identities and transformations · StandardProve the identity and preserve its original excluded inputs.Original teaching material · Reasoning checklist Trigonometric identities and transformations · StandardFor acute α and β, find sin(α−β) from the given values.Original teaching material · Answer check Trigonometric identities and transformations · TransferGiven tan α=2, find tan 2α.Original teaching material · Answer check Trigonometric identities and transformations · TransferDerive the product-to-sum identity for cos x cos y.Original teaching material · Reasoning checklist The function y=A sin(ωx+φ) · FoundationFind the amplitude.Original teaching material · Worked example The function y=A sin(ωx+φ) · StandardWrite the least positive period as cπ; find c.Original teaching material · Worked example The function y=A sin(ωx+φ) · TransferGive a positive-amplitude expression for the same graph.Original teaching material · Worked example The function y=A sin(ωx+φ) · FoundationFind the amplitude.Original teaching material · Answer check The function y=A sin(ωx+φ) · FoundationWrite the least positive period as cπ; find c.Original teaching material · Answer check The function y=A sin(ωx+φ) · FoundationState the horizontal shift of this graph relative to y=sin(3x).Original teaching material · Reasoning checklist The function y=A sin(ωx+φ) · FoundationFind the value at x=0.Original teaching material · Answer check The function y=A sin(ωx+φ) · StandardFind A>0 from the maximum and minimum.Original teaching material · Answer check The function y=A sin(ωx+φ) · StandardWhat function remains when the frequency is zero? Does it have a least positive period?Original teaching material · Reasoning checklist The function y=A sin(ωx+φ) · StandardGive a positive-amplitude expression for the same graph.Original teaching material · Reasoning checklist The function y=A sin(ωx+φ) · StandardFind the amplitude.Original teaching material · Answer check The function y=A sin(ωx+φ) · TransferWrite the least positive period as cπ; find c.Original teaching material · Answer check The function y=A sin(ωx+φ) · TransferState the horizontal shift of this graph relative to y=sin(4x).Original teaching material · Reasoning checklist Applications of trigonometric functions · FoundationA 4-metre ladder makes 30° with level ground. Find the top height.Original teaching material · Worked example Applications of trigonometric functions · StandardFind the period in hours.Original teaching material · Worked example Applications of trigonometric functions · TransferTwo daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period.Original teaching material · Worked example Applications of trigonometric functions · FoundationA 6-metre ladder makes 30° with level ground. Find the top height.Original teaching material · Answer check Applications of trigonometric functions · FoundationFind the period in hours.Original teaching material · Answer check Applications of trigonometric functions · FoundationA wheel has radius 3 m and its centre is 5 m above ground. Find the greatest passenger height.Original teaching material · Answer check Applications of trigonometric functions · FoundationAt which first nonnegative time does the model reach its maximum?Original teaching material · Answer check Applications of trigonometric functions · StandardA tide has maximum 8 m and minimum 4 m. Find its mean level in a symmetric sinusoidal model.Original teaching material · Answer check Applications of trigonometric functions · StandardA student enters 30 into a calculator in radian mode to calculate sin 30°. Explain and repair the error.Original teaching material · Reasoning checklist Applications of trigonometric functions · StandardCould h(t)=1+2cos t model the height above ground of an object that must always stay above ground? Explain.Original teaching material · Reasoning checklist Applications of trigonometric functions · StandardA 8-metre ladder makes 30° with level ground. Find the top height.Original teaching material · Answer check Applications of trigonometric functions · TransferFind the period in hours.Original teaching material · Answer check Applications of trigonometric functions · TransferA wheel has radius 4 m and its centre is 6 m above ground. Find the greatest passenger height.Original teaching material · Answer check Concept of plane vectors · FoundationFind the vector magnitude.Original teaching material · Worked example Concept of plane vectors · StandardFind the displacement vector from A to B.Original teaching material · Worked example Concept of plane vectors · TransferA walker travels 2 m east then 2 m west. Find the magnitude of displacement.Original teaching material · Worked example Concept of plane vectors · FoundationFind the vector magnitude.Original teaching material · Answer check Concept of plane vectors · FoundationFind the displacement vector from A to B.Original teaching material · Reasoning checklist Concept of plane vectors · FoundationDo equal magnitudes guarantee equal vectors? Give a counterexample.Original teaching material · Reasoning checklist Concept of plane vectors · FoundationGive a unit vector in the same direction.Original teaching material · Reasoning checklist Concept of plane vectors · StandardExplain whether the zero vector has a unique direction.Original teaching material · Reasoning checklist Concept of plane vectors · StandardFind the opposite vector.Original teaching material · Reasoning checklist Concept of plane vectors · StandardA walker travels 3 m east then 3 m west. Find the magnitude of displacement.Original teaching material · Answer check Concept of plane vectors · StandardFind the vector magnitude.Original teaching material · Answer check Concept of plane vectors · TransferFind the displacement vector from A to B.Original teaching material · Reasoning checklist Concept of plane vectors · TransferDo equal magnitudes guarantee equal vectors? Give a counterexample.Original teaching material · Reasoning checklist Operations on plane vectors · FoundationFind the x-coordinate of a+b.Original teaching material · Worked example Operations on plane vectors · StandardCalculate the dot product.Original teaching material · Worked example Operations on plane vectors · TransferIs vector dot product associative as (a·b)·c=a·(b·c)? Explain the type issue.Original teaching material · Worked example Operations on plane vectors · FoundationFind the x-coordinate of a+b.Original teaching material · Answer check Operations on plane vectors · FoundationCalculate the dot product.Original teaching material · Answer check Operations on plane vectors · FoundationFind the magnitude after scalar multiplication.Original teaching material · Answer check Operations on plane vectors · FoundationSimplify using the triangle rule.Original teaching material · Reasoning checklist Operations on plane vectors · StandardFind the angle in degrees between these nonzero vectors.Original teaching material · Answer check Operations on plane vectors · StandardExpand the squared norm and explain the cross term.Original teaching material · Reasoning checklist Operations on plane vectors · StandardFor perpendicular vectors of lengths 3 and 4, compare |a+b| with |a|+|b|.Original teaching material · Reasoning checklist Operations on plane vectors · StandardFind the x-coordinate of a+b.Original teaching material · Answer check Operations on plane vectors · TransferCalculate the dot product.Original teaching material · Answer check Operations on plane vectors · TransferFind the magnitude after scalar multiplication.Original teaching material · Answer check Basis theorem and coordinate representation · FoundationFind the coefficient of e₁ in the given basis representation.Original teaching material · Worked example Basis theorem and coordinate representation · StandardFind the midpoint x-coordinate.Original teaching material · Worked example Basis theorem and coordinate representation · TransferFind the distance between the two points.Original teaching material · Worked example Basis theorem and coordinate representation · FoundationFind the coefficient of e₁ in the given basis representation.Original teaching material · Answer check Basis theorem and coordinate representation · FoundationFind the midpoint x-coordinate.Original teaching material · Answer check Basis theorem and coordinate representation · FoundationFind m for the two vectors to be parallel.Original teaching material · Answer check Basis theorem and coordinate representation · FoundationFind m for the vectors to be perpendicular.Original teaching material · Answer check Basis theorem and coordinate representation · StandardExplain why parallel vectors cannot form a basis for the plane.Original teaching material · Reasoning checklist Basis theorem and coordinate representation · StandardFind the signed scalar projection on the positive x-axis.Original teaching material · Answer check Basis theorem and coordinate representation · StandardFind the distance between the two points.Original teaching material · Answer check Basis theorem and coordinate representation · StandardFind the coefficient of e₁ in the given basis representation.Original teaching material · Answer check Basis theorem and coordinate representation · TransferFind the midpoint x-coordinate.Original teaching material · Answer check Basis theorem and coordinate representation · TransferFind m for the two vectors to be parallel.Original teaching material · Answer check Applications of plane vectors · FoundationFind the third side using the cosine rule.Original teaching material · Worked example Applications of plane vectors · StandardFind b using the sine rule.Original teaching material · Worked example Applications of plane vectors · TransferExplain why the sine rule in an SSA problem may give two triangles.Original teaching material · Worked example Applications of plane vectors · FoundationFind the third side using the cosine rule.Original teaching material · Answer check Applications of plane vectors · FoundationFind b using the sine rule.Original teaching material · Answer check Applications of plane vectors · FoundationFind the triangle area.Original teaching material · Answer check Applications of plane vectors · FoundationFind the x-coordinate of the centroid.Original teaching material · Answer check Applications of plane vectors · StandardFind the work done by the constant force, in joules.Original teaching material · Answer check Applications of plane vectors · StandardFind the magnitude of the resultant force.Original teaching material · Answer check Applications of plane vectors · StandardCan side lengths 1,1,3 form a triangle to which the cosine rule applies? Explain.Original teaching material · Reasoning checklist Applications of plane vectors · StandardFind the third side using the cosine rule.Original teaching material · Answer check Applications of plane vectors · TransferFind b using the sine rule.Original teaching material · Answer check Applications of plane vectors · TransferFind the triangle area.Original teaching material · Answer check Concept of complex numbers · FoundationFind the imaginary part.Original teaching material · Worked example Concept of complex numbers · StandardFind m so the complex number is real.Original teaching material · Worked example Concept of complex numbers · TransferFind the sum a+b when the complex number is zero.Original teaching material · Worked example Concept of complex numbers · FoundationFind the imaginary part.Original teaching material · Answer check Concept of complex numbers · FoundationFind m so the complex number is real.Original teaching material · Answer check Concept of complex numbers · FoundationFind the modulus.Original teaching material · Answer check Concept of complex numbers · FoundationWrite the conjugate.Original teaching material · Reasoning checklist Concept of complex numbers · StandardState the quadrant of the complex-plane point.Original teaching material · Reasoning checklist Concept of complex numbers · StandardSolve over the complex numbers.Original teaching material · Reasoning checklist Concept of complex numbers · StandardFind the sum a+b when the complex number is zero.Original teaching material · Answer check Concept of complex numbers · StandardFind the imaginary part.Original teaching material · Answer check Concept of complex numbers · TransferFind m so the complex number is real.Original teaching material · Answer check Concept of complex numbers · TransferFind the modulus.Original teaching material · Answer check Arithmetic of complex numbers · FoundationFind the real part of the product.Original teaching material · Worked example Arithmetic of complex numbers · StandardSimplify the quotient.Original teaching material · Worked example Arithmetic of complex numbers · TransferSimplify the sum and explain the cancellation.Original teaching material · Worked example Arithmetic of complex numbers · FoundationFind the real part of the product.Original teaching material · Answer check Arithmetic of complex numbers · FoundationSimplify the quotient.Original teaching material · Reasoning checklist Arithmetic of complex numbers · FoundationCalculate z times its conjugate.Original teaching material · Answer check Arithmetic of complex numbers · FoundationEvaluate the power.Original teaching material · Answer check Arithmetic of complex numbers · StandardSolve the quadratic over the complex numbers.Original teaching material · Reasoning checklist Arithmetic of complex numbers · StandardFind the imaginary part of the reciprocal.Original teaching material · Answer check Arithmetic of complex numbers · StandardExplain why the conjugate of a complex root is also a root of a polynomial with real coefficients.Original teaching material · Reasoning checklist Arithmetic of complex numbers · StandardFind the real part of the product.Original teaching material · Answer check Arithmetic of complex numbers · TransferSimplify the quotient.Original teaching material · Reasoning checklist Arithmetic of complex numbers · TransferCalculate z times its conjugate.Original teaching material · Answer check Trigonometric representation of complex numbers (starred) · FoundationGive a polar representation.Original teaching material · Worked example Trigonometric representation of complex numbers (starred) · StandardFind the modulus of the product.Original teaching material · Worked example Trigonometric representation of complex numbers (starred) · TransferWhy does zero have no uniquely defined argument?Original teaching material · Worked example Trigonometric representation of complex numbers (starred) · FoundationGive a polar representation.Original teaching material · Reasoning checklist Trigonometric representation of complex numbers (starred) · FoundationFind the modulus of the product.Original teaching material · Answer check Trigonometric representation of complex numbers (starred) · FoundationFind the real part using de Moivre’s formula.Original teaching material · Answer check Trigonometric representation of complex numbers (starred) · FoundationList the two square roots of 1 in polar or algebraic form.Original teaching material · Reasoning checklist Trigonometric representation of complex numbers (starred) · StandardFind the modulus of the quotient.Original teaching material · Answer check Trigonometric representation of complex numbers (starred) · StandardAre arguments θ and θ+2π different complex numbers at the same positive modulus? Explain.Original teaching material · Reasoning checklist Trigonometric representation of complex numbers (starred) · StandardList all cube roots of unity.Original teaching material · Reasoning checklist Trigonometric representation of complex numbers (starred) · StandardGive a polar representation.Original teaching material · Reasoning checklist Trigonometric representation of complex numbers (starred) · TransferFind the modulus of the product.Original teaching material · Answer check Trigonometric representation of complex numbers (starred) · TransferFind the real part using de Moivre’s formula.Original teaching material · Answer check Basic solid figures · FoundationHow many vertices does an 5-gonal prism have?Original teaching material · Worked example Basic solid figures · StandardHow many edges does an 5-gonal pyramid have?Original teaching material · Worked example Basic solid figures · TransferVerify Euler’s relation for an 5-gonal prism by giving V,E,F.Original teaching material · Worked example Basic solid figures · FoundationHow many vertices does an 6-gonal prism have?Original teaching material · Answer check Basic solid figures · FoundationHow many edges does an 6-gonal pyramid have?Original teaching material · Answer check Basic solid figures · FoundationFind the slant height of the right circular cone.Original teaching material · Answer check Basic solid figures · FoundationA plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area.Original teaching material · Answer check Basic solid figures · StandardDoes having two parallel faces alone guarantee a prism? Explain.Original teaching material · Reasoning checklist Basic solid figures · StandardA sphere has radius r. What is the distance of every surface point from its centre?Original teaching material · Answer check Basic solid figures · StandardVerify Euler’s relation for an 6-gonal prism by giving V,E,F.Original teaching material · Reasoning checklist Basic solid figures · StandardHow many vertices does an 7-gonal prism have?Original teaching material · Answer check Basic solid figures · TransferHow many edges does an 7-gonal pyramid have?Original teaching material · Answer check Basic solid figures · TransferFind the slant height of the right circular cone.Original teaching material · Answer check Oblique drawings of solids · FoundationIn the 45° half-depth oblique convention, an original depth is d. Find its drawn length.Original teaching material · Worked example Oblique drawings of solids · StandardRecover the actual depth from its half-scale drawn length.Original teaching material · Worked example Oblique drawings of solids · TransferCan you read a cube’s true space-diagonal length directly with a ruler from an oblique drawing? Explain a reliable alternative.Original teaching material · Worked example Oblique drawings of solids · FoundationIn the 45° half-depth oblique convention, an original depth is d. Find its drawn length.Original teaching material · Answer check Oblique drawings of solids · FoundationRecover the actual depth from its half-scale drawn length.Original teaching material · Answer check Oblique drawings of solids · FoundationA horizontal rectangle has side lengths 2 and 4. Its receding side is drawn at half length and 45°. The drawn area is c√2. Find c.Original teaching material · Answer check Oblique drawings of solids · FoundationDoes a right angle in space have to appear as a right angle in an oblique drawing? Explain.Original teaching material · Reasoning checklist Oblique drawings of solids · StandardA vertical edge has actual length 5 and vertical scale one. Find its drawn length.Original teaching material · Answer check Oblique drawings of solids · StandardHow should hidden edges be distinguished in a conventional solid sketch? Explain the purpose.Original teaching material · Reasoning checklist Oblique drawings of solids · StandardUnder a half-depth 45° oblique drawing, why is the horizontal area scale √2/4?Original teaching material · Reasoning checklist Oblique drawings of solids · StandardIn the 45° half-depth oblique convention, an original depth is d. Find its drawn length.Original teaching material · Answer check Oblique drawings of solids · TransferRecover the actual depth from its half-scale drawn length.Original teaching material · Answer check Oblique drawings of solids · TransferA horizontal region has oblique drawn area 9√2 using the half-depth 45° convention. Find its true area.Original teaching material · Answer check Surface areas and volumes of simple solids · FoundationFind the volume of the prism.Original teaching material · Worked example Surface areas and volumes of simple solids · StandardA cone has radius r and height h. Write its volume as cπ and find c.Original teaching material · Worked example Surface areas and volumes of simple solids · TransferSimilar solids have volume ratio 64 from large to small. Find the linear scale factor.Original teaching material · Worked example Surface areas and volumes of simple solids · FoundationFind the volume of the prism.Original teaching material · Answer check Surface areas and volumes of simple solids · FoundationA cone has radius r and height h. Write its volume as cπ and find c.Original teaching material · Answer check Surface areas and volumes of simple solids · FoundationA sphere has radius r. Write its volume as cπ and find c.Original teaching material · Answer check Surface areas and volumes of simple solids · FoundationFind c when the total surface area of this closed cylinder is cπ.Original teaching material · Answer check Surface areas and volumes of simple solids · StandardFind the volume of a pyramid frustum with parallel bases.Original teaching material · Answer check Surface areas and volumes of simple solids · StandardSimilar solids have linear scale factor 3 from small to large. Find the surface-area ratio.Original teaching material · Answer check Surface areas and volumes of simple solids · StandardA sphere is replaced by eight similar smaller spheres with the same total volume. Find the ratio of total new surface area to the original area.Original teaching material · Answer check Surface areas and volumes of simple solids · StandardFind the volume of the prism.Original teaching material · Answer check Surface areas and volumes of simple solids · TransferA cone has radius r and height h. Write its volume as cπ and find c.Original teaching material · Answer check Surface areas and volumes of simple solids · TransferA sphere has radius r. Write its volume as cπ and find c.Original teaching material · Answer check Positions of points, lines and planes in space · FoundationHow many planes are determined by three noncollinear points?Original teaching material · Worked example Positions of points, lines and planes in space · StandardHow many planes are determined by choosing three of four noncoplanar points?Original teaching material · Worked example Positions of points, lines and planes in space · TransferHow many planes contain two distinct parallel lines? Explain.Original teaching material · Worked example Positions of points, lines and planes in space · FoundationHow many planes pass through three distinct collinear points? Explain.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · FoundationHow many planes contain a given line and a point outside it? Explain.Original teaching material · Answer check Positions of points, lines and planes in space · FoundationIn the coordinate cube, classify AB and CC′.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · FoundationState the intersection of two distinct planes known to share two distinct points.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · StandardDo three pairwise intersecting lines always lie in one plane? Give a counterexample.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · StandardProve that a line with two distinct points in a plane lies entirely in the plane.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · StandardCan two distinct planes have exactly one common point? Explain.Original teaching material · Reasoning checklist Positions of points, lines and planes in space · StandardFive points have no three collinear and no four coplanar. How many different planes do their triples determine?Original teaching material · Answer check Positions of points, lines and planes in space · TransferHow many regions do two distinct intersecting planes divide space into?Original teaching material · Answer check Positions of points, lines and planes in space · TransferIn the coordinate cube, classify AB and CC′.Original teaching material · Reasoning checklist Parallel lines and planes in space · FoundationA line l is outside plane α and parallel to a line m in α. State and justify the conclusion.Original teaching material · Worked example Parallel lines and planes in space · StandardIn tetrahedron ABCD, E and F are midpoints of AB and AC. Prove EF is parallel to plane BCD.Original teaching material · Worked example Parallel lines and planes in space · TransferTwo parallel planes have equations z=a and z=b. For a=0 and b=k, find their distance.Original teaching material · Worked example Parallel lines and planes in space · FoundationIn a cube with base z=0 and top z=4, prove a base edge directed along x is parallel to the top plane.Original teaching material · Reasoning checklist Parallel lines and planes in space · FoundationIn tetrahedron ABCD, E,F,G are the midpoints of AB,AC,AD. Prove plane EFG is parallel to plane BCD.Original teaching material · Reasoning checklist Parallel lines and planes in space · FoundationParallel planes are cut by a third plane in two lines. What is the relation between those lines?Original teaching material · Reasoning checklist Parallel lines and planes in space · FoundationDo two lines both parallel to a plane have to be parallel to each other? Give a counterexample.Original teaching material · Reasoning checklist Parallel lines and planes in space · StandardState a sufficient criterion for two distinct planes to be parallel using lines in one plane.Original teaching material · Reasoning checklist Parallel lines and planes in space · StandardWhy are two parallel lines in α, both parallel to β, insufficient to prove α∥β?Original teaching material · Reasoning checklist Parallel lines and planes in space · StandardTwo parallel planes have equations z=a and z=b. For a=0 and b=k, find their distance.Original teaching material · Answer check Parallel lines and planes in space · StandardIf a line is parallel to plane α and plane β is parallel to α, what are the possible positions of the line relative to β?Original teaching material · Reasoning checklist Parallel lines and planes in space · TransferDoes a line parallel to a plane have to be parallel to every line in that plane? Give a counterexample.Original teaching material · Reasoning checklist Parallel lines and planes in space · TransferA plane parallel to a pyramid base cuts at two-thirds of the full height measured from the apex. Find the section-to-base area ratio.Original teaching material · Answer check Perpendicular lines and planes in space · FoundationState a line-plane perpendicularity criterion.Original teaching material · Worked example Perpendicular lines and planes in space · StandardFind the distance from P to the plane.Original teaching material · Worked example Perpendicular lines and planes in space · TransferExplain the projection criterion: a line in a plane is perpendicular to an oblique line exactly when it is perpendicular to its nonzero orthogonal projection.Original teaching material · Worked example Perpendicular lines and planes in space · FoundationShow why perpendicularity to one line in a plane is insufficient to prove line-plane perpendicularity.Original teaching material · Reasoning checklist Perpendicular lines and planes in space · FoundationFind the distance from P to the plane.Original teaching material · Answer check Perpendicular lines and planes in space · FoundationA segment makes 30° with a plane and has length 2k. Find its perpendicular component.Original teaching material · Answer check Perpendicular lines and planes in space · FoundationFind sin²θ for a cube’s space diagonal making angle θ with the base.Original teaching material · Answer check Perpendicular lines and planes in space · StandardDo perpendicular planes force every line in one plane to be perpendicular to the other? Give a counterexample.Original teaching material · Reasoning checklist Perpendicular lines and planes in space · StandardFind the dihedral angle between the coordinate planes x=0 and y=0.Original teaching material · Answer check Perpendicular lines and planes in space · StandardIn two perpendicular planes α and β, a line m in α is perpendicular to their intersection line. State its relation to β and justify by a perpendicular cross-section.Original teaching material · Reasoning checklist Perpendicular lines and planes in space · StandardIf l is perpendicular to α and m is parallel to l, prove m is perpendicular to α.Original teaching material · Reasoning checklist Perpendicular lines and planes in space · TransferFind the distance from P to the plane.Original teaching material · Answer check Perpendicular lines and planes in space · TransferA segment makes 30° with a plane and has length 2k. Find its perpendicular component.Original teaching material · Answer check Random sampling · FoundationA population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.Original teaching material · Worked example Random sampling · StandardA stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.Original teaching material · Worked example Random sampling · TransferA school studies all 100k students but measures everyone’s height with a ruler biased by +1 cm. Is there sampling error? Is there measurement bias?Original teaching material · Worked example Random sampling · FoundationA population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.Original teaching material · Answer check Random sampling · FoundationA stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.Original teaching material · Answer check Random sampling · FoundationSystematically select 10k from an ordered population of 100k. Find the interval.Original teaching material · Answer check Random sampling · FoundationA voluntary web poll receives 20k replies. Does the large reply count remove self-selection bias?Original teaching material · Reasoning checklist Random sampling · StandardA selected sample has 10k people but only 6k reply. Find the response rate.Original teaching material · Answer check Random sampling · StandardSampling with replacement chooses 2 people from a population of k+2. Find the probability the same person appears twice.Original teaching material · Answer check Random sampling · StandardA school studies all 100k students but measures everyone’s height with a ruler biased by +1 cm. Is there sampling error? Is there measurement bias?Original teaching material · Reasoning checklist Random sampling · StandardA population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.Original teaching material · Answer check Random sampling · TransferA stratified sample has total 10k from a school with 60k junior and 40k senior students. Find the senior sample size under proportional allocation.Original teaching material · Answer check Random sampling · TransferSystematically select 10k from an ordered population of 100k. Find the interval.Original teaching material · Answer check Estimating a population from a sample · FoundationIn a sample of 20k, 5k use public transport. Estimate the population proportion.Original teaching material · Worked example Estimating a population from a sample · StandardA representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.Original teaching material · Worked example Estimating a population from a sample · TransferA sample’s mean is k and one extreme observation 100k is added. Must the median change as much as the mean? Explain.Original teaching material · Worked example Estimating a population from a sample · FoundationIn a sample of 20k, 5k use public transport. Estimate the population proportion.Original teaching material · Answer check Estimating a population from a sample · FoundationA representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.Original teaching material · Answer check Estimating a population from a sample · FoundationA histogram class [0,2k) has frequency 6k. Find its frequency density (count per unit).Original teaching material · Answer check Estimating a population from a sample · FoundationFor observations k,k+2,k+4, find the sample mean.Original teaching material · Answer check Estimating a population from a sample · StandardFor observations k,k+2,k+4, find variance using denominator 3 as specified.Original teaching material · Answer check Estimating a population from a sample · StandardThe sample proportions for three exhaustive categories are 1/4,1/2,p. Find p.Original teaching material · Answer check Estimating a population from a sample · StandardA sample’s mean is k and one extreme observation 100k is added. Must the median change as much as the mean? Explain.Original teaching material · Reasoning checklist Estimating a population from a sample · StandardIn a sample of 20k, 5k use public transport. Estimate the population proportion.Original teaching material · Answer check Estimating a population from a sample · TransferA representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.Original teaching material · Answer check Estimating a population from a sample · TransferA histogram class [0,2k) has frequency 6k. Find its frequency density (count per unit).Original teaching material · Answer check Statistical case study · FoundationAn anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.Original teaching material · Worked example Statistical case study · StandardTwo strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.Original teaching material · Worked example Statistical case study · TransferA report gives only the estimated proportion k/(k+2). Name two additional pieces of information needed to judge it.Original teaching material · Worked example Statistical case study · FoundationAn anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.Original teaching material · Answer check Statistical case study · FoundationTwo strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.Original teaching material · Answer check Statistical case study · FoundationA questionnaire asks only classmates arriving before 8 am. Explain a coverage issue when estimating the whole school’s commute time.Original teaching material · Reasoning checklist Statistical case study · FoundationThe class counts for commute time <20,20–40,>40 minutes are k,2k,k. Find the fraction above 40.Original teaching material · Answer check Statistical case study · StandardSurvey responses 5k contain 2k reports in seconds although the question asked minutes. Is direct pooling valid?Original teaching material · Reasoning checklist Statistical case study · StandardA survey finds longer commute is associated with lower attendance. Does the survey prove that commute caused the attendance pattern?Original teaching material · Reasoning checklist Statistical case study · StandardA report gives only the estimated proportion k/(k+2). Name two additional pieces of information needed to judge it.Original teaching material · Reasoning checklist Statistical case study · StandardAn anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.Original teaching material · Answer check Statistical case study · TransferTwo strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.Original teaching material · Answer check Statistical case study · TransferA questionnaire asks only classmates arriving before 8 am. Explain a coverage issue when estimating the whole school’s commute time.Original teaching material · Reasoning checklist Random events and probability · FoundationA fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.Original teaching material · Worked example Random events and probability · StandardThe spinner has labels 1,…,2k. Find the probability of an even label.Original teaching material · Worked example Random events and probability · TransferA claim assigns probability 1+1/k to an event. Explain why it cannot be valid.Original teaching material · Worked example Random events and probability · FoundationA fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.Original teaching material · Answer check Random events and probability · FoundationThe spinner has labels 1,…,2k. Find the probability of an even label.Original teaching material · Answer check Random events and probability · FoundationP(A)=k/(k+2). Find P(Aᶜ).Original teaching material · Answer check Random events and probability · FoundationDisjoint events have probabilities 1/(k+3) and 2/(k+3). Find their union probability.Original teaching material · Answer check Random events and probability · StandardP(A)=1/2,P(B)=1/3,P(A∩B)=1/(6k). Find P(A∪B).Original teaching material · Answer check Random events and probability · StandardA fair coin is tossed twice. List the event “exactly one head”.Original teaching material · Reasoning checklist Random events and probability · StandardA claim assigns probability 1+1/k to an event. Explain why it cannot be valid.Original teaching material · Reasoning checklist Random events and probability · StandardA fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.Original teaching material · Answer check Random events and probability · TransferThe spinner has labels 1,…,2k. Find the probability of an even label.Original teaching material · Answer check Random events and probability · TransferP(A)=k/(k+2). Find P(Aᶜ).Original teaching material · Answer check Independence of events · FoundationIndependent A,B have probabilities 1/2 and 1/(k+2). Find P(A∩B).Original teaching material · Worked example Independence of events · StandardTwo independent components each work with probability k/(k+1). A series system needs both. Find success probability.Original teaching material · Worked example Independence of events · TransferThe probability of each component failing is 1/(k+1), but common power failures can affect both. Can the independent-series formula be used without further evidence?Original teaching material · Worked example Independence of events · FoundationIndependent A,B have probabilities 1/2 and 1/(k+2). Find P(A∩B).Original teaching material · Answer check Independence of events · FoundationTwo independent components each work with probability k/(k+1). A series system needs both. Find success probability.Original teaching material · Answer check Independence of events · FoundationTwo independent components each fail with probability 1/(k+1). A parallel system works if at least one works. Find success probability.Original teaching material · Answer check Independence of events · FoundationA,B are independent and P(B)>0. Compare P(A|B) with P(A).Original teaching material · Reasoning checklist Independence of events · StandardA and B are disjoint with P(A)=P(B)=1/(k+2). Are they independent?Original teaching material · Reasoning checklist Independence of events · StandardA bag has k red and 1 blue; draw twice without replacement. Are the two red-draw events independent?Original teaching material · Reasoning checklist Independence of events · StandardThe probability of each component failing is 1/(k+1), but common power failures can affect both. Can the independent-series formula be used without further evidence?Original teaching material · Reasoning checklist Independence of events · StandardIndependent A,B have probabilities 1/2 and 1/(k+2). Find P(A∩B).Original teaching material · Answer check Independence of events · TransferTwo independent components each work with probability k/(k+1). A series system needs both. Find success probability.Original teaching material · Answer check Independence of events · TransferTwo independent components each fail with probability 1/(k+1). A parallel system works if at least one works. Find success probability.Original teaching material · Answer check Frequency and probability · FoundationAn event occurs 3k times in 10k trials. Find its relative frequency.Original teaching material · Worked example Frequency and probability · StandardTwo batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.Original teaching material · Worked example Frequency and probability · TransferIn k trials, the event never occurs. Must its true probability be zero? Explain.Original teaching material · Worked example Frequency and probability · FoundationAn event occurs 3k times in 10k trials. Find its relative frequency.Original teaching material · Answer check Frequency and probability · FoundationTwo batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.Original teaching material · Answer check Frequency and probability · FoundationA model has p=1/4. In 20k independent trials, find the expected number of occurrences.Original teaching material · Answer check Frequency and probability · FoundationAfter k consecutive heads from an independent fair coin, what is the probability the next toss is heads?Original teaching material · Answer check Frequency and probability · StandardA simulation makes 10k equally likely selections from labels 1,…,k. What model probability does label 1 have?Original teaching material · Answer check Frequency and probability · StandardA program’s pseudo-random outputs are reused with the same seed. Does that create new independent evidence each rerun?Original teaching material · Reasoning checklist Frequency and probability · StandardIn k trials, the event never occurs. Must its true probability be zero? Explain.Original teaching material · Reasoning checklist Frequency and probability · StandardAn event occurs 3k times in 10k trials. Find its relative frequency.Original teaching material · Answer check Frequency and probability · TransferTwo batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.Original teaching material · Answer check Frequency and probability · TransferA model has p=1/4. In 20k independent trials, find the expected number of occurrences.Original teaching material · Answer check Space vectors and their operations · FoundationFind the first component of a+b.Original teaching material · Worked example Space vectors and their operations · StandardFind cos θ between a and b.Original teaching material · Worked example Space vectors and their operations · TransferThree mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.Original teaching material · Worked example Space vectors and their operations · FoundationFind the first component of a+b.Original teaching material · Answer check Space vectors and their operations · FoundationFind the third component of 2a−3b.Original teaching material · Answer check Space vectors and their operations · FoundationFind the squared length of a.Original teaching material · Answer check Space vectors and their operations · FoundationCalculate a·b.Original teaching material · Answer check Space vectors and their operations · StandardCalculate a·b.Original teaching material · Answer check Space vectors and their operations · StandardFind cos θ between a and b.Original teaching material · Answer check Space vectors and their operations · StandardAre a and b parallel? Justify your answer.Original teaching material · Reasoning checklist Space vectors and their operations · StandardThree mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.Original teaching material · Answer check Space vectors and their operations · TransferAre a and b parallel? Justify your answer.Original teaching material · Reasoning checklist Space vectors and their operations · TransferThree mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.Original teaching material · Answer check The fundamental theorem of space vectors · FoundationIn the standard basis, find the coefficient of e3.Original teaching material · Worked example The fundamental theorem of space vectors · StandardP=αA+βB+γC is affine, α=1/(t+1), β=1/2. Find γ.Original teaching material · Worked example The fundamental theorem of space vectors · TransferExplain why these three vectors do not give unique coefficients.Original teaching material · Worked example The fundamental theorem of space vectors · FoundationIn the standard basis, find the coefficient of e3.Original teaching material · Answer check The fundamental theorem of space vectors · FoundationFind the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).Original teaching material · Answer check The fundamental theorem of space vectors · FoundationDecide whether e1,e2,e3 form a basis.Original teaching material · Reasoning checklist The fundamental theorem of space vectors · FoundationFind λ so that the third vector lies in the plane spanned by the first two.Original teaching material · Answer check The fundamental theorem of space vectors · StandardFind λ so that the third vector lies in the plane spanned by the first two.Original teaching material · Answer check The fundamental theorem of space vectors · StandardP=αA+βB+γC is affine, α=1/(t+1), β=1/2. Find γ.Original teaching material · Answer check The fundamental theorem of space vectors · StandardFind the representation of v in the given basis.Original teaching material · Reasoning checklist The fundamental theorem of space vectors · StandardExplain why these three vectors do not give unique coefficients.Original teaching material · Reasoning checklist The fundamental theorem of space vectors · TransferFind the representation of v in the given basis.Original teaching material · Reasoning checklist The fundamental theorem of space vectors · TransferExplain why these three vectors do not give unique coefficients.Original teaching material · Reasoning checklist Coordinate representation of space vectors · FoundationFind AB².Original teaching material · Worked example Coordinate representation of space vectors · StandardFind the signed scalar projection of v onto u.Original teaching material · Worked example Coordinate representation of space vectors · TransferFind a nonzero vector perpendicular to both a and b, with third component 1.Original teaching material · Worked example Coordinate representation of space vectors · FoundationFind AB².Original teaching material · Answer check Coordinate representation of space vectors · FoundationFind the z-coordinate of the midpoint of A and B.Original teaching material · Answer check Coordinate representation of space vectors · FoundationFind λ so that a and b are perpendicular.Original teaching material · Answer check Coordinate representation of space vectors · FoundationFind cos θ between a and b.Original teaching material · Answer check Coordinate representation of space vectors · StandardFind cos θ between a and b.Original teaching material · Answer check Coordinate representation of space vectors · StandardFind the signed scalar projection of v onto u.Original teaching material · Answer check Coordinate representation of space vectors · StandardA unit direction has cos²α=1/t², cos²β=1/9. Find cos²γ.Original teaching material · Answer check Coordinate representation of space vectors · StandardFind a nonzero vector perpendicular to both a and b, with third component 1.Original teaching material · Reasoning checklist Coordinate representation of space vectors · TransferA unit direction has cos²α=1/t², cos²β=1/9. Find cos²γ.Original teaching material · Answer check Coordinate representation of space vectors · TransferFind a nonzero vector perpendicular to both a and b, with third component 1.Original teaching material · Reasoning checklist Applications of space vectors · FoundationFind the cosine of the acute angle between the lines.Original teaching material · Worked example Applications of space vectors · StandardFind the distance from P to the plane.Original teaching material · Worked example Applications of space vectors · TransferFind the plane through P with normal n.Original teaching material · Worked example Applications of space vectors · FoundationFind the cosine of the acute angle between the lines.Original teaching material · Answer check Applications of space vectors · FoundationFind sin α for the angle of the line with the plane.Original teaching material · Answer check Applications of space vectors · FoundationFind the cosine of the acute angle between the planes.Original teaching material · Answer check Applications of space vectors · FoundationFind the distance from P to the plane.Original teaching material · Answer check Applications of space vectors · StandardFind the distance from P to the plane.Original teaching material · Answer check Applications of space vectors · StandardFind the distance from P to the plane.Original teaching material · Answer check Applications of space vectors · StandardDoes the entire line lie in the plane?Original teaching material · Reasoning checklist Applications of space vectors · StandardFind the plane through P with normal n.Original teaching material · Reasoning checklist Applications of space vectors · TransferDoes the entire line lie in the plane?Original teaching material · Reasoning checklist Applications of space vectors · TransferFind the plane through P with normal n.Original teaching material · Reasoning checklist Inclination and slope of a line · FoundationFind the slope of AB.Original teaching material · Worked example Inclination and slope of a line · StandardFind a so that the two nonvertical lines are parallel.Original teaching material · Worked example Inclination and slope of a line · TransferLines through O meet the segment from A=(t,1) to B=(t,3). Find the slope range.Original teaching material · Worked example Inclination and slope of a line · FoundationFind the slope of AB.Original teaching material · Answer check Inclination and slope of a line · FoundationA line has inclination 135°. Find its slope.Original teaching material · Answer check Inclination and slope of a line · FoundationDescribe the inclination and slope of the line.Original teaching material · Reasoning checklist Inclination and slope of a line · FoundationFind the slope of a line perpendicular to y=tx+1.Original teaching material · Answer check Inclination and slope of a line · StandardFind the slope of a line perpendicular to y=tx+1.Original teaching material · Answer check Inclination and slope of a line · StandardFind a so that the two nonvertical lines are parallel.Original teaching material · Answer check Inclination and slope of a line · StandardFind the tangent of the acute angle between slopes 0 and t.Original teaching material · Answer check Inclination and slope of a line · StandardLines through O meet the segment from A=(t,1) to B=(t,3). Find the slope range.Original teaching material · Reasoning checklist Inclination and slope of a line · TransferFind the tangent of the acute angle between slopes 0 and t.Original teaching material · Answer check Inclination and slope of a line · TransferLines through O meet the segment from A=(t,1) to B=(t,3). Find the slope range.Original teaching material · Reasoning checklist Equations of a line · FoundationFind the line through (2,3) with slope t.Original teaching material · Worked example Equations of a line · StandardFind the perpendicular bisector of (0,0) and (2t,0).Original teaching material · Worked example Equations of a line · TransferFind the fixed point shared by this line family.Original teaching material · Worked example Equations of a line · FoundationFind the line through (2,3) with slope t.Original teaching material · Reasoning checklist Equations of a line · FoundationFind the line through (t,1) and (t,5).Original teaching material · Reasoning checklist Equations of a line · FoundationFind the y-intercept of the line with x-intercept t and passing through (0,4).Original teaching material · Answer check Equations of a line · FoundationFind C so that Ax+By+C=0 passes through P.Original teaching material · Answer check Equations of a line · StandardFind C so that Ax+By+C=0 passes through P.Original teaching material · Answer check Equations of a line · StandardFind the perpendicular bisector of (0,0) and (2t,0).Original teaching material · Reasoning checklist Equations of a line · StandardFind the reflection of the line in the y-axis.Original teaching material · Reasoning checklist Equations of a line · StandardFind the fixed point shared by this line family.Original teaching material · Reasoning checklist Equations of a line · TransferFind the reflection of the line in the y-axis.Original teaching material · Reasoning checklist Equations of a line · TransferFind the fixed point shared by this line family.Original teaching material · Reasoning checklist Intersections and distance formulas · FoundationFind the x-coordinate of the intersection.Original teaching material · Worked example Intersections and distance formulas · StandardFind the foot of the perpendicular from P to the line.Original teaching material · Worked example Intersections and distance formulas · TransferA=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.Original teaching material · Worked example Intersections and distance formulas · FoundationFind the x-coordinate of the intersection.Original teaching material · Answer check Intersections and distance formulas · FoundationFind the distance from (0,0) to x=t.Original teaching material · Answer check Intersections and distance formulas · FoundationFind the distance from P to the line.Original teaching material · Answer check Intersections and distance formulas · FoundationFind the distance between the parallel lines.Original teaching material · Answer check Intersections and distance formulas · StandardFind the distance between the parallel lines.Original teaching material · Answer check Intersections and distance formulas · StandardFind the foot of the perpendicular from P to the line.Original teaching material · Reasoning checklist Intersections and distance formulas · StandardReflect P in the line y=x.Original teaching material · Reasoning checklist Intersections and distance formulas · StandardA=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.Original teaching material · Answer check Intersections and distance formulas · TransferReflect P in the line y=x.Original teaching material · Reasoning checklist Intersections and distance formulas · TransferA=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.Original teaching material · Answer check Equations of a circle · FoundationFind the radius of the circle.Original teaching material · Worked example Equations of a circle · StandardFind the parameter range for a real nondegenerate circle.Original teaching material · Worked example Equations of a circle · TransferThe circle center is (t,0), radius 2. Find the largest x-coordinate on it.Original teaching material · Worked example Equations of a circle · FoundationFind the radius of the circle.Original teaching material · Answer check Equations of a circle · FoundationFind the x-coordinate of the center.Original teaching material · Answer check Equations of a circle · FoundationFind the squared radius of the circle with diameter endpoints A and B.Original teaching material · Answer check Equations of a circle · FoundationFind the circle centered at (t,−1) and passing through (t+3,3).Original teaching material · Reasoning checklist Equations of a circle · StandardFind the circle centered at (t,−1) and passing through (t+3,3).Original teaching material · Reasoning checklist Equations of a circle · StandardFind the parameter range for a real nondegenerate circle.Original teaching material · Reasoning checklist Equations of a circle · StandardFind the circle through O=(0,0), A=(2t,0), B=(0,2).Original teaching material · Reasoning checklist Equations of a circle · StandardThe circle center is (t,0), radius 2. Find the largest x-coordinate on it.Original teaching material · Answer check Equations of a circle · TransferFind the circle through O=(0,0), A=(2t,0), B=(0,2).Original teaching material · Reasoning checklist Equations of a circle · TransferThe circle center is (t,0), radius 2. Find the largest x-coordinate on it.Original teaching material · Answer check Relations between lines and circles · FoundationFind the number of common points.Original teaching material · Worked example Relations between lines and circles · StandardFind the radical axis and state when it is a common chord line.Original teaching material · Worked example Relations between lines and circles · TransferFind all values of a for which y=a is tangent to x²+y²=t².Original teaching material · Worked example Relations between lines and circles · FoundationFind the number of common points.Original teaching material · Answer check Relations between lines and circles · FoundationFind the length of the chord.Original teaching material · Answer check Relations between lines and circles · FoundationFind the tangent length from P to the circle.Original teaching material · Answer check Relations between lines and circles · FoundationClassify the relation of these two circles.Original teaching material · Reasoning checklist Relations between lines and circles · StandardClassify the relation of these two circles.Original teaching material · Reasoning checklist Relations between lines and circles · StandardFind the radical axis and state when it is a common chord line.Original teaching material · Reasoning checklist Relations between lines and circles · StandardFind the tangent at the given point.Original teaching material · Reasoning checklist Relations between lines and circles · StandardFind all values of a for which y=a is tangent to x²+y²=t².Original teaching material · Reasoning checklist Relations between lines and circles · TransferFind the tangent at the given point.Original teaching material · Reasoning checklist Relations between lines and circles · TransferFind all values of a for which y=a is tangent to x²+y²=t².Original teaching material · Reasoning checklist Ellipse · FoundationFind the major-axis length.Original teaching material · Worked example Ellipse · StandardFind the horizontal ellipse with a=t+1 and c=t.Original teaching material · Worked example Ellipse · TransferFind the largest value of 3x+4y on the ellipse.Original teaching material · Worked example Ellipse · FoundationFind the major-axis length.Original teaching material · Answer check Ellipse · FoundationFind the squared focal distance c².Original teaching material · Answer check Ellipse · FoundationFind the eccentricity.Original teaching material · Answer check Ellipse · FoundationP is on the ellipse and PF1=t. Find PF2.Original teaching material · Answer check Ellipse · StandardP is on the ellipse and PF1=t. Find PF2.Original teaching material · Answer check Ellipse · StandardFind the horizontal ellipse with a=t+1 and c=t.Original teaching material · Reasoning checklist Ellipse · StandardFind the length of the chord cut by x=0.Original teaching material · Answer check Ellipse · StandardFind the largest value of 3x+4y on the ellipse.Original teaching material · Answer check Ellipse · TransferFind the length of the chord cut by x=0.Original teaching material · Answer check Ellipse · TransferFind the largest value of 3x+4y on the ellipse.Original teaching material · Answer check Hyperbola · FoundationFind c².Original teaching material · Worked example Hyperbola · StandardFind the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).Original teaching material · Worked example Hyperbola · TransferShow there is one common point and explain why this line is not tangent.Original teaching material · Worked example Hyperbola · FoundationFind c².Original teaching material · Answer check Hyperbola · FoundationFind the length of the transverse axis.Original teaching material · Answer check Hyperbola · FoundationFind the positive asymptote slope.Original teaching material · Answer check Hyperbola · FoundationFind the eccentricity.Original teaching material · Answer check Hyperbola · StandardFind the eccentricity.Original teaching material · Answer check Hyperbola · StandardFind the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).Original teaching material · Reasoning checklist Hyperbola · StandardFind the intersection with x=0.Original teaching material · Reasoning checklist Hyperbola · StandardShow there is one common point and explain why this line is not tangent.Original teaching material · Reasoning checklist Hyperbola · TransferFind the intersection with x=0.Original teaching material · Reasoning checklist Hyperbola · TransferShow there is one common point and explain why this line is not tangent.Original teaching material · Reasoning checklist Parabola · FoundationFind the x-coordinate of the focus.Original teaching material · Worked example Parabola · StandardFind the length of the focal chord perpendicular to the axis.Original teaching material · Worked example Parabola · TransferFind all k giving exactly one common point.Original teaching material · Worked example Parabola · FoundationFind the x-coordinate of the focus.Original teaching material · Answer check Parabola · FoundationFind the x-coordinate of the directrix.Original teaching material · Answer check Parabola · FoundationFind the parabola with vertex O and focus (0,−t).Original teaching material · Reasoning checklist Parabola · FoundationOn y²=4tx, P has x-coordinate 2t. Find PF.Original teaching material · Answer check Parabola · StandardOn y²=4tx, P has x-coordinate 2t. Find PF.Original teaching material · Answer check Parabola · StandardFind the length of the focal chord perpendicular to the axis.Original teaching material · Answer check Parabola · StandardFind the tangent of slope 1 to y²=4tx.Original teaching material · Reasoning checklist Parabola · StandardFind all k giving exactly one common point.Original teaching material · Reasoning checklist Parabola · TransferFind the tangent of slope 1 to y²=4tx.Original teaching material · Reasoning checklist Parabola · TransferFind all k giving exactly one common point.Original teaching material · Reasoning checklist Concept of a sequence · FoundationFind a3.Original teaching material · Worked example Concept of a sequence · StandardFind the smallest positive index with an>t.Original teaching material · Worked example Concept of a sequence · TransferProve that the sequence is strictly increasing.Original teaching material · Worked example Concept of a sequence · FoundationFind a3.Original teaching material · Answer check Concept of a sequence · FoundationFind a4 from the recurrence.Original teaching material · Answer check Concept of a sequence · FoundationFind a1.Original teaching material · Answer check Concept of a sequence · FoundationFind a5.Original teaching material · Answer check Concept of a sequence · StandardFind a5.Original teaching material · Answer check Concept of a sequence · StandardFind the smallest positive index with an>t.Original teaching material · Answer check Concept of a sequence · StandardFind the sum of the first t terms.Original teaching material · Answer check Concept of a sequence · StandardProve that the sequence is strictly increasing.Original teaching material · Reasoning checklist Concept of a sequence · TransferFind the sum of the first t terms.Original teaching material · Answer check Concept of a sequence · TransferProve that the sequence is strictly increasing.Original teaching material · Reasoning checklist Arithmetic sequences · FoundationFind a6.Original teaching material · Worked example Arithmetic sequences · StandardHow many terms of 2,5,8,… are at most 3t+2?Original teaching material · Worked example Arithmetic sequences · TransferA theatre has t rows; the first has 10 seats and each next row has 2 more. Find the total seats.Original teaching material · Worked example Arithmetic sequences · FoundationFind a6.Original teaching material · Answer check Arithmetic sequences · FoundationFind the common difference.Original teaching material · Answer check Arithmetic sequences · FoundationFind S4.Original teaching material · Answer check Arithmetic sequences · FoundationFind a5 when S9 is given.Original teaching material · Answer check Arithmetic sequences · StandardFind a5 when S9 is given.Original teaching material · Answer check Arithmetic sequences · StandardHow many terms of 2,5,8,… are at most 3t+2?Original teaching material · Answer check Arithmetic sequences · StandardFind the sum of all positive terms in t,t−1,t−2,… .Original teaching material · Answer check Arithmetic sequences · StandardA theatre has t rows; the first has 10 seats and each next row has 2 more. Find the total seats.Original teaching material · Answer check Arithmetic sequences · TransferFind the sum of all positive terms in t,t−1,t−2,… .Original teaching material · Answer check Arithmetic sequences · TransferA theatre has t rows; the first has 10 seats and each next row has 2 more. Find the total seats.Original teaching material · Answer check Geometric sequences · FoundationFind a4.Original teaching material · Worked example Geometric sequences · StandardFind a3a7 given a5=t.Original teaching material · Worked example Geometric sequences · TransferA geometric sequence has S2=t and S4=5t. Find q².Original teaching material · Worked example Geometric sequences · FoundationFind a4.Original teaching material · Answer check Geometric sequences · FoundationFind q.Original teaching material · Answer check Geometric sequences · FoundationFind S3.Original teaching material · Answer check Geometric sequences · FoundationFind S5 when q=1.Original teaching material · Answer check Geometric sequences · StandardFind S5 when q=1.Original teaching material · Answer check Geometric sequences · StandardFind a3a7 given a5=t.Original teaching material · Answer check Geometric sequences · StandardFind all real geometric means between t and 4t.Original teaching material · Reasoning checklist Geometric sequences · StandardA geometric sequence has S2=t and S4=5t. Find q².Original teaching material · Answer check Geometric sequences · TransferFind all real geometric means between t and 4t.Original teaching material · Reasoning checklist Geometric sequences · TransferA geometric sequence has S2=t and S4=5t. Find q².Original teaching material · Answer check Mathematical induction · FoundationProve the shifted-integer sum by induction.Original teaching material · Worked example Mathematical induction · StandardProve 2ⁿ≥n+1 for every integer n≥t, using t as the base.Original teaching material · Worked example Mathematical induction · TransferDisprove the claim that n²+n+t is prime for every nonnegative integer n.Original teaching material · Worked example Mathematical induction · FoundationProve the shifted-integer sum by induction.Original teaching material · Reasoning checklist Mathematical induction · FoundationProve the odd-number sum identity.Original teaching material · Reasoning checklist Mathematical induction · FoundationProve the finite power-of-two identity.Original teaching material · Reasoning checklist Mathematical induction · FoundationProve n³−n+3tn is divisible by 3 for all n≥1.Original teaching material · Reasoning checklist Mathematical induction · StandardProve n³−n+3tn is divisible by 3 for all n≥1.Original teaching material · Reasoning checklist Mathematical induction · StandardProve 2ⁿ≥n+1 for every integer n≥t, using t as the base.Original teaching material · Reasoning checklist Mathematical induction · StandardProve the scaled telescoping identity by induction.Original teaching material · Reasoning checklist Mathematical induction · StandardDisprove the claim that n²+n+t is prime for every nonnegative integer n.Original teaching material · Reasoning checklist Mathematical induction · TransferProve the scaled telescoping identity by induction.Original teaching material · Reasoning checklist Mathematical induction · TransferDisprove the claim that n²+n+t is prime for every nonnegative integer n.Original teaching material · Reasoning checklist Derivative: concept and meaning · FoundationFind the derivative of x² at x=t from the limit.Original teaching material · Worked example Derivative: concept and meaning · StandardIs |x−t| differentiable at x=t?Original teaching material · Worked example Derivative: concept and meaning · TransferFind the normal to y=x² at x=t.Original teaching material · Worked example Derivative: concept and meaning · FoundationFind the derivative of x² at x=t from the limit.Original teaching material · Answer check Derivative: concept and meaning · FoundationFind the average rate of x² from t to t+1.Original teaching material · Answer check Derivative: concept and meaning · FoundationPosition is s(u)=u²+3u. Find velocity at u=t.Original teaching material · Answer check Derivative: concept and meaning · FoundationFind the tangent to y=x² at x=t.Original teaching material · Reasoning checklist Derivative: concept and meaning · StandardFind the tangent to y=x² at x=t.Original teaching material · Reasoning checklist Derivative: concept and meaning · StandardIs |x−t| differentiable at x=t?Original teaching material · Reasoning checklist Derivative: concept and meaning · StandardA tank volume V(h)=3h². Find dV/dh at h=t.Original teaching material · Answer check Derivative: concept and meaning · StandardFind the normal to y=x² at x=t.Original teaching material · Reasoning checklist Derivative: concept and meaning · TransferA tank volume V(h)=3h². Find dV/dh at h=t.Original teaching material · Answer check Derivative: concept and meaning · TransferFind the normal to y=x² at x=t.Original teaching material · Reasoning checklist Derivative operations · FoundationFind f′(1).Original teaching material · Worked example Derivative operations · StandardFind f′(0).Original teaching material · Worked example Derivative operations · TransferFind f′(0), using radians.Original teaching material · Worked example Derivative operations · FoundationFind f′(1).Original teaching material · Answer check Derivative operations · FoundationFind f′(1) using the product rule.Original teaching material · Answer check Derivative operations · FoundationFind f′(0).Original teaching material · Answer check Derivative operations · FoundationFind f′(0) using the chain rule.Original teaching material · Answer check Derivative operations · StandardFind f′(0) using the chain rule.Original teaching material · Answer check Derivative operations · StandardFind f′(0).Original teaching material · Answer check Derivative operations · StandardFind f′(1).Original teaching material · Answer check Derivative operations · StandardFind f′(0), using radians.Original teaching material · Answer check Derivative operations · TransferFind f′(1).Original teaching material · Answer check Derivative operations · TransferFind f′(0), using radians.Original teaching material · Answer check Applications of derivatives · FoundationFind the minimizing x over all real x.Original teaching material · Worked example Applications of derivatives · StandardA rectangle has perimeter 4t. Find its maximum area.Original teaching material · Worked example Applications of derivatives · TransferProfit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.Original teaching material · Worked example Applications of derivatives · FoundationFind the minimizing x over all real x.Original teaching material · Answer check Applications of derivatives · FoundationFind the maximum of f on [0,t].Original teaching material · Answer check Applications of derivatives · FoundationFind the increasing and decreasing intervals.Original teaching material · Reasoning checklist Applications of derivatives · FoundationIs x=t a local extremum of (x−t)³?Original teaching material · Reasoning checklist Applications of derivatives · StandardIs x=t a local extremum of (x−t)³?Original teaching material · Reasoning checklist Applications of derivatives · StandardA rectangle has perimeter 4t. Find its maximum area.Original teaching material · Answer check Applications of derivatives · StandardFind the minimum of x+t²/x for x>0.Original teaching material · Answer check Applications of derivatives · StandardProfit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.Original teaching material · Answer check Applications of derivatives · TransferFind the minimum of x+t²/x for x>0.Original teaching material · Answer check Applications of derivatives · TransferProfit is P(x)=−x²+2tx−3, for 0≤x≤t+2. Find maximum profit.Original teaching material · Answer check Addition and multiplication counting principles · FoundationChoose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?Original teaching material · Worked example Addition and multiplication counting principles · StandardThere are t first-stage choices with 2 continuations each, and 3 other choices with 5 continuations each. Count paths.Original teaching material · Worked example Addition and multiplication counting principles · TransferUse digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.Original teaching material · Worked example Addition and multiplication counting principles · FoundationChoose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?Original teaching material · Answer check Addition and multiplication counting principles · FoundationChoose one of t shirts and one of 4 trousers. How many outfits?Original teaching material · Answer check Addition and multiplication counting principles · FoundationA code is one of t letters followed by a digit 0–9. How many codes?Original teaching material · Answer check Addition and multiplication counting principles · FoundationAmong t+8 students, t study art, 8 study music and 3 study both. How many study either subject?Original teaching material · Answer check Addition and multiplication counting principles · StandardAmong t+8 students, t study art, 8 study music and 3 study both. How many study either subject?Original teaching material · Answer check Addition and multiplication counting principles · StandardThere are t first-stage choices with 2 continuations each, and 3 other choices with 5 continuations each. Count paths.Original teaching material · Answer check Addition and multiplication counting principles · StandardPasswords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.Original teaching material · Answer check Addition and multiplication counting principles · StandardUse digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.Original teaching material · Answer check Addition and multiplication counting principles · TransferPasswords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.Original teaching material · Answer check Addition and multiplication counting principles · TransferUse digits 0,…,t−1 to form two-digit numbers without repeated digits, where 2≤t≤9. Count them.Original teaching material · Answer check Permutations and combinations · FoundationChoose a president and secretary from m distinct people. Count assignments.Original teaching material · Worked example Permutations and combinations · StandardFrom m people, form a 3-person committee that includes a specified person.Original teaching material · Worked example Permutations and combinations · TransferArrange m distinct people in a line with A and B adjacent.Original teaching material · Worked example Permutations and combinations · FoundationChoose a president and secretary from m distinct people. Count assignments.Original teaching material · Answer check Permutations and combinations · FoundationChoose an unordered committee of 3 from m people.Original teaching material · Answer check Permutations and combinations · FoundationArrange m distinct people in a circle, considering rotations identical but reflections different.Original teaching material · Answer check Permutations and combinations · FoundationCount arrangements containing r identical As, two identical Bs and one C.Original teaching material · Answer check Permutations and combinations · StandardCount arrangements containing r identical As, two identical Bs and one C.Original teaching material · Answer check Permutations and combinations · StandardFrom m people, form a 3-person committee that includes a specified person.Original teaching material · Answer check Permutations and combinations · StandardPlace t identical balls in 3 labeled boxes; empty boxes are allowed. Count distributions.Original teaching material · Answer check Permutations and combinations · StandardArrange m distinct people in a line with A and B adjacent.Original teaching material · Answer check Permutations and combinations · TransferPlace t identical balls in 3 labeled boxes; empty boxes are allowed. Count distributions.Original teaching material · Answer check Permutations and combinations · TransferArrange m distinct people in a line with A and B adjacent.Original teaching material · Answer check Binomial theorem · FoundationFind the coefficient of x in (1+x)^m.Original teaching material · Worked example Binomial theorem · StandardFind the constant term in (x+1/x)^2m.Original teaching material · Worked example Binomial theorem · TransferFind the remainder when (1+10)^m is divided by 100.Original teaching material · Worked example Binomial theorem · FoundationFind the coefficient of x in (1+x)^m.Original teaching material · Answer check Binomial theorem · FoundationFind the coefficient of x² in (1+x)^m.Original teaching material · Answer check Binomial theorem · FoundationFind the sum of all coefficients of (2+x)^m.Original teaching material · Answer check Binomial theorem · FoundationFind the coefficient of x² in (1−2x)^m.Original teaching material · Answer check Binomial theorem · StandardFind the coefficient of x² in (1−2x)^m.Original teaching material · Answer check Binomial theorem · StandardFind the constant term in (x+1/x)^2m.Original teaching material · Answer check Binomial theorem · StandardFind the alternating sum of binomial coefficients for m≥1.Original teaching material · Answer check Binomial theorem · StandardFind the remainder when (1+10)^m is divided by 100.Original teaching material · Answer check Binomial theorem · TransferFind the alternating sum of binomial coefficients for m≥1.Original teaching material · Answer check Binomial theorem · TransferFind the remainder when (1+10)^m is divided by 100.Original teaching material · Answer check Conditional and total probability · FoundationFind P(A|B) from the given probabilities.Original teaching material · Worked example Conditional and total probability · StandardChoose machine 1 with probability 1/(t+2), otherwise machine 2; their failure rates are 1/10 and 1/5. Given failure, find P(machine 1).Original teaching material · Worked example Conditional and total probability · TransferA fair coin selects a bag: bag 1 has t red and 1 blue; bag 2 has 1 red and t blue. Given a red draw, find the probability bag 1 was chosen.Original teaching material · Worked example Conditional and total probability · FoundationFind P(A|B) from the given probabilities.Original teaching material · Answer check Conditional and total probability · FoundationFrom a bag of t red and 2 blue balls, two are drawn without replacement. Given the first is red, find the probability the second is blue.Original teaching material · Answer check Conditional and total probability · FoundationEvents A,B are independent with P(A)=1/3 and P(B)=t/(t+1). Find P(A∩B).Original teaching material · Answer check Conditional and total probability · FoundationChoose machine 1 with probability 1/(t+2) and machine 2 otherwise. Failure rates are 1/10 and 1/5. Find total failure probability.Original teaching material · Answer check Conditional and total probability · StandardChoose machine 1 with probability 1/(t+2) and machine 2 otherwise. Failure rates are 1/10 and 1/5. Find total failure probability.Original teaching material · Answer check Conditional and total probability · StandardChoose machine 1 with probability 1/(t+2), otherwise machine 2; their failure rates are 1/10 and 1/5. Given failure, find P(machine 1).Original teaching material · Answer check Conditional and total probability · StandardA and B are disjoint and both have positive probability. Can they be independent?Original teaching material · Reasoning checklist Conditional and total probability · StandardA fair coin selects a bag: bag 1 has t red and 1 blue; bag 2 has 1 red and t blue. Given a red draw, find the probability bag 1 was chosen.Original teaching material · Answer check Conditional and total probability · TransferA and B are disjoint and both have positive probability. Can they be independent?Original teaching material · Reasoning checklist Conditional and total probability · TransferA fair coin selects a bag: bag 1 has t red and 1 blue; bag 2 has 1 red and t blue. Given a red draw, find the probability bag 1 was chosen.Original teaching material · Answer check Discrete random variables and distributions · FoundationFind the missing probability p.Original teaching material · Worked example Discrete random variables and distributions · StandardFind p when P(X=0)=p, P(X=1)=2p and P(X=2)=tp.Original teaching material · Worked example Discrete random variables and distributions · TransferA fair die is rolled; X=1 for a face at most d, and X=0 otherwise. Find P(X=0).Original teaching material · Worked example Discrete random variables and distributions · FoundationFind the missing probability p.Original teaching material · Answer check Discrete random variables and distributions · FoundationFind P(X≥t).Original teaching material · Answer check Discrete random variables and distributions · FoundationToss r independent fair coins; X counts heads. Find P(X=1).Original teaching material · Answer check Discrete random variables and distributions · FoundationGive the distribution of Y=2X+1.Original teaching material · Reasoning checklist Discrete random variables and distributions · StandardGive the distribution of Y=2X+1.Original teaching material · Reasoning checklist Discrete random variables and distributions · StandardFind p when P(X=0)=p, P(X=1)=2p and P(X=2)=tp.Original teaching material · Answer check Discrete random variables and distributions · StandardCan these numbers be a probability distribution? Explain.Original teaching material · Reasoning checklist Discrete random variables and distributions · StandardA fair die is rolled; X=1 for a face at most d, and X=0 otherwise. Find P(X=0).Original teaching material · Answer check Discrete random variables and distributions · TransferCan these numbers be a probability distribution? Explain.Original teaching material · Reasoning checklist Discrete random variables and distributions · TransferA fair die is rolled; X=1 for a face at most d, and X=0 otherwise. Find P(X=0).Original teaching material · Answer check Numerical characteristics of random variables · FoundationFind E(X).Original teaching material · Worked example Numerical characteristics of random variables · StandardFind Var(3X+2).Original teaching material · Worked example Numerical characteristics of random variables · TransferA game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.Original teaching material · Worked example Numerical characteristics of random variables · FoundationFind E(X).Original teaching material · Answer check Numerical characteristics of random variables · FoundationFind E(X²).Original teaching material · Answer check Numerical characteristics of random variables · FoundationFind Var(X).Original teaching material · Answer check Numerical characteristics of random variables · FoundationFind E(3X+2).Original teaching material · Answer check Numerical characteristics of random variables · StandardFind E(3X+2).Original teaching material · Answer check Numerical characteristics of random variables · StandardFind Var(3X+2).Original teaching material · Answer check Numerical characteristics of random variables · StandardIndependent X,Y have variances t and 2. Find Var(X+Y).Original teaching material · Answer check Numerical characteristics of random variables · StandardA game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.Original teaching material · Answer check Numerical characteristics of random variables · TransferIndependent X,Y have variances t and 2. Find Var(X+Y).Original teaching material · Answer check Numerical characteristics of random variables · TransferA game pays t with probability 1/4 and 0 otherwise; entry fee is 2. Find expected net gain.Original teaching material · Answer check Binomial and hypergeometric distributions · FoundationX~Bin(m,1/2). Find P(X=0).Original teaching material · Worked example Binomial and hypergeometric distributions · StandardA bag has t red and 3 blue balls; select 2 without replacement. Find the probability both are blue.Original teaching material · Worked example Binomial and hypergeometric distributions · TransferIndependent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).Original teaching material · Worked example Binomial and hypergeometric distributions · FoundationX~Bin(m,1/2). Find P(X=0).Original teaching material · Answer check Binomial and hypergeometric distributions · FoundationX~Bin(m,1/2). Find P(X=1).Original teaching material · Answer check Binomial and hypergeometric distributions · FoundationX~Bin(m,1/4). Find E(X).Original teaching material · Answer check Binomial and hypergeometric distributions · FoundationX~Bin(m,1/4). Find Var(X).Original teaching material · Answer check Binomial and hypergeometric distributions · StandardX~Bin(m,1/4). Find Var(X).Original teaching material · Answer check Binomial and hypergeometric distributions · StandardA bag has t red and 3 blue balls; select 2 without replacement. Find the probability both are blue.Original teaching material · Answer check Binomial and hypergeometric distributions · StandardFrom t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.Original teaching material · Answer check Binomial and hypergeometric distributions · StandardIndependent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).Original teaching material · Answer check Binomial and hypergeometric distributions · TransferFrom t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.Original teaching material · Answer check Binomial and hypergeometric distributions · TransferIndependent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).Original teaching material · Answer check Normal distribution · FoundationX~N(t,4). Find its standard deviation.Original teaching material · Worked example Normal distribution · StandardX~N(t,9), Φ(2)=0.9772. Find P(X>t+6).Original teaching material · Worked example Normal distribution · TransferModels A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).Original teaching material · Worked example Normal distribution · FoundationX~N(t,4). Find its standard deviation.Original teaching material · Answer check Normal distribution · FoundationStandardise x=t+4 under X~N(t,4). Find z.Original teaching material · Answer check Normal distribution · FoundationX~N(t,4), Φ(1)=0.8413. Find P(X≤t+2).Original teaching material · Answer check Normal distribution · FoundationX~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).Original teaching material · Answer check Normal distribution · StandardX~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).Original teaching material · Answer check Normal distribution · StandardX~N(t,9), Φ(2)=0.9772. Find P(X>t+6).Original teaching material · Answer check Normal distribution · StandardX~N(t,4). Find P(X=t).Original teaching material · Answer check Normal distribution · StandardModels A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).Original teaching material · Reasoning checklist Normal distribution · TransferX~N(t,4). Find P(X=t).Original teaching material · Answer check Normal distribution · TransferModels A~N(t,4), B~N(t+10,9). Compare P(A≤t+2) and P(B≤t+13).Original teaching material · Reasoning checklist Correlation of paired data · FoundationFind r from centered summaries.Original teaching material · Worked example Correlation of paired data · StandardExplain whether adding t to all y-values changes r.Original teaching material · Worked example Correlation of paired data · TransferCan Pearson r be computed when all x-values equal t? Explain.Original teaching material · Worked example Correlation of paired data · FoundationFind r from centered summaries.Original teaching material · Answer check Correlation of paired data · FoundationFind r when y=tx+2 and x varies.Original teaching material · Answer check Correlation of paired data · FoundationFind r when y=−tx+2 and x varies.Original teaching material · Answer check Correlation of paired data · FoundationFor pairs (−1,t),(0,0),(1,t), find Sxy.Original teaching material · Answer check Correlation of paired data · StandardFor pairs (−1,t),(0,0),(1,t), find Sxy.Original teaching material · Answer check Correlation of paired data · StandardExplain whether adding t to all y-values changes r.Original teaching material · Reasoning checklist Correlation of paired data · StandardAn observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?Original teaching material · Reasoning checklist Correlation of paired data · StandardCan Pearson r be computed when all x-values equal t? Explain.Original teaching material · Reasoning checklist Correlation of paired data · TransferAn observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?Original teaching material · Reasoning checklist Correlation of paired data · TransferCan Pearson r be computed when all x-values equal t? Explain.Original teaching material · Reasoning checklist Linear regression and applications · FoundationFind the fitted slope.Original teaching material · Worked example Linear regression and applications · StandardFor x=−1,0,1 and y=−t,0,t, find the least-squares slope.Original teaching material · Worked example Linear regression and applications · TransferFor x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.Original teaching material · Worked example Linear regression and applications · FoundationFind the fitted slope.Original teaching material · Answer check Linear regression and applications · FoundationFind the intercept.Original teaching material · Answer check Linear regression and applications · FoundationUse the fitted line to predict y at x=2.Original teaching material · Answer check Linear regression and applications · FoundationAn observation at x=2 has y=2t+5. Find its residual under ŷ=3+tx.Original teaching material · Answer check Linear regression and applications · StandardAn observation at x=2 has y=2t+5. Find its residual under ŷ=3+tx.Original teaching material · Answer check Linear regression and applications · StandardFor x=−1,0,1 and y=−t,0,t, find the least-squares slope.Original teaching material · Answer check Linear regression and applications · StandardData were observed only for 0≤x≤t. Assess a prediction at x=10t.Original teaching material · Reasoning checklist Linear regression and applications · StandardFor x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.Original teaching material · Reasoning checklist Linear regression and applications · TransferData were observed only for 0≤x≤t. Assess a prediction at x=10t.Original teaching material · Reasoning checklist Linear regression and applications · TransferFor x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.Original teaching material · Reasoning checklist Contingency tables and independence tests · FoundationFind the first row total.Original teaching material · Worked example Contingency tables and independence tests · StandardUnder perfect independence in a table with all four cells t, find χ².Original teaching material · Worked example Contingency tables and independence tests · TransferThe displayed test statistic is below the supplied 5% threshold 3.841. State the defensible conclusion.Original teaching material · Worked example Contingency tables and independence tests · FoundationFind the first row total.Original teaching material · Answer check Contingency tables and independence tests · FoundationUnder independence, find E11.Original teaching material · Answer check Contingency tables and independence tests · FoundationCalculate χ² for the table.Original teaching material · Answer check Contingency tables and independence tests · FoundationUse the supplied 5% critical value 3.841 (one degree of freedom) to assess independence.Original teaching material · Reasoning checklist Contingency tables and independence tests · StandardUse the supplied 5% critical value 3.841 (one degree of freedom) to assess independence.Original teaching material · Reasoning checklist Contingency tables and independence tests · StandardUnder perfect independence in a table with all four cells t, find χ².Original teaching material · Answer check Contingency tables and independence tests · StandardA before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?Original teaching material · Reasoning checklist Contingency tables and independence tests · StandardThe displayed test statistic is below the supplied 5% threshold 3.841. State the defensible conclusion.Original teaching material · Reasoning checklist Contingency tables and independence tests · TransferA before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?Original teaching material · Reasoning checklist Contingency tables and independence tests · TransferThe displayed test statistic is below the supplied 5% threshold 3.841. State the defensible conclusion.Original teaching material · Reasoning checklist Polynomial division and partial fractions · FoundationFind the remainder when x²+t is divided by x−2.Original teaching material · Worked example Polynomial division and partial fractions · StandardDecompose the rational expression with its repeated denominator factor.Original teaching material · Worked example Polynomial division and partial fractions · TransferExplain the difference between f and g as functions.Original teaching material · Worked example Polynomial division and partial fractions · FoundationFind the remainder when x²+t is divided by x−2.Original teaching material · Answer check Polynomial division and partial fractions · FoundationDivide x²+t x+1 by x.Original teaching material · Reasoning checklist Polynomial division and partial fractions · FoundationIn t/[x(x+1)]=A/x+B/(x+1), find A.Original teaching material · Answer check Polynomial division and partial fractions · FoundationIn (x+t)/[(x−1)(x+1)]=A/(x−1)+B/(x+1), find A.Original teaching material · Answer check Polynomial division and partial fractions · StandardIn (x+t)/[(x−1)(x+1)]=A/(x−1)+B/(x+1), find A.Original teaching material · Answer check Polynomial division and partial fractions · StandardDecompose the rational expression with its repeated denominator factor.Original teaching material · Reasoning checklist Polynomial division and partial fractions · StandardFind quotient and remainder, then express the improper fraction.Original teaching material · Reasoning checklist Polynomial division and partial fractions · StandardExplain the difference between f and g as functions.Original teaching material · Reasoning checklist Polynomial division and partial fractions · TransferFind quotient and remainder, then express the improper fraction.Original teaching material · Reasoning checklist Polynomial division and partial fractions · TransferExplain the difference between f and g as functions.Original teaching material · Reasoning checklist Variation and financial models · Foundationy varies directly with x; y=t when x=2. Find y when x=6.Original teaching material · Worked example Variation and financial models · StandardA hypothetical balance starts at 100t and compounds annually at 10% for 2 years. Find final balance.Original teaching material · Worked example Variation and financial models · TransferA hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.Original teaching material · Worked example Variation and financial models · Foundationy varies directly with x; y=t when x=2. Find y when x=6.Original teaching material · Answer check Variation and financial models · Foundationy varies inversely with x; y=t at x=2. Find y at x=4.Original teaching material · Answer check Variation and financial models · Foundationy varies jointly with x,z and equals t at x=z=1. Find y at x=2,z=3.Original teaching material · Answer check Variation and financial models · Foundationy=a+kx; y=t at x=0 and y=t+6 at x=2. Find y at x=3.Original teaching material · Answer check Variation and financial models · Standardy=a+kx; y=t at x=0 and y=t+6 at x=2. Find y at x=3.Original teaching material · Answer check Variation and financial models · StandardA hypothetical balance starts at 100t and compounds annually at 10% for 2 years. Find final balance.Original teaching material · Answer check Variation and financial models · StandardAn asset worth 100t loses 20% each year for 2 years. Find final value.Original teaching material · Answer check Variation and financial models · StandardA hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.Original teaching material · Answer check Variation and financial models · TransferAn asset worth 100t loses 20% each year for 2 years. Find final value.Original teaching material · Answer check Variation and financial models · TransferA hypothetical amount 100t grows continuously at rate ln2 per year. Find its value after one year.Original teaching material · Answer check Euclidean figures and circle theorems · FoundationA convex polygon has t+3 sides. Find its interior-angle sum in degrees.Original teaching material · Worked example Euclidean figures and circle theorems · StandardA cyclic quadrilateral has one interior angle a°. Find its opposite interior angle.Original teaching material · Worked example Euclidean figures and circle theorems · TransferA chord of a circle radius 5t is distance 3t from the center. Find its length.Original teaching material · Worked example Euclidean figures and circle theorems · FoundationA convex polygon has t+3 sides. Find its interior-angle sum in degrees.Original teaching material · Answer check Euclidean figures and circle theorems · FoundationA right triangle has legs 3t,4t. Find the hypotenuse.Original teaching material · Answer check Euclidean figures and circle theorems · FoundationSimilar triangles have length ratio 2:3. The smaller area is 4t. Find the larger area.Original teaching material · Answer check Euclidean figures and circle theorems · FoundationA central angle subtends a minor arc of 2a°. Find an inscribed angle subtending that same arc.Original teaching material · Answer check Euclidean figures and circle theorems · StandardA central angle subtends a minor arc of 2a°. Find an inscribed angle subtending that same arc.Original teaching material · Answer check Euclidean figures and circle theorems · StandardA cyclic quadrilateral has one interior angle a°. Find its opposite interior angle.Original teaching material · Answer check Euclidean figures and circle theorems · StandardA circle of radius t has a sector angle π/3 radians. Find its sector area.Original teaching material · Answer check Euclidean figures and circle theorems · StandardA chord of a circle radius 5t is distance 3t from the center. Find its length.Original teaching material · Answer check Euclidean figures and circle theorems · TransferA circle of radius t has a sector angle π/3 radians. Find its sector area.Original teaching material · Answer check Euclidean figures and circle theorems · TransferA chord of a circle radius 5t is distance 3t from the center. Find its length.Original teaching material · Answer check Linear programming and discrete feasibility · FoundationIs (t,0) feasible? Enter 1 for yes, 0 for no.Original teaching material · Worked example Linear programming and discrete feasibility · StandardDoes the displayed objective have a maximum over x,y≥0?Original teaching material · Worked example Linear programming and discrete feasibility · TransferProduction uses 2 units per x item and 1 per y item, with 2t units available. Profit is 3x+y. Find maximum continuous profit.Original teaching material · Worked example Linear programming and discrete feasibility · FoundationIs (t,0) feasible? Enter 1 for yes, 0 for no.Original teaching material · Answer check Linear programming and discrete feasibility · FoundationMaximise z=x+y over x,y≥0 and x+y≤t.Original teaching material · Answer check Linear programming and discrete feasibility · FoundationMaximise z=2x+y over x,y≥0 and x+y≤t.Original teaching material · Answer check Linear programming and discrete feasibility · FoundationMinimise x+y for x,y≥0 and x+y≥t.Original teaching material · Answer check Linear programming and discrete feasibility · StandardMinimise x+y for x,y≥0 and x+y≥t.Original teaching material · Answer check Linear programming and discrete feasibility · StandardDoes the displayed objective have a maximum over x,y≥0?Original teaching material · Reasoning checklist Linear programming and discrete feasibility · StandardFor integer x,y≥0 and 2x+2y≤2t+1, find max(x+y).Original teaching material · Answer check Linear programming and discrete feasibility · StandardProduction uses 2 units per x item and 1 per y item, with 2t units available. Profit is 3x+y. Find maximum continuous profit.Original teaching material · Answer check Linear programming and discrete feasibility · TransferFor integer x,y≥0 and 2x+2y≤2t+1, find max(x+y).Original teaching material · Answer check Linear programming and discrete feasibility · TransferProduction uses 2 units per x item and 1 per y item, with 2t units available. Profit is 3x+y. Find maximum continuous profit.Original teaching material · Answer check Matrices, determinants and linear systems · FoundationFind det A.Original teaching material · Worked example Matrices, determinants and linear systems · StandardFind the determinant of the triangular 3×3 matrix.Original teaching material · Worked example Matrices, determinants and linear systems · TransferShow AB≠BA for the specified matrices.Original teaching material · Worked example Matrices, determinants and linear systems · FoundationFind det A.Original teaching material · Answer check Matrices, determinants and linear systems · FoundationFind the (1,2) entry of AB.Original teaching material · Answer check Matrices, determinants and linear systems · FoundationFind x in the system.Original teaching material · Answer check Matrices, determinants and linear systems · FoundationFind A⁻¹.Original teaching material · Reasoning checklist Matrices, determinants and linear systems · StandardFind A⁻¹.Original teaching material · Reasoning checklist Matrices, determinants and linear systems · StandardFind the determinant of the triangular 3×3 matrix.Original teaching material · Answer check Matrices, determinants and linear systems · StandardClassify the solutions of the system.Original teaching material · Reasoning checklist Matrices, determinants and linear systems · StandardShow AB≠BA for the specified matrices.Original teaching material · Reasoning checklist Matrices, determinants and linear systems · TransferClassify the solutions of the system.Original teaching material · Reasoning checklist Matrices, determinants and linear systems · TransferShow AB≠BA for the specified matrices.Original teaching material · Reasoning checklist Polar coordinates · FoundationFind x for the polar point.Original teaching material · Worked example Polar coordinates · StandardConvert the polar curve to Cartesian form.Original teaching material · Worked example Polar coordinates · TransferDescribe the displayed polar locus with r≥0, and compare it with its full supporting line.Original teaching material · Worked example Polar coordinates · FoundationFind x for the polar point.Original teaching material · Answer check Polar coordinates · FoundationFind y for the polar point.Original teaching material · Answer check Polar coordinates · FoundationFind r≥0 for the Cartesian point.Original teaching material · Answer check Polar coordinates · FoundationGive a positive-radius polar representation of (−t,0).Original teaching material · Reasoning checklist Polar coordinates · StandardGive a positive-radius polar representation of (−t,0).Original teaching material · Reasoning checklist Polar coordinates · StandardConvert the polar curve to Cartesian form.Original teaching material · Reasoning checklist Polar coordinates · StandardUnder the signed-radius convention, show that the two polar pairs represent the same point.Original teaching material · Reasoning checklist Polar coordinates · StandardDescribe the displayed polar locus with r≥0, and compare it with its full supporting line.Original teaching material · Reasoning checklist Polar coordinates · TransferUnder the signed-radius convention, show that the two polar pairs represent the same point.Original teaching material · Reasoning checklist Polar coordinates · TransferDescribe the displayed polar locus with r≥0, and compare it with its full supporting line.Original teaching material · Reasoning checklist Integration, area and school extensions · FoundationFind all antiderivatives of tx².Original teaching material · Worked example Integration, area and school extensions · StandardFind the area between y=t and y=x for 0≤x≤t.Original teaching material · Worked example Integration, area and school extensions · TransferSchool extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.Original teaching material · Worked example Integration, area and school extensions · FoundationFind all antiderivatives of tx².Original teaching material · Reasoning checklist Integration, area and school extensions · FoundationEvaluate the definite integral.Original teaching material · Answer check Integration, area and school extensions · FoundationEvaluate the signed integral.Original teaching material · Answer check Integration, area and school extensions · FoundationFind the geometric area between y=x and the x-axis on [−t,t].Original teaching material · Answer check Integration, area and school extensions · StandardFind the geometric area between y=x and the x-axis on [−t,t].Original teaching material · Answer check Integration, area and school extensions · StandardFind the area between y=t and y=x for 0≤x≤t.Original teaching material · Answer check Integration, area and school extensions · StandardF′(x)=2x and F(0)=t. Find F(2).Original teaching material · Answer check Integration, area and school extensions · StandardSchool extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.Original teaching material · Answer check Integration, area and school extensions · TransferF′(x)=2x and F(0)=t. Find F(2).Original teaching material · Answer check Integration, area and school extensions · TransferSchool extension: revolve y=x, 0≤x≤t, about the x-axis. Use the disk model to find volume.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · FoundationFind the plane through (t,0,0) with normal (1,2,3).Original teaching material · Worked example Spatial lines, planes, spheres and cross products · StandardCalculate u×v.Original teaching material · Worked example Spatial lines, planes, spheres and cross products · TransferClassify the two spatial lines.Original teaching material · Worked example Spatial lines, planes, spheres and cross products · FoundationFind the plane through (t,0,0) with normal (1,2,3).Original teaching material · Reasoning checklist Spatial lines, planes, spheres and cross products · FoundationFind the sphere radius.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · FoundationFind the line through P with direction u.Original teaching material · Reasoning checklist Spatial lines, planes, spheres and cross products · FoundationThe line X=(0,0,t)+s(1,0,−1) meets z=0. Find s.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · StandardThe line X=(0,0,t)+s(1,0,−1) meets z=0. Find s.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · StandardCalculate u×v.Original teaching material · Reasoning checklist Spatial lines, planes, spheres and cross products · StandardFind the parallelogram area spanned by u,v.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · StandardClassify the two spatial lines.Original teaching material · Reasoning checklist Spatial lines, planes, spheres and cross products · TransferFind the parallelogram area spanned by u,v.Original teaching material · Answer check Spatial lines, planes, spheres and cross products · TransferClassify the two spatial lines.Original teaching material · Reasoning checklist Parametric curves and preserved domains · FoundationEliminate s from the line representation.Original teaching material · Worked example Parametric curves and preserved domains · StandardA point on x=t cos s,y=2sin s maximises x+ty. Find the maximum.Original teaching material · Worked example Parametric curves and preserved domains · TransferEliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.Original teaching material · Worked example Parametric curves and preserved domains · FoundationEliminate s from the line representation.Original teaching material · Reasoning checklist Parametric curves and preserved domains · FoundationEliminate s and retain the range.Original teaching material · Reasoning checklist Parametric curves and preserved domains · FoundationFind the Cartesian equation of the parametrised circle.Original teaching material · Reasoning checklist Parametric curves and preserved domains · FoundationEliminate s, retaining the excluded point.Original teaching material · Reasoning checklist Parametric curves and preserved domains · StandardEliminate s, retaining the excluded point.Original teaching material · Reasoning checklist Parametric curves and preserved domains · StandardA point on x=t cos s,y=2sin s maximises x+ty. Find the maximum.Original teaching material · Answer check Parametric curves and preserved domains · StandardDescribe the locus when x=t cos s,y=t sin s and 0≤s≤π.Original teaching material · Reasoning checklist Parametric curves and preserved domains · StandardEliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.Original teaching material · Reasoning checklist Parametric curves and preserved domains · TransferDescribe the locus when x=t cos s,y=t sin s and 0≤s≤π.Original teaching material · Reasoning checklist Parametric curves and preserved domains · TransferEliminate s from x=t+s,y=t−s and find the range if 0≤s≤2.Original teaching material · Reasoning checklist Infinite geometric series and error control · FoundationFind the infinite geometric sum.Original teaching material · Worked example Infinite geometric series and error control · StandardFind the tail after the first 3 terms of a geometric series.Original teaching material · Worked example Infinite geometric series and error control · TransferEvaluate the convergent telescoping series.Original teaching material · Worked example Infinite geometric series and error control · FoundationFind the infinite geometric sum.Original teaching material · Answer check Infinite geometric series and error control · FoundationFind the infinite sum with alternating signs.Original teaching material · Answer check Infinite geometric series and error control · FoundationDoes this geometric series converge? Explain.Original teaching material · Reasoning checklist Infinite geometric series and error control · FoundationExpress the repeating decimal 0.dddd… as a fraction value, where the single digit d repeats.Original teaching material · Answer check Infinite geometric series and error control · StandardExpress the repeating decimal 0.dddd… as a fraction value, where the single digit d repeats.Original teaching material · Answer check Infinite geometric series and error control · StandardFind the tail after the first 3 terms of a geometric series.Original teaching material · Answer check Infinite geometric series and error control · StandardFind the least n so the tail of 1+1/2+1/4+… is at most 2^(1−t).Original teaching material · Answer check Infinite geometric series and error control · StandardEvaluate the convergent telescoping series.Original teaching material · Answer check Infinite geometric series and error control · TransferFind the least n so the tail of 1+1/2+1/4+… is at most 2^(1−t).Original teaching material · Answer check Infinite geometric series and error control · TransferEvaluate the convergent telescoping series.Original teaching material · Answer check Radicals, rational equations and simultaneous equations · FoundationSimplify the radical.Original teaching material · Worked example Radicals, rational equations and simultaneous equations · StandardSolve the rational equation.Original teaching material · Worked example Radicals, rational equations and simultaneous equations · TransferExplain why squaring both sides may introduce a root, using this equation.Original teaching material · Worked example Radicals, rational equations and simultaneous equations · FoundationSimplify the radical.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · FoundationSolve the rational equation.Original teaching material · Answer check Radicals, rational equations and simultaneous equations · FoundationSolve the radical equation and reject any extraneous candidate.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · FoundationSolve the linear system; enter x.Original teaching material · Answer check Radicals, rational equations and simultaneous equations · StandardSolve the linear–quadratic system completely.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · StandardIs cancelling x−3 sufficient to conclude that the original equation has every real solution? Explain.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · StandardExplain why squaring both sides may introduce a root, using this equation.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · StandardSimplify the radical.Original teaching material · Reasoning checklist Radicals, rational equations and simultaneous equations · TransferSolve the rational equation.Original teaching material · Answer check Radicals, rational equations and simultaneous equations · TransferSolve the radical equation and reject any extraneous candidate.Original teaching material · Reasoning checklist Absolute values and solution sets · FoundationSolve the strict absolute-value inequality.Original teaching material · Worked example Absolute values and solution sets · StandardSolve the exterior distance inequality.Original teaching material · Worked example Absolute values and solution sets · TransferSolve by comparing distances to 0 and 2.Original teaching material · Worked example Absolute values and solution sets · FoundationSolve the strict absolute-value inequality.Original teaching material · Reasoning checklist Absolute values and solution sets · FoundationSolve the exterior distance inequality.Original teaching material · Reasoning checklist Absolute values and solution sets · FoundationSolve the simultaneous inequalities.Original teaching material · Reasoning checklist Absolute values and solution sets · FoundationCount the integer solutions.Original teaching material · Answer check Absolute values and solution sets · StandardSolve the inequality with a negative right side.Original teaching material · Reasoning checklist Absolute values and solution sets · StandardFind the minimum of the sum of distances and every point attaining it.Original teaching material · Reasoning checklist Absolute values and solution sets · StandardSolve by comparing distances to 0 and 3.Original teaching material · Reasoning checklist Absolute values and solution sets · StandardSolve the strict absolute-value inequality.Original teaching material · Reasoning checklist Absolute values and solution sets · TransferSolve the exterior distance inequality.Original teaching material · Reasoning checklist Absolute values and solution sets · TransferSolve the simultaneous inequalities.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · FoundationFind the inverse of the linear function.Original teaching material · Worked example Inverse functions and domain restrictions · StandardFind the inverse and its domain.Original teaching material · Worked example Inverse functions and domain restrictions · TransferFind the inverse of the fractional linear function with all exclusions.Original teaching material · Worked example Inverse functions and domain restrictions · FoundationFind the inverse of the linear function.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · FoundationFind the inverse and its domain.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · FoundationFind the inverse on the stated restricted domain.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · FoundationEvaluate the inverse composition at 5.Original teaching material · Answer check Inverse functions and domain restrictions · StandardCan f(x)=x² on the real line have an inverse function? Give a counterexample.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · StandardFind both inverse branches by choosing x≤3 or x≥3.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · StandardFind the inverse of the fractional linear function with all exclusions.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · StandardFind the inverse of the linear function.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · TransferFind the inverse and its domain.Original teaching material · Reasoning checklist Inverse functions and domain restrictions · TransferFind the inverse on the stated restricted domain.Original teaching material · Reasoning checklist Exponential and logarithmic equations · FoundationSolve the exponential equation.Original teaching material · Worked example Exponential and logarithmic equations · StandardSolve the logarithmic equation.Original teaching material · Worked example Exponential and logarithmic equations · TransferUse x=−2 to explain why log(x²)=2 log(x) is not an identity for every nonzero real x.Original teaching material · Worked example Exponential and logarithmic equations · FoundationSolve the exponential equation.Original teaching material · Answer check Exponential and logarithmic equations · FoundationSolve the logarithmic equation.Original teaching material · Answer check Exponential and logarithmic equations · FoundationSolve the equation, with the domain restriction.Original teaching material · Answer check Exponential and logarithmic equations · FoundationSolve the exponential quadratic completely.Original teaching material · Reasoning checklist Exponential and logarithmic equations · StandardSolve the logarithmic inequality with a base below one.Original teaching material · Reasoning checklist Exponential and logarithmic equations · StandardA population doubles every 3 years. How many years are needed for an eightfold increase?Original teaching material · Answer check Exponential and logarithmic equations · StandardUse x=−3 to explain why log(x²)=2 log(x) is not an identity for every nonzero real x.Original teaching material · Reasoning checklist Exponential and logarithmic equations · StandardSolve the exponential equation.Original teaching material · Answer check Exponential and logarithmic equations · TransferSolve the logarithmic equation.Original teaching material · Answer check Exponential and logarithmic equations · TransferSolve the equation, with the domain restriction.Original teaching material · Answer check Sets and commonly used logic: mixed assessment · FoundationCount the distinct elements.Original teaching material · Answer check Sets and commonly used logic: mixed assessment · FoundationHow many proper subsets does A have?Original teaching material · Answer check Sets and commonly used logic: mixed assessment · FoundationFind the cardinality of the complement.Original teaching material · Answer check Sets and commonly used logic: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: Repair this proposed implication. Part B: Disprove: a positive product forces both factors to be positive.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · FoundationCount the possible integer witnesses.Original teaching material · Answer check Sets and commonly used logic: mixed assessment · FoundationAre the two sets equal? Explain.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Prove that the empty set is a subset of every set. Part B: How many subsets does A have?Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardFind the interval intersection.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Disprove: a positive product forces both factors to be positive. Part B: Give a necessary and sufficient condition for the product to vanish.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardNegate this statement.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Does “interesting books” specify a well-defined set? Repair the description. Part B: Count the distinct elements.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · StandardHow many subsets does A have?Original teaching material · Answer check Sets and commonly used logic: mixed assessment · TransferFind the cardinality of the union.Original teaching material · Answer check Sets and commonly used logic: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Give a necessary and sufficient condition for the product to vanish. Part B: Repair this proposed implication.Original teaching material · Reasoning checklist Sets and commonly used logic: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Give an integer counterexample. Part B: Count the possible integer witnesses.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · FoundationSolve and justify the sign direction.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · FoundationPositive a,b have sum 14; maximize their product.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · FoundationCalculate the discriminant and interpret its sign.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · FoundationGive a counterexample to squaring as a general order-preserving rule.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · FoundationDoes the function attain a minimum on this domain?Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · FoundationFind the length of the x-interval where the line is above or on the parabola.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Explain how the sign of a affects multiplying x<b by a. Part B: For which m is this positive for every real x?Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · StandardFind the minimum for x>0.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · StandardCount the integer solutions.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · StandardFind the least integer greater than every admissible sum a+b.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · StandardMinimize the expression over positive x.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · StandardFor which m is this positive for every real x?Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Does a<b and c<b guarantee a<c? Explain. Part B: Calculate the discriminant and interpret its sign.Original teaching material · Reasoning checklist Quadratic functions, equations and inequalities: mixed assessment · TransferMinimize the sum of two positive numbers with this product.Original teaching material · Answer check Quadratic functions, equations and inequalities: mixed assessment · TransferSolve the quadratic inequality.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · FoundationEvaluate the function at the stated input.Original teaching material · Answer check Functions: concept and properties: mixed assessment · FoundationClassify parity on the real line.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · FoundationEvaluate the real cube root.Original teaching material · Answer check Functions: concept and properties: mixed assessment · FoundationA service costs C=7+4n for n whole items. A bill is 27. Find n.Original teaching material · Answer check Functions: concept and properties: mixed assessment · FoundationEvaluate the piecewise rule at the breakpoint.Original teaching material · Answer check Functions: concept and properties: mixed assessment · FoundationHow does adding a constant affect monotonicity? Prove your answer.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · StandardFind m so that this is a power function with coefficient one.Original teaching material · Answer check Functions: concept and properties: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: A learner fits a straight line through two observations and claims the model is valid forever. Evaluate the claim. Part B: Find the greatest value on the stated closed interval.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · StandardRecover f(x) from the shifted-input identity.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · StandardState the monotonicity on the real line and justify it.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Give the real domain and range. Part B: Evaluate the function at the stated input.Original teaching material · Reasoning checklist Functions: concept and properties: mixed assessment · StandardA journey covers equal distances at speeds 18 and 54. Find its average speed.Original teaching material · Answer check Functions: concept and properties: mixed assessment · TransferFind the greatest value on the stated closed interval.Original teaching material · Answer check Functions: concept and properties: mixed assessment · TransferFind the minimum on the stated interval.Original teaching material · Answer check Functions: concept and properties: mixed assessment · TransferClassify the parity of this power function on the reals.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · FoundationSimplify the product.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · FoundationSolve the exponential equation.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · FoundationEvaluate by change of base.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · FoundationSolve the inequality.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique. Part B: A population is 800 and increases by 10% each year. Find it after two years.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · FoundationSimplify for a>0.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · StandardFind the y-intercept.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · StandardEvaluate the logarithm with a base below one.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Explain why the graphs y=log₂x and y=2ˣ are reflections in y=x. Part B: Find the inverse function and its domain.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · StandardA population is 800 and increases by 10% each year. Find it after two years.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · StandardCompare the powers without decimal approximations.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · StandardFind the positive exponential base.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · TransferSimplify using the product rule.Original teaching material · Answer check Exponential and logarithmic functions: mixed assessment · TransferFind the inverse function and its domain.Original teaching material · Reasoning checklist Exponential and logarithmic functions: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Find the base-10 logarithmic increase when a positive quantity is multiplied by 1000. Part B: Show that f(x)=2ˣ−3 has a zero between 1 and 2, and explain what else makes the zero unique.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · FoundationWrite the angle as cπ radians; find c.Original teaching material · Answer check Trigonometric functions: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: Find cos θ when θ is in quadrant II. Part B: Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · FoundationFind tangent after a half turn.Original teaching material · Answer check Trigonometric functions: mixed assessment · FoundationCount the zeros on the closed interval.Original teaching material · Answer check Trigonometric functions: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: For π<α<3π/2 and cos α=−3/5, determine the sign and value of sin(α/2). Part B: Evaluate without a calculator.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · FoundationWhat function remains when the frequency is zero? Does it have a least positive period?Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period. Part B: Given an acute angle, find sin 2α.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · StandardFind the sector area for the stated radian angle.Original teaching material · Answer check Trigonometric functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Determine whether the two claimed trigonometric values can occur together. Part B: A wheel has radius 8 m and its centre is 10 m above ground. Find the greatest passenger height.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Evaluate without a calculator. Part B: How many coterminal representatives of 30° lie in this closed interval?Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Is sine increasing on the whole real line? Give a counterexample. Part B: Write the angle as cπ radians; find c.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Given an acute angle, find sin 2α. Part B: Find cos θ when θ is in quadrant II.Original teaching material · Reasoning checklist Trigonometric functions: mixed assessment · TransferWrite the least positive period as cπ; find c.Original teaching material · Answer check Trigonometric functions: mixed assessment · TransferA wheel has radius 8 m and its centre is 10 m above ground. Find the greatest passenger height.Original teaching material · Answer check Trigonometric functions: mixed assessment · TransferHow many coterminal representatives of 30° lie in this closed interval?Original teaching material · Answer check Plane vectors and applications: mixed assessment · FoundationFind the vector magnitude.Original teaching material · Answer check Plane vectors and applications: mixed assessment · FoundationCalculate the dot product.Original teaching material · Answer check Plane vectors and applications: mixed assessment · FoundationFind m for the two vectors to be parallel.Original teaching material · Answer check Plane vectors and applications: mixed assessment · FoundationFind the x-coordinate of the centroid.Original teaching material · Answer check Plane vectors and applications: mixed assessment · FoundationGive a unit vector in the same direction.Original teaching material · Reasoning checklist Plane vectors and applications: mixed assessment · FoundationFind the angle in degrees between these nonzero vectors.Original teaching material · Answer check Plane vectors and applications: mixed assessment · StandardFind the signed scalar projection on the positive x-axis.Original teaching material · Answer check Plane vectors and applications: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Explain why the sine rule in an SSA problem may give two triangles. Part B: Do equal magnitudes guarantee equal vectors? Give a counterexample.Original teaching material · Reasoning checklist Plane vectors and applications: mixed assessment · StandardA walker travels 9 m east then 9 m west. Find the magnitude of displacement.Original teaching material · Answer check Plane vectors and applications: mixed assessment · StandardFind the x-coordinate of a+b.Original teaching material · Answer check Plane vectors and applications: mixed assessment · StandardFind the midpoint x-coordinate.Original teaching material · Answer check Plane vectors and applications: mixed assessment · StandardFind the triangle area.Original teaching material · Answer check Plane vectors and applications: mixed assessment · TransferDo equal magnitudes guarantee equal vectors? Give a counterexample.Original teaching material · Reasoning checklist Plane vectors and applications: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Simplify using the triangle rule. Part B: Find the x-coordinate of the centroid.Original teaching material · Reasoning checklist Plane vectors and applications: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Explain why parallel vectors cannot form a basis for the plane. Part B: Give a unit vector in the same direction.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · FoundationFind the imaginary part.Original teaching material · Answer check Complex numbers: mixed assessment · FoundationSimplify the quotient.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · FoundationFind the real part using de Moivre’s formula.Original teaching material · Answer check Complex numbers: mixed assessment · FoundationWrite the conjugate.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · FoundationSolve the quadratic over the complex numbers.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: Are arguments θ and θ+2π different complex numbers at the same positive modulus? Explain. Part B: Evaluate the power.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · StandardFind the sum a+b when the complex number is zero.Original teaching material · Answer check Complex numbers: mixed assessment · StandardFind the real part of the product.Original teaching material · Answer check Complex numbers: mixed assessment · StandardFind the modulus of the product.Original teaching material · Answer check Complex numbers: mixed assessment · StandardFind the modulus.Original teaching material · Answer check Complex numbers: mixed assessment · StandardEvaluate the power.Original teaching material · Answer check Complex numbers: mixed assessment · StandardFind the modulus of the quotient.Original teaching material · Answer check Complex numbers: mixed assessment · TransferSolve over the complex numbers.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: Simplify the sum and explain the cancellation. Part B: Write the conjugate.Original teaching material · Reasoning checklist Complex numbers: mixed assessment · TransferGive a polar representation.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · FoundationHow many vertices does an 10-gonal prism have?Original teaching material · Answer check Introduction to solid geometry: mixed assessment · FoundationRecover the actual depth from its half-scale drawn length.Original teaching material · Answer check Introduction to solid geometry: mixed assessment · FoundationA sphere has radius r. Write its volume as cπ and find c.Original teaching material · Answer check Introduction to solid geometry: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: State the intersection of two distinct planes known to share two distinct points. Part B: Similar solids have linear scale factor 3 from small to large. Find the surface-area ratio.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · FoundationSolve both parts and justify the conditions used. Part A: Find the dihedral angle between the coordinate planes x=0 and y=0. Part B: A line l is outside plane α and parallel to a line m in α. State and justify the conclusion.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · StandardA vertical edge has actual length 10 and vertical scale one. Find its drawn length.Original teaching material · Answer check Introduction to solid geometry: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: Similar solids have linear scale factor 3 from small to large. Find the surface-area ratio. Part B: In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: How many planes contain two distinct parallel lines? Explain. Part B: A cone has radius r and height h. Write its volume as cπ and find c.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · StandardSolve both parts and justify the conditions used. Part A: A line l is outside plane α and parallel to a line m in α. State and justify the conclusion. Part B: How many vertices does an 10-gonal prism have?Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · StandardFind the distance from P to the plane.Original teaching material · Answer check Introduction to solid geometry: mixed assessment · TransferVerify Euler’s relation for an 12-gonal prism by giving V,E,F.Original teaching material · Reasoning checklist Introduction to solid geometry: mixed assessment · TransferIn the 45° half-depth oblique convention, an original depth is d. Find its drawn length.Original teaching material · Answer check Introduction to solid geometry: mixed assessment · TransferA cone has radius r and height h. Write its volume as cπ and find c.Original teaching material · Answer check Statistics: mixed assessment · FoundationA population has 100k students and a simple random sample has 10k. Find each student’s inclusion probability.Original teaching material · Answer check Statistics: mixed assessment · FoundationA representative sample estimates a proportion 1/4 in a population of 80k. Estimate the corresponding count.Original teaching material · Answer check Statistics: mixed assessment · FoundationA questionnaire asks only classmates arriving before 8 am. Explain a coverage issue when estimating the whole school’s commute time.Original teaching material · Reasoning checklist Statistics: mixed assessment · FoundationA voluntary web poll receives 20k replies. Does the large reply count remove self-selection bias?Original teaching material · Reasoning checklist Statistics: mixed assessment · FoundationFor observations k,k+2,k+4, find variance using denominator 3 as specified.Original teaching material · Answer check Statistics: mixed assessment · FoundationA survey finds longer commute is associated with lower attendance. Does the survey prove that commute caused the attendance pattern?Original teaching material · Reasoning checklist Statistics: mixed assessment · StandardA school studies all 100k students but measures everyone’s height with a ruler biased by +1 cm. Is there sampling error? Is there measurement bias?Original teaching material · Reasoning checklist Statistics: mixed assessment · StandardIn a sample of 20k, 5k use public transport. Estimate the population proportion.Original teaching material · Answer check Statistics: mixed assessment · StandardTwo strata have mean commute times 20 and 40 minutes and sizes 3k and k. Find the overall mean.Original teaching material · Answer check Statistics: mixed assessment · StandardSystematically select 10k from an ordered population of 100k. Find the interval.Original teaching material · Answer check Statistics: mixed assessment · StandardFor observations k,k+2,k+4, find the sample mean.Original teaching material · Answer check Statistics: mixed assessment · StandardSurvey responses 5k contain 2k reports in seconds although the question asked minutes. Is direct pooling valid?Original teaching material · Reasoning checklist Statistics: mixed assessment · TransferSampling with replacement chooses 2 people from a population of k+2. Find the probability the same person appears twice.Original teaching material · Answer check Statistics: mixed assessment · TransferA sample’s mean is k and one extreme observation 100k is added. Must the median change as much as the mean? Explain.Original teaching material · Reasoning checklist Statistics: mixed assessment · TransferAn anonymous commute survey records 20k students; 12k travel at most 30 minutes. Find the sample proportion.Original teaching material · Answer check Probability: mixed assessment · FoundationA fair spinner has k+3 equally likely labels 1,…,k+3. Find the probability of label 1.Original teaching material · Answer check Probability: mixed assessment · FoundationTwo independent components each work with probability k/(k+1). A series system needs both. Find success probability.Original teaching material · Answer check Probability: mixed assessment · FoundationA model has p=1/4. In 20k independent trials, find the expected number of occurrences.Original teaching material · Answer check Probability: mixed assessment · FoundationDisjoint events have probabilities 1/(k+3) and 2/(k+3). Find their union probability.Original teaching material · Answer check Probability: mixed assessment · FoundationA and B are disjoint with P(A)=P(B)=1/(k+2). Are they independent?Original teaching material · Reasoning checklist Probability: mixed assessment · FoundationA program’s pseudo-random outputs are reused with the same seed. Does that create new independent evidence each rerun?Original teaching material · Reasoning checklist Probability: mixed assessment · StandardA claim assigns probability 1+1/k to an event. Explain why it cannot be valid.Original teaching material · Reasoning checklist Probability: mixed assessment · StandardIndependent A,B have probabilities 1/2 and 1/(k+2). Find P(A∩B).Original teaching material · Answer check Probability: mixed assessment · StandardTwo batches have 2k successes in 5k trials and 3k in 10k. Find pooled relative frequency.Original teaching material · Answer check Probability: mixed assessment · StandardP(A)=k/(k+2). Find P(Aᶜ).Original teaching material · Answer check Probability: mixed assessment · StandardA,B are independent and P(B)>0. Compare P(A|B) with P(A).Original teaching material · Reasoning checklist Probability: mixed assessment · StandardA simulation makes 10k equally likely selections from labels 1,…,k. What model probability does label 1 have?Original teaching material · Answer check Probability: mixed assessment · TransferSolve both parts and justify the conditions used. Part A: A fair coin is tossed twice. List the event “exactly one head”. Part B: A model has p=1/4. In 20k independent trials, find the expected number of occurrences.Original teaching material · Reasoning checklist Probability: mixed assessment · TransferThe probability of each component failing is 1/(k+1), but common power failures can affect both. Can the independent-series formula be used without further evidence?Original teaching material · Reasoning checklist Probability: mixed assessment · TransferAn event occurs 3k times in 10k trials. Find its relative frequency.Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind the first component of a+b.Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind λ so that a and b are perpendicular.Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind the distance from P to the plane.Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind the first component of a+b.Original teaching material · Answer check Space vectors and solid geometry: mixed review · FoundationFind the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).Original teaching material · Answer check Space vectors and solid geometry: mixed review · StandardFind cos θ between a and b.Original teaching material · Answer check Space vectors and solid geometry: mixed review · StandardFind the distance from P to the plane.Original teaching material · Answer check Space vectors and solid geometry: mixed review · StandardAre a and b parallel? Justify your answer.Original teaching material · Reasoning checklist Space vectors and solid geometry: mixed review · StandardFind λ so that the third vector lies in the plane spanned by the first two.Original teaching material · Answer check Space vectors and solid geometry: mixed review · StandardFind the signed scalar projection of v onto u.Original teaching material · Answer check Space vectors and solid geometry: mixed review · StandardDoes the entire line lie in the plane?Original teaching material · Reasoning checklist Space vectors and solid geometry: mixed review · TransferAre a and b parallel? Justify your answer.Original teaching material · Reasoning checklist Space vectors and solid geometry: mixed review · TransferExplain why these three vectors do not give unique coefficients.Original teaching material · Reasoning checklist Space vectors and solid geometry: mixed review · TransferA unit direction has cos²α=1/t², cos²β=1/9. Find cos²γ.Original teaching material · Answer check Equations of lines and circles: mixed review · FoundationFind the slope of AB.Original teaching material · Answer check Equations of lines and circles: mixed review · FoundationFind the line through (t,1) and (t,5).Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · FoundationFind the distance from P to the line.Original teaching material · Answer check Equations of lines and circles: mixed review · FoundationFind the circle centered at (t,−1) and passing through (t+3,3).Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · FoundationFind the number of common points.Original teaching material · Answer check Equations of lines and circles: mixed review · FoundationA line has inclination 135°. Find its slope.Original teaching material · Answer check Equations of lines and circles: mixed review · StandardFind C so that Ax+By+C=0 passes through P.Original teaching material · Answer check Equations of lines and circles: mixed review · StandardFind the foot of the perpendicular from P to the line.Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · StandardFind the circle through O=(0,0), A=(2t,0), B=(0,2).Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · StandardClassify the relation of these two circles.Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · StandardFind a so that the two nonvertical lines are parallel.Original teaching material · Answer check Equations of lines and circles: mixed review · StandardFind the reflection of the line in the y-axis.Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · TransferReflect P in the line y=x.Original teaching material · Reasoning checklist Equations of lines and circles: mixed review · TransferThe circle center is (t,0), radius 2. Find the largest x-coordinate on it.Original teaching material · Answer check Equations of lines and circles: mixed review · TransferFind the tangent at the given point.Original teaching material · Reasoning checklist Conic sections: mixed review · FoundationFind the major-axis length.Original teaching material · Answer check Conic sections: mixed review · FoundationFind the length of the transverse axis.Original teaching material · Answer check Conic sections: mixed review · FoundationFind the parabola with vertex O and focus (0,−t).Original teaching material · Reasoning checklist Conic sections: mixed review · FoundationP is on the ellipse and PF1=t. Find PF2.Original teaching material · Answer check Conic sections: mixed review · FoundationFind c².Original teaching material · Answer check Conic sections: mixed review · FoundationFind the x-coordinate of the directrix.Original teaching material · Answer check Conic sections: mixed review · StandardP is on the ellipse and PF1=t. Find PF2.Original teaching material · Answer check Conic sections: mixed review · StandardFind the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).Original teaching material · Reasoning checklist Conic sections: mixed review · StandardFind the tangent of slope 1 to y²=4tx.Original teaching material · Reasoning checklist Conic sections: mixed review · StandardP is on the ellipse and PF1=t. Find PF2.Original teaching material · Answer check Conic sections: mixed review · StandardFind the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).Original teaching material · Reasoning checklist Conic sections: mixed review · StandardFind the tangent of slope 1 to y²=4tx.Original teaching material · Reasoning checklist Conic sections: mixed review · TransferFind the length of the chord cut by x=0.Original teaching material · Answer check Conic sections: mixed review · TransferShow there is one common point and explain why this line is not tangent.Original teaching material · Reasoning checklist Conic sections: mixed review · TransferFind the tangent of slope 1 to y²=4tx.Original teaching material · Reasoning checklist Sequences: mixed review · FoundationFind a3.Original teaching material · Answer check Sequences: mixed review · FoundationFind the common difference.Original teaching material · Answer check Sequences: mixed review · FoundationFind S3.Original teaching material · Answer check Sequences: mixed review · FoundationProve n³−n+3tn is divisible by 3 for all n≥1.Original teaching material · Reasoning checklist Sequences: mixed review · FoundationFind a3.Original teaching material · Answer check Sequences: mixed review · FoundationFind the common difference.Original teaching material · Answer check Sequences: mixed review · StandardFind S5 when q=1.Original teaching material · Answer check Sequences: mixed review · StandardProve 2ⁿ≥n+1 for every integer n≥t, using t as the base.Original teaching material · Reasoning checklist Sequences: mixed review · StandardFind the sum of the first t terms.Original teaching material · Answer check Sequences: mixed review · StandardFind a5 when S9 is given.Original teaching material · Answer check Sequences: mixed review · StandardFind a3a7 given a5=t.Original teaching material · Answer check Sequences: mixed review · StandardProve the scaled telescoping identity by induction.Original teaching material · Reasoning checklist Sequences: mixed review · TransferFind the sum of the first t terms.Original teaching material · Answer check Sequences: mixed review · TransferA theatre has t rows; the first has 10 seats and each next row has 2 more. Find the total seats.Original teaching material · Answer check Sequences: mixed review · TransferFind all real geometric means between t and 4t.Original teaching material · Reasoning checklist Derivatives and their applications: mixed review · FoundationFind the derivative of x² at x=t from the limit.Original teaching material · Answer check Derivatives and their applications: mixed review · FoundationFind f′(1) using the product rule.Original teaching material · Answer check Derivatives and their applications: mixed review · FoundationFind the increasing and decreasing intervals.Original teaching material · Reasoning checklist Derivatives and their applications: mixed review · FoundationFind the tangent to y=x² at x=t.Original teaching material · Reasoning checklist Derivatives and their applications: mixed review · FoundationFind f′(1).Original teaching material · Answer check Derivatives and their applications: mixed review · FoundationFind the maximum of f on [0,t].Original teaching material · Answer check Derivatives and their applications: mixed review · StandardFind the tangent to y=x² at x=t.Original teaching material · Reasoning checklist Derivatives and their applications: mixed review · StandardFind f′(0).Original teaching material · Answer check Derivatives and their applications: mixed review · StandardFind the minimum of x+t²/x for x>0.Original teaching material · Answer check Derivatives and their applications: mixed review · StandardFind the tangent to y=x² at x=t.Original teaching material · Reasoning checklist Derivatives and their applications: mixed review · StandardFind f′(0).Original teaching material · Answer check Derivatives and their applications: mixed review · StandardFind the minimum of x+t²/x for x>0.Original teaching material · Answer check Derivatives and their applications: mixed review · TransferA tank volume V(h)=3h². Find dV/dh at h=t.Original teaching material · Answer check Derivatives and their applications: mixed review · TransferFind f′(0), using radians.Original teaching material · Answer check Derivatives and their applications: mixed review · TransferFind the minimum of x+t²/x for x>0.Original teaching material · Answer check Counting principles: mixed review · FoundationChoose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?Original teaching material · Answer check Counting principles: mixed review · FoundationChoose an unordered committee of 3 from m people.Original teaching material · Answer check Counting principles: mixed review · FoundationFind the sum of all coefficients of (2+x)^m.Original teaching material · Answer check Counting principles: mixed review · FoundationAmong t+8 students, t study art, 8 study music and 3 study both. How many study either subject?Original teaching material · Answer check Counting principles: mixed review · FoundationChoose a president and secretary from m distinct people. Count assignments.Original teaching material · Answer check Counting principles: mixed review · FoundationFind the coefficient of x² in (1+x)^m.Original teaching material · Answer check Counting principles: mixed review · StandardAmong t+8 students, t study art, 8 study music and 3 study both. How many study either subject?Original teaching material · Answer check Counting principles: mixed review · StandardFrom m people, form a 3-person committee that includes a specified person.Original teaching material · Answer check Counting principles: mixed review · StandardFind the alternating sum of binomial coefficients for m≥1.Original teaching material · Answer check Counting principles: mixed review · StandardAmong t+8 students, t study art, 8 study music and 3 study both. How many study either subject?Original teaching material · Answer check Counting principles: mixed review · StandardFrom m people, form a 3-person committee that includes a specified person.Original teaching material · Answer check Counting principles: mixed review · StandardFind the alternating sum of binomial coefficients for m≥1.Original teaching material · Answer check Counting principles: mixed review · TransferPasswords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.Original teaching material · Answer check Counting principles: mixed review · TransferArrange m distinct people in a line with A and B adjacent.Original teaching material · Answer check Counting principles: mixed review · TransferFind the alternating sum of binomial coefficients for m≥1.Original teaching material · Answer check Random variables and distributions: mixed review · FoundationFind P(A|B) from the given probabilities.Original teaching material · Answer check Random variables and distributions: mixed review · FoundationFind P(X≥t).Original teaching material · Answer check Random variables and distributions: mixed review · FoundationFind Var(X).Original teaching material · Answer check Random variables and distributions: mixed review · FoundationX~Bin(m,1/4). Find Var(X).Original teaching material · Answer check Random variables and distributions: mixed review · FoundationX~N(t,4). Find its standard deviation.Original teaching material · Answer check Random variables and distributions: mixed review · FoundationFrom a bag of t red and 2 blue balls, two are drawn without replacement. Given the first is red, find the probability the second is blue.Original teaching material · Answer check Random variables and distributions: mixed review · StandardGive the distribution of Y=2X+1.Original teaching material · Reasoning checklist Random variables and distributions: mixed review · StandardFind Var(3X+2).Original teaching material · Answer check Random variables and distributions: mixed review · StandardFrom t red and 3 blue balls, select 2 without replacement. Find the probability exactly one is red.Original teaching material · Answer check Random variables and distributions: mixed review · StandardX~N(t,4), Φ(1)=0.8413. Find P(t−2≤X≤t+2).Original teaching material · Answer check Random variables and distributions: mixed review · StandardChoose machine 1 with probability 1/(t+2), otherwise machine 2; their failure rates are 1/10 and 1/5. Given failure, find P(machine 1).Original teaching material · Answer check Random variables and distributions: mixed review · StandardCan these numbers be a probability distribution? Explain.Original teaching material · Reasoning checklist Random variables and distributions: mixed review · TransferIndependent X,Y have variances t and 2. Find Var(X+Y).Original teaching material · Answer check Random variables and distributions: mixed review · TransferIndependent trials succeed with probability 1/2. Find the least n for success at least once with probability ≥1−2^(−m).Original teaching material · Answer check Random variables and distributions: mixed review · TransferX~N(t,4). Find P(X=t).Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationFind r from centered summaries.Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationFind the intercept.Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationCalculate χ² for the table.Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationFor pairs (−1,t),(0,0),(1,t), find Sxy.Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationFind the fitted slope.Original teaching material · Answer check Statistical analysis of paired data: mixed review · FoundationUnder independence, find E11.Original teaching material · Answer check Statistical analysis of paired data: mixed review · StandardFor pairs (−1,t),(0,0),(1,t), find Sxy.Original teaching material · Answer check Statistical analysis of paired data: mixed review · StandardFor x=−1,0,1 and y=−t,0,t, find the least-squares slope.Original teaching material · Answer check Statistical analysis of paired data: mixed review · StandardA before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?Original teaching material · Reasoning checklist Statistical analysis of paired data: mixed review · StandardFor pairs (−1,t),(0,0),(1,t), find Sxy.Original teaching material · Answer check Statistical analysis of paired data: mixed review · StandardFor x=−1,0,1 and y=−t,0,t, find the least-squares slope.Original teaching material · Answer check Statistical analysis of paired data: mixed review · StandardA before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?Original teaching material · Reasoning checklist Statistical analysis of paired data: mixed review · TransferAn observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?Original teaching material · Reasoning checklist Statistical analysis of paired data: mixed review · TransferFor x=−1,0,1 and y=t,0,t, find the fitted line and explain what it misses.Original teaching material · Reasoning checklist Statistical analysis of paired data: mixed review · TransferA before/after category table records the same t people twice. Can a standard independent-observation chi-square analysis treat the records as separate people?Original teaching material · Reasoning checklist 2026 JM01 · StandardIntersect the quadratic-inequality set with the absolute-value set.jm01-2026 · Answer check 2026 JM01 · StandardDetermine x/y from the given rational equation.jm01-2026 · Answer check 2026 JM01 · StandardFind a−b when the linear equation is an identity in x.jm01-2026 · Answer check 2026 JM01 · StandardFind the parameter range for two distinct real roots.jm01-2026 · Answer check 2026 JM01 · StandardFind the remainder after division by x+3.jm01-2026 · Answer check 2026 JM01 · StandardFind the sum of the squared roots without solving the quadratic.jm01-2026 · Answer check 2026 JM01 · StandardExpress log base b of ab using x.jm01-2026 · Answer check 2026 JM01 · StandardFind the tangent at the given point on the circle.jm01-2026 · Answer check 2026 JM01 · StandardRecover f(−2) using the cancellation of odd powers.jm01-2026 · Answer check 2026 JM01 · StandardCount five-person committees from six boys and four girls, with at least three boys and one girl.jm01-2026 · Answer check 2026 JM01 · StandardFind all real solutions of the radical equation.jm01-2026 · Answer check 2026 JM01 · StandardEvaluate the double-angle quotient using the given tangent.jm01-2026 · Answer check 2026 JM01 · StandardFind a₅+a₇ for the geometric sequence with the two given constraints.jm01-2026 · Answer check 2026 JM01 · StandardFind the parameter range making the function strictly decreasing on (−1,∞).jm01-2026 · Answer check 2026 JM01 · StandardDetermine the range of a quadratic g when the composite has range [0,∞).jm01-2026 · Answer check 2026 JM01 · StandardComplete the distribution of team A’s score for three independent shots with success probabilities 3/4, 2/3 and 1/2.jm01-2026 · Reasoning checklist 2026 JM01 · StandardTeam B also has three independent shots, each with probability 3/4. Find the probability that the total score is three and B beats A.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFind the y-intercept P and vertex Q.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFor M on the x-axis, minimise PM+QM using P=(0,5), Q=(2,9), and find M.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFind the general term of an arithmetic sequence with a₅=44 and S₅=140.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFor the preceding sequence, sum bₙ=1/Sₙ to obtain Tₙ.jm01-2026 · Reasoning checklist 2026 JM01 · StandardProve both bounds for Tₙ, for every positive integer n.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFind the horizontal hyperbola through (2√3,2) with eccentricity √6/2.jm01-2026 · Reasoning checklist 2026 JM01 · StandardThe line y=x−6 meets this hyperbola at A,B. With M=(2,0), prove AM is perpendicular to BM.jm01-2026 · Reasoning checklist 2026 JM01 · StandardCalculate the next two terms of the recurrence.jm01-2026 · Reasoning checklist 2026 JM01 · StandardProve the recurrence formula by induction and give f(xₙ), as requested by the printed wording.jm01-2026 · Reasoning checklist 2026 JM01 · StandardFind the maximum for x>0.jm01-2026 · Reasoning checklist 2026 JM02 · StandardIn the unit cube shown in the official diagram, M is on EC and GM⊥EC. Prove triangles CMG, CMB and CMD are congruent.jm02-2026 · Reasoning checklist 2026 JM02 · StandardProve that M lies in plane BDG and CM is perpendicular to it.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the volume of the cone with vertex E and base circle through B,D,G.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the first and second derivatives.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind and classify the local extrema.jm02-2026 · Reasoning checklist 2026 JM02 · StandardLocate and justify the inflection point.jm02-2026 · Reasoning checklist 2026 JM02 · StandardSketch the cubic on the stated closed interval.jm02-2026 · Reasoning checklist 2026 JM02 · StandardSketch the absolute-value reflection of the cubic.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the total area of both regions between the cubic and y=x.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the focal distance a and first-quadrant intersection Q of the ellipse and the circle whose diameter joins its foci.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the other intersection M of QF₂ with the ellipse.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind tan∠QMF₁ for the preceding points.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind the perpendicular distance from Q to line MF₁.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind both roots in unit polar form with principal argument in (−π,π].jm02-2026 · Reasoning checklist 2026 JM02 · StandardUse the preceding quadratic to find all four roots, with principal arguments.jm02-2026 · Reasoning checklist 2026 JM02 · StandardDerive the real quadratic factorisation from the four roots.jm02-2026 · Reasoning checklist 2026 JM02 · StandardDeduce the positive exact value of cos(π/12) from the factorisation.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFactor the determinant completely.jm02-2026 · Reasoning checklist 2026 JM02 · StandardGive the complete real solution set of the two linear equations.jm02-2026 · Reasoning checklist 2026 JM02 · StandardFind a,b,c so that every solution of the preceding linear system also satisfies the quadratic constraint.jm02-2026 · Reasoning checklist 2026 JM02 · StandardWith a=b=c=1, solve the three-equation system.jm02-2026 · Reasoning checklist 2025 JM01 · StandardFind the intersection with the finite set.jm01-2025 · Answer check 2025 JM01 · StandardThe reciprocal numbers 1/α and 1/β are the roots; evaluate the exponential product.jm01-2025 · Answer check 2025 JM01 · StandardA cylinder’s radius increases 30% while its height decreases 30%. Find the volume change.jm01-2025 · Answer check 2025 JM01 · StandardSimplify the expression on its real domain.jm01-2025 · Answer check 2025 JM01 · StandardExpress the logarithm using p and q.jm01-2025 · Answer check 2025 JM01 · StandardSolve the strict absolute-value inequality.jm01-2025 · Answer check 2025 JM01 · StandardArrange four distinct girls and three distinct boys in a row with no adjacent boys.jm01-2025 · Answer check 2025 JM01 · StandardSix shifted numbers have median 8. Find their mean.jm01-2025 · Answer check 2025 JM01 · StandardFind the coefficient of x⁴y⁵ in the product.jm01-2025 · Answer check 2025 JM01 · StandardFind the shortest chord cut by a line through (1,1).jm01-2025 · Answer check 2025 JM01 · StandardA point on the downward parabola has focal distance 15 and distance 7 from the x-axis. Find p.jm01-2025 · Answer check 2025 JM01 · StandardJohn and Anna each shoot three times, with independent hit probabilities 1/3 and 2/3. Find the probability of four total hits.jm01-2025 · Answer check 2025 JM01 · StandardFind the seventh term of the arithmetic sequence.jm01-2025 · Answer check 2025 JM01 · StandardMaximise the linear objective over the feasible region.jm01-2025 · Answer check 2025 JM01 · StandardUse the periodic extension to identify the incorrect inequality.jm01-2025 · Answer check 2025 JM01 · StandardFind the general terms of both sequences.jm01-2025 · Reasoning checklist 2025 JM01 · StandardSum cₙ=aₙbₙ for the preceding two sequences.jm01-2025 · Reasoning checklist 2025 JM01 · StandardDetermine the three parameters in the factorisation using the leading coefficient, constant term and remainder.jm01-2025 · Reasoning checklist 2025 JM01 · StandardFind all real zeros of the factorised polynomial.jm01-2025 · Reasoning checklist 2025 JM01 · StandardIn a nondegenerate triangle, determine cos C from the half-angle relation.jm01-2025 · Reasoning checklist 2025 JM01 · StandardWith A=45° and cos C=4/5, calculate sin(2B).jm01-2025 · Reasoning checklist 2025 JM01 · StandardUse the isosceles triangle ABC with D on AB and A,E,F,C in order on AC; AD=AE, G=DF∩BE and ∠AFD=∠DEB. Prove △DEG∼△DFE.jm01-2025 · Reasoning checklist 2025 JM01 · StandardUse the isosceles triangle ABC with D on AB and A,E,F,C in order on AC; AD=AE, G=DF∩BE and ∠AFD=∠DEB. Prove △DEF∼△BDE.jm01-2025 · Reasoning checklist 2025 JM01 · StandardUse the two established similarities to prove the product identity.jm01-2025 · Reasoning checklist 2025 JM01 · StandardFind the locus equation, retaining the condition that both slopes exist.jm01-2025 · Reasoning checklist 2025 JM01 · StandardA secant y=kx+b has nonzero k,b and chord midpoint D on the preceding ellipse. Find the slope of OD.jm01-2025 · Reasoning checklist 2025 JM02 · StandardA square pyramid V–ABCD has base side a>0. Face VAD is equilateral and perpendicular to the base; M is the midpoint of AD. Find VM.jm02-2025 · Reasoning checklist 2025 JM02 · StandardA square pyramid V–ABCD has base side a>0. Face VAD is equilateral and perpendicular to the base; M is the midpoint of AD. Find the pyramid volume.jm02-2025 · Reasoning checklist 2025 JM02 · StandardA square pyramid V–ABCD has base side a>0. Face VAD is equilateral and perpendicular to the base; M is the midpoint of AD. Find the tangent of the dihedral angle between VAD and VDB.jm02-2025 · Reasoning checklist 2025 JM02 · StandardA square pyramid V–ABCD has base side a>0. Face VAD is equilateral and perpendicular to the base; M is the midpoint of AD. Find the total surface area.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind all real zeros, including multiplicity.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind the first two derivatives.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind and classify the local extrema.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind the inflection point and verify the concavity change.jm02-2025 · Reasoning checklist 2025 JM02 · StandardSketch the cubic on −3≤x≤1.2.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind the area enclosed by the two cubics.jm02-2025 · Reasoning checklist 2025 JM02 · StandardThe line through the right focus and P=(2,1) also passes through A=(0,−1). Find its equation.jm02-2025 · Reasoning checklist 2025 JM02 · StandardThe horizontal ellipse has right focus F₁ on y=x−1. Find both foci.jm02-2025 · Reasoning checklist 2025 JM02 · StandardUse A=(0,−1) and the foci to determine the positive semiaxes a,b.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind both intercepts m for a tangent parallel to y=x−1.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind the nonreal cube roots of eight.jm02-2025 · Reasoning checklist 2025 JM02 · StandardCompute the 2024th power for each of the two nonreal roots from part (a).jm02-2025 · Reasoning checklist 2025 JM02 · StandardProve the finite complex-sum identity for positive even n under the stated angle restrictions.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind every solution in the open interval.jm02-2025 · Reasoning checklist 2025 JM02 · StandardCalculate and factor the determinant.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind all real triples for which the determinant vanishes.jm02-2025 · Reasoning checklist 2025 JM02 · StandardFind the parameter values giving a unique solution.jm02-2025 · Reasoning checklist 2025 JM02 · StandardClassify all cases with more than one solution, including conditions on p,q and the general solutions.jm02-2025 · Reasoning checklist 2024 JM01 · StandardFind a from the stated intersection.jm01-2024 · Answer check 2024 JM01 · StandardFind f(15).jm01-2024 · Answer check 2024 JM01 · StandardIf x increases by 4, how does y change?jm01-2024 · Answer check 2024 JM01 · StandardFind the coefficient of x.jm01-2024 · Answer check 2024 JM01 · StandardA moving point stays a fixed distance from (2,3), and its locus passes through the origin. Find the locus.jm01-2024 · Answer check 2024 JM01 · StandardEvaluate using common logarithms.jm01-2024 · Answer check 2024 JM01 · StandardA real geometric sequence satisfies a₂+a₅=9 and a₇+a₁₀=288. Find a₂₀.jm01-2024 · Answer check 2024 JM01 · StandardThe mean of the data is 6.8. Find the median.jm01-2024 · Answer check 2024 JM01 · StandardSimplify for real m≠0.jm01-2024 · Answer check 2024 JM01 · StandardIn acute triangle ABC, AB=8, AC=7 and sin C=4√3/7. Find BC.jm01-2024 · Answer check 2024 JM01 · StandardA parabola passes through (−2,0), (6,0), and (0,4). Find its maximum ordinate.jm01-2024 · Answer check 2024 JM01 · StandardWhich listed interval is entirely an interval of increase?jm01-2024 · Answer check 2024 JM01 · StandardSolve on the stated interval.jm01-2024 · Answer check 2024 JM01 · StandardEvaluate the symmetric expression.jm01-2024 · Answer check 2024 JM01 · StandardAn even function decreases strictly on (−∞,0). Choose the correct ordering; write a=2^(−7/3), b=3^(−2/7), c=log₃(2/7).jm01-2024 · Answer check 2024 JM01 · StandardFour items are chosen uniformly without replacement from ten, of which three are defective. Find the probability of at least two defectives.jm01-2024 · Reasoning checklist 2024 JM01 · StandardFor the same sample, find the expected number of defective items.jm01-2024 · Reasoning checklist 2024 JM01 · StandardFind tan(α+β).jm01-2024 · Reasoning checklist 2024 JM01 · StandardFind cos(α+2β).jm01-2024 · Reasoning checklist 2024 JM01 · StandardAn arithmetic sequence starts at 3, and a₁,a₂,a₅ form a geometric sequence. Find every possible general term.jm01-2024 · Reasoning checklist 2024 JM01 · StandardFor each sequence in part (a), decide whether Sₙ≥12n+36 holds for a positive integer n, and find the least such n.jm01-2024 · Reasoning checklist 2024 JM01 · StandardSolve f(x)≥0 when a=8.jm01-2024 · Reasoning checklist 2024 JM01 · StandardFor the same f, g(x)=xf(x) has minimum −1 on [−1,1]. Find all a.jm01-2024 · Reasoning checklist 2024 JM01 · StandardA horizontal hyperbola has eccentricity √10/2 and contains (2√2,3). Find its equation.jm01-2024 · Reasoning checklist 2024 JM01 · StandardThe line y=x+m meets this hyperbola at P,Q, and OP⊥OQ. Find m.jm01-2024 · Reasoning checklist 2024 JM02 · StandardA,B,C,D are coplanar as in the official diagram, with B and D on opposite sides of AC. ABC is equilateral; PAD is right isosceles at A; ∠PAC=π/2, ∠PCA=π/6, PA=1 and CD=2. Prove DA⊥PC.jm02-2024 · Reasoning checklist 2024 JM02 · StandardA,B,C,D are coplanar as in the official diagram, with B and D on opposite sides of AC. ABC is equilateral; PAD is right isosceles at A; ∠PAC=π/2, ∠PCA=π/6, PA=1 and CD=2. Find the volume of PBCD.jm02-2024 · Reasoning checklist 2024 JM02 · StandardA,B,C,D are coplanar as in the official diagram, with B and D on opposite sides of AC. ABC is equilateral; PAD is right isosceles at A; ∠PAC=π/2, ∠PCA=π/6, PA=1 and CD=2. Find the cosine of the dihedral angle between APD and CPD.jm02-2024 · Reasoning checklist 2024 JM02 · StandardA right cone has slant height 1 m and base radius x m. Derive V²(x) and its radius domain.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind all local extreme values of V² on 0<x<1.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind the inflection point of V² within its radius domain.jm02-2024 · Reasoning checklist 2024 JM02 · StandardSketch y=V²(x) on the radius domain using the extrema and inflection.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind the maximum possible cone volume.jm02-2024 · Reasoning checklist 2024 JM02 · StandardIn the first quadrant, the region bounded by y=x, y=x²/k and y=x²/(2k) has area 1, with k>0. Find k.jm02-2024 · Reasoning checklist 2024 JM02 · StandardShow the equation satisfied by the abscissae of the two intersections.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind all real slopes m giving two distinct intersections.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind slopes for which OA⊥OB, where A,B are the intersections and O is the origin.jm02-2024 · Reasoning checklist 2024 JM02 · StandardWhen m=√5, find the area of triangle AOB.jm02-2024 · Reasoning checklist 2024 JM02 · StandardExpress the quotient in polar form with principal argument −π<θ≤π.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind z²⁶ in Cartesian form.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFor ω=cosα+i sinα, prove the cosine/sine power formulae and deduce the displayed product identity.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind all real solutions.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFactor the determinant.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind the values of k giving a unique solution.jm02-2024 · Reasoning checklist 2024 JM02 · StandardFind the general solution when k=3.jm02-2024 · Reasoning checklist 2024 JM02 · StandardAdd 2x+2√y+3z=a to the k=3 system. Find the largest possible a and the solution attaining it.jm02-2024 · Reasoning checklist 2023 JM01 · StandardFind the intersection.jm01-2023 · Answer check 2023 JM01 · StandardDivision of f(x) by x²−x−6 leaves remainder 3x−2. Find f(3).jm01-2023 · Answer check 2023 JM01 · StandardEvaluate the logarithm product.jm01-2023 · Answer check 2023 JM01 · StandardFind the complete real solution set.jm01-2023 · Answer check 2023 JM01 · StandardThe stated equation is quadratic and has exactly one real root. Find a.jm01-2023 · Answer check 2023 JM01 · StandardFind the constant term.jm01-2023 · Answer check 2023 JM01 · StandardFind all a for which f is increasing throughout (2,4).jm01-2023 · Answer check 2023 JM01 · StandardSolve f(1/2−3|x|)+f(5)>0.jm01-2023 · Answer check 2023 JM01 · StandardA vertical cylindrical tank has radius 3 m, height 8 m and initial water depth 5 m. A radius-2 m sphere is fully submerged. Find the water rise.jm01-2023 · Answer check 2023 JM01 · StandardAn arithmetic sequence has a₇=80 and a₁₆=26. Find a₃₄.jm01-2023 · Answer check 2023 JM01 · StandardFind the line through the midpoint of A(3,−8), B(−7,4), perpendicular to 3x−4y+14=0.jm01-2023 · Answer check 2023 JM01 · StandardThe hyperbola has eccentricity 3. Minimise (b²+2)/a for a,b>0.jm01-2023 · Answer check 2023 JM01 · StandardA,B lie in quadrant II, with sin A=2/5 and sin B=4/5. Find sin(A+B).jm01-2023 · Answer check 2023 JM01 · StandardThe official graph has minimum 0, maximum 4, and a maximum at x=5π/3. Find a,b.jm01-2023 · Answer check 2023 JM01 · StandardRotate A(−2,3) clockwise by 90° about the origin, reflect in the x-axis, then move down 3 units. Find the final point.jm01-2023 · Answer check 2023 JM01 · StandardIndependent tosses have P(head)=1/4. Find the probability of at most one head in ten tosses.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFor this coin, find the probability that the first head occurs on toss 10.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFind the probability that the third head occurs on toss 10.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFind the focus of x²=4y.jm01-2023 · Reasoning checklist 2023 JM01 · StandardA slope-3/4 line through the focus meets x²=4y at A,B. Find AB.jm01-2023 · Reasoning checklist 2023 JM01 · StandardA slope-1 line meets the parabola at C,D and the y-axis at M, with M between C,D as in the diagram. If DM=3CM, find CD.jm01-2023 · Reasoning checklist 2023 JM01 · StandardSₙ=3ⁿ⁺¹−2k is the partial sum of a geometric sequence. Find k and aₙ.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFind the first n-term sum Tₙ of bₙ.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFor positive integer n, find the n maximising −5cₙ²+cₙ, where cₙ=2/aₙ and aₙ=2·3ⁿ.jm01-2023 · Reasoning checklist 2023 JM01 · StandardDetermine f from its least positive period 3π. The printed question does not state the sign of w.jm01-2023 · Reasoning checklist 2023 JM01 · StandardIn triangle ABC, f(C)=0 and 2sin²B=cosB+cos(A−C), with f from part (a). Find sin A.jm01-2023 · Reasoning checklist 2023 JM01 · StandardDraw the feasible region, including its boundary.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFind the range of z=y/x over the feasible region.jm01-2023 · Reasoning checklist 2023 JM01 · StandardFind the minimum of t=x²+y² over the feasible region.jm01-2023 · Reasoning checklist 2023 JM02 · StandardE–ABCD has rhombus base AB=2√2 and ∠DAB=π/3. EB=EC, ∠BEC=π/2, DE=√5. F is the midpoint of BC; G is the perpendicular foot from E to DF. Show cos∠DFE=√3/4.jm02-2023 · Reasoning checklist 2023 JM02 · StandardE–ABCD has rhombus base AB=2√2 and ∠DAB=π/3. EB=EC, ∠BEC=π/2, DE=√5. F is the midpoint of BC; G is the perpendicular foot from E to DF. Prove EG is perpendicular to plane ABCD.jm02-2023 · Reasoning checklist 2023 JM02 · StandardE–ABCD has rhombus base AB=2√2 and ∠DAB=π/3. EB=EC, ∠BEC=π/2, DE=√5. F is the midpoint of BC; G is the perpendicular foot from E to DF. Find AE.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFind the first two derivatives.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFind and classify the local extrema.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFind and justify the inflection point.jm02-2023 · Reasoning checklist 2023 JM02 · StandardSketch the cubic using the preceding derivative information.jm02-2023 · Reasoning checklist 2023 JM02 · StandardThe line y=x+4 is tangent to y=x³+3x²+x at A. Find A.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFind the area enclosed by y=x+4 and y=x³+3x²+x.jm02-2023 · Reasoning checklist 2023 JM02 · StandardShow that the nonvertical line through A is tangent exactly when 9m²−(k−mh)²+4=0.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFor the two roots m₁,m₂ of the slope quadratic, derive their sum and product.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFind the locus of A when its two tangents to the ellipse are perpendicular, including vertical-tangent cases.jm02-2023 · Reasoning checklist 2023 JM02 · StandardAt A=(5,4), find the acute angle between the tangents, expressed using arctan.jm02-2023 · Reasoning checklist 2023 JM02 · StandardSolve the complex equation.jm02-2023 · Reasoning checklist 2023 JM02 · StandardUse De Moivre to derive the sine, cosine and tangent triple-angle formulae.jm02-2023 · Reasoning checklist 2023 JM02 · StandardSolve the cubic with roots expressed as tangents.jm02-2023 · Reasoning checklist 2023 JM02 · StandardDerive the two difference-to-product identities from the angle addition and subtraction formulae.jm02-2023 · Reasoning checklist 2023 JM02 · StandardProve D=−4sinθ sin2θ sin3θ.jm02-2023 · Reasoning checklist 2023 JM02 · StandardFor 0≤θ≤π/2, the displayed system has more than one solution. Find θ and its full solution.jm02-2023 · Reasoning checklist 2022 JM01 · StandardWhich set is empty? Variables are real.jm01-2022 · Answer check 2022 JM01 · StandardFor a>b>0 and m>0, which inequality always holds?jm01-2022 · Answer check 2022 JM01 · StandardThe polynomial has equal remainders on division by x−2 and x+1. Find a.jm01-2022 · Answer check 2022 JM01 · StandardSimplify the principal square root.jm01-2022 · Answer check 2022 JM01 · StandardFind all p for which the equation has a real solution.jm01-2022 · Answer check 2022 JM01 · StandardFind 1/a+1/b.jm01-2022 · Answer check 2022 JM01 · StandardTwo externally tangent circles of radii 3 and 4 touch the same straight line. A smaller circle occupies the bounded gap and is tangent to both and to the line. Find its radius.jm01-2022 · Answer check 2022 JM01 · StandardEach of three independent games is won with probability 1/4. Find the probability of at least one win.jm01-2022 · Answer check 2022 JM01 · StandardChoose the labelled shaded region. The four original diagrams are described equivalently by their sides of the boundary lines below.jm01-2022 · Answer check 2022 JM01 · StandardA line through (1,2) is perpendicular to 2x−3y=4. Find its y-intercept.jm01-2022 · Answer check 2022 JM01 · StandardThere are 6400 seats in rows of 32. No row may contain more than four consecutive occupied seats. Find the maximum occupancy.jm01-2022 · Answer check 2022 JM01 · StandardThe real roots x₁,x₂ of 3x²−8x+m=0 have reciprocal arithmetic mean 2. Find m.jm01-2022 · Answer check 2022 JM01 · StandardChoose two distinct digits uniformly in order from 1,2,3,4,5 to form a two-digit number. Find the probability it is below 40.jm01-2022 · Answer check 2022 JM01 · StandardSolve the exponential equation.jm01-2022 · Answer check 2022 JM01 · StandardA person walks n km east, turns right 150°, then walks 3 km. The final distance from the start is √3 km. Find n.jm01-2022 · Answer check 2022 JM01 · StandardA quadratic passes through (5,0), has symmetry axis x=2 and minimum −9. Find a,b,c.jm01-2022 · Reasoning checklist 2022 JM01 · StandardTranslate that graph 3 units left, then 3 units up. Find its new function.jm01-2022 · Reasoning checklist 2022 JM01 · StandardFind the maximum and minimum of g(x)=f(3sin x).jm01-2022 · Reasoning checklist 2022 JM01 · StandardA geometric sequence has a₁=1; a₁,4a₂,16a₃ form an arithmetic sequence. Given bₙ=naₙ/4, find both general terms.jm01-2022 · Reasoning checklist 2022 JM01 · StandardFind Sₙ=Σaⱼ and Tₙ=Σbⱼ for the sequences in part (a).jm01-2022 · Reasoning checklist 2022 JM01 · StandardThe vertical ellipse has eccentricity √2/2 and contains (2√2,4). Find its equation.jm01-2022 · Reasoning checklist 2022 JM01 · StandardA secant y=k₁x+d, with k₁d≠0, meets that ellipse at A,B. M is the midpoint, and OM has slope k₂. Prove k₁k₂=−2.jm01-2022 · Reasoning checklist 2022 JM01 · StandardWrite √3cosθ−sinθ as r cos(θ+β), and give its range.jm01-2022 · Reasoning checklist 2022 JM01 · StandardSolve in radians on 0≤θ≤2π.jm01-2022 · Reasoning checklist 2022 JM01 · StandardIf 0≤θ≤π and √3cosθ−sinθ=−1/2, find sin(θ/2+π/12).jm01-2022 · Reasoning checklist 2022 JM01 · StandardProve by mathematical induction that 3⁴ⁿ⁺²+5²ⁿ⁺¹ is divisible by 14 for every positive integer n.jm01-2022 · Reasoning checklist 2022 JM02 · StandardIn pyramid P–ABC, PC⊥plane ABC, PC=3, ∠BCA=π/2. D lies on AB and E on BC; CD=DE=√2, CE=2 and EB=1. Find AC.jm02-2022 · Reasoning checklist 2022 JM02 · StandardIn pyramid P–ABC, PC⊥plane ABC, PC=3, ∠BCA=π/2. D lies on AB and E on BC; CD=DE=√2, CE=2 and EB=1. Prove ∠CDE=π/2.jm02-2022 · Reasoning checklist 2022 JM02 · StandardIn pyramid P–ABC, PC⊥plane ABC, PC=3, ∠BCA=π/2. D lies on AB and E on BC; CD=DE=√2, CE=2 and EB=1. Find ∠CDB using arccos.jm02-2022 · Reasoning checklist 2022 JM02 · StandardIn pyramid P–ABC, PC⊥plane ABC, PC=3, ∠BCA=π/2. D lies on AB and E on BC; CD=DE=√2, CE=2 and EB=1. Find the dihedral angle P–AB–C using arctan.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind all real zeros, using the given f(1)=0.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind the first two derivatives.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind the local maximum and minimum.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind and justify the inflection point.jm02-2022 · Reasoning checklist 2022 JM02 · StandardSketch the cubic using its zeros, extrema and inflection.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind the area between the two parabolas.jm02-2022 · Reasoning checklist 2022 JM02 · StandardA=(−1,0), B=(1,0). Prove the locus condition describes y²=4x, including the converse.jm02-2022 · Reasoning checklist 2022 JM02 · StandardA nonvertical line y=ax+b is tangent to y²=4x. Prove ab=1.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFor m>0, find the non-origin intersection P of y=mx and y²=4x.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind the slope of the tangent at P=(4/m²,4/m).jm02-2022 · Reasoning checklist 2022 JM02 · StandardThe angle between the secant and tangent from (i)–(ii) is arctan(1/4). Find m>0.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFor z=x+yi, derive the Cartesian equation of its locus.jm02-2022 · Reasoning checklist 2022 JM02 · StandardDraw z₁, z₂ and the locus in an Argand diagram.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind the minimum of |z−z₁| on the locus.jm02-2022 · Reasoning checklist 2022 JM02 · StandardShow ω⁷=1 and deduce 1+ω+⋯+ω⁶=0.jm02-2022 · Reasoning checklist 2022 JM02 · StandardProve the conjugate-power identity for positive integers n.jm02-2022 · Reasoning checklist 2022 JM02 · StandardUse the preceding identities to evaluate the cosine-square sum.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFactor the determinant.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind all p for which the system has a unique solution.jm02-2022 · Reasoning checklist 2022 JM02 · StandardFind every p,q giving more than one solution, and give each full solution family.jm02-2022 · Reasoning checklist 2021 JM01 · StandardFind the number of elements in P∩Q.jm01-2021 · Answer check 2021 JM01 · StandardTwo workers take 4 and 3 hours separately for one job. At constant additive rates, how long do they take together for five identical jobs?jm01-2021 · Answer check 2021 JM01 · StandardAn arithmetic sequence has partial sum Sₙ=n². Find its tenth term.jm01-2021 · Answer check 2021 JM01 · StandardFind every m for which the expression is strictly positive for all real x.jm01-2021 · Answer check 2021 JM01 · StandardThe roots of the given polynomial are α,β. Choose an equation whose roots are their reciprocals.jm01-2021 · Answer check 2021 JM01 · StandardTwo equal disks inside a 7 cm by 4 cm rectangle touch its top and bottom sides. Their overlap has area 3 cm². Find the shaded area outside both disks.jm01-2021 · Answer check 2021 JM01 · StandardThe equation has a real solution. Choose the official best answer describing the attainable range of k.jm01-2021 · Answer check 2021 JM01 · StandardFind m so that the polynomial is divisible by 2x+1.jm01-2021 · Answer check 2021 JM01 · StandardFind the tens digit of the integer.jm01-2021 · Answer check 2021 JM01 · StandardSolve for all positive x satisfying both equations.jm01-2021 · Answer check 2021 JM01 · StandardChoose the function matching the displayed graph.jm01-2021 · Answer check 2021 JM01 · StandardChoose the perpendicular bisector of PQ.jm01-2021 · Answer check 2021 JM01 · StandardA dataset has mean 1 and variance 0.01. Find the mean and variance after every value is multiplied by 10.jm01-2021 · Answer check 2021 JM01 · StandardSimplify the radical expression exactly.jm01-2021 · Answer check 2021 JM01 · StandardFind the minimum of ab under the positive-variable condition.jm01-2021 · Answer check 2021 JM01 · StandardNine distinct books consist of four Chinese, two English and three mathematics books. Three are chosen uniformly without replacement. Find the probability of one of each kind.jm01-2021 · Reasoning checklist 2021 JM01 · StandardArrange the same nine distinct books uniformly at random. Find the probability that books of each category occur together in one block.jm01-2021 · Reasoning checklist 2021 JM01 · StandardA circle centred at M(4,4) is tangent to y=2x. Find its equation.jm01-2021 · Reasoning checklist 2021 JM01 · StandardThe second distinct tangent from the origin to the circle is y=mx. Find m.jm01-2021 · Reasoning checklist 2021 JM01 · StandardDerive the completed-square identity from the stated relation.jm01-2021 · Reasoning checklist 2021 JM01 · StandardShow that y reaches its largest feasible value when y=x/m.jm01-2021 · Reasoning checklist 2021 JM01 · StandardFind the x-coordinate, in terms of a and m, where y is maximal.jm01-2021 · Reasoning checklist 2021 JM01 · StandardAn arithmetic sequence satisfies a₂=3 and a₂₀=39. Find its general term.jm01-2021 · Reasoning checklist 2021 JM01 · StandardFor the same arithmetic sequence, Sₙ is the sum of 1/(aₖaₖ₊₁) from k=1 to n. Find n when Sₙ=10/21.jm01-2021 · Reasoning checklist 2021 JM01 · StandardFor triangle ABC with the stated conditions, find sin²C.jm01-2021 · Reasoning checklist 2021 JM01 · StandardUnder the same angle conditions, AC=5. Find the triangle area.jm01-2021 · Reasoning checklist 2021 JM02 · StandardABCD is a trapezoid with ∠DAB=∠ABC=π/2, AD=1 and PA=AB=BC=2; PA is perpendicular to plane ABCD. M is the midpoint of PC. Find the area of triangle PBD.jm02-2021 · Reasoning checklist 2021 JM02 · StandardABCD is a trapezoid with ∠DAB=∠ABC=π/2, AD=1 and PA=AB=BC=2; PA is perpendicular to plane ABCD. M is the midpoint of PC. Find the volume of P–ABD, then the distance from A to plane PBD.jm02-2021 · Reasoning checklist 2021 JM02 · StandardABCD is a trapezoid with ∠DAB=∠ABC=π/2, AD=1 and PA=AB=BC=2; PA is perpendicular to plane ABCD. M is the midpoint of PC. Prove CB is perpendicular to plane PAB.jm02-2021 · Reasoning checklist 2021 JM02 · StandardABCD is a trapezoid with ∠DAB=∠ABC=π/2, AD=1 and PA=AB=BC=2; PA is perpendicular to plane ABCD. M is the midpoint of PC. Prove DM is parallel to plane PAB. Use N, the midpoint of PB, and show ADMN is a rectangle.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFind the first and second derivatives of f.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFind the local maximum and minimum values of f.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFind the inflection point of y=f(x).jm02-2021 · Reasoning checklist 2021 JM02 · StandardSketch y=f(x) for −1≤x≤3.jm02-2021 · Reasoning checklist 2021 JM02 · StandardSketch y=f(|x|)−1 for −1≤x≤3.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFind the total area enclosed by y=−x²+3x and y=2x³−x²−5x.jm02-2021 · Reasoning checklist 2021 JM02 · StandardIf y=mx+c is tangent to x²=8y, prove c=−2m².jm02-2021 · Reasoning checklist 2021 JM02 · StandardTwo distinct tangents to x²=8y have slopes m₁,m₂ and meet at A(h,k). Find A in terms of the slopes.jm02-2021 · Reasoning checklist 2021 JM02 · StandardIf the two tangents in part (b) are perpendicular, find the complete locus of A.jm02-2021 · Reasoning checklist 2021 JM02 · StandardThe angle between the two tangents is π/4 and m₁=2. Find all possible A.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFrom the sine addition formula, prove sin A+sin B=2sin((A+B)/2)cos((A−B)/2).jm02-2021 · Reasoning checklist 2021 JM02 · StandardGiven A+B+C=π, prove sin A+sin B+sin C=4cos(A/2)cos(B/2)cos(C/2).jm02-2021 · Reasoning checklist 2021 JM02 · StandardUse induction to prove the identity for every positive integer n.jm02-2021 · Reasoning checklist 2021 JM02 · StandardUse part (i) to solve cos x+cos3x+cos5x=0 for 0≤x≤2π.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFactorize the displayed determinant.jm02-2021 · Reasoning checklist 2021 JM02 · StandardFor the displayed system with constants k,p,q,r, find all k for which the solution is unique.jm02-2021 · Reasoning checklist 2021 JM02 · StandardSet k=2 and suppose the system has more than one solution. Find the relation among p,q,r; then solve when p=5,q=1,r=−4.jm02-2021 · Reasoning checklist Quadratic equations and functions · FoundationComplete the square. Locate the vertex and find the minimum value of f over all real x.Original teaching material · Worked example Quadratic equations and functions · StandardFind the value of k for which the graph touches the x-axis at exactly one point. Give that point as well.Original teaching material · Worked example Quadratic equations and functions · TransferA rectangular display has perimeter 28 m. Let one side be x metres. Form its area function and find the greatest possible area in square metres.Original teaching material · Worked example Quadratic equations and functions · FoundationWhich point is the vertex of this graph?Original teaching material · Answer check Quadratic equations and functions · StandardHow many distinct real roots does this equation have?Original teaching material · Answer check Quadratic equations and functions · StandardWhich pair gives all the roots?Original teaching material · Answer check Quadratic equations and functions · TransferTranslate this graph 3 units to the left and 4 units down. Which equation describes the new graph?Original teaching material · Answer check Quadratic equations and functions · StandardFind the minimum value over all real x. Enter the minimum value, not the x-coordinate.Original teaching material · Answer check Quadratic equations and functions · TransferLet α and β be the roots. Find the sum of their squares without calculating the individual roots.Original teaching material · Answer check Quadratic equations and functions · FoundationFind the positive root.Original teaching material · Answer check Quadratic equations and functions · FoundationFind the maximum value over real x.Original teaching material · Answer check Quadratic equations and functions · FoundationCalculate the discriminant.Original teaching material · Answer check Quadratic equations and functions · StandardClassify the real solutions for every real m.Original teaching material · Reasoning checklist Trigonometric ratios and function graphs · FoundationConvert 150° to radians. Use its position on the unit circle to find its exact sine.Original teaching material · Worked example Trigonometric ratios and function graphs · StandardFor x in radians, find the amplitude, period and range. Explain what the minus sign does to the graph.Original teaching material · Worked example Trigonometric ratios and function graphs · TransferIn this idealised height model, t is hours after observation begins and h is metres. All trigonometric arguments are in radians. Find the first time the maximum height occurs for 0 ≤ t ≤ 14.Original teaching material · Worked example Trigonometric ratios and function graphs · FoundationThe angle is in radians. What is its exact cosine?Original teaching material · Answer check Trigonometric ratios and function graphs · StandardWith x measured in radians, what is the least positive period?Original teaching material · Answer check Trigonometric ratios and function graphs · StandardUsing the displayed bracketed form and radians, how is this graph obtained from y = 2 sin x?Original teaching material · Answer check Trigonometric ratios and function graphs · TransferA cosine curve has a maximum of 8 at x = 0, a minimum of 2, and successive maxima 4π apart. With x in radians, which equation fits all three features?Original teaching material · Answer check Trigonometric ratios and function graphs · FoundationHere x is a number of degrees, so the whole angle 4x is measured in degrees. Find the least positive period in degrees. Enter only the number.Original teaching material · Answer check Trigonometric ratios and function graphs · TransferLet x be measured in radians. Find the least positive x at which y reaches its maximum. Write x = cπ and enter the number c.Original teaching material · Answer check Trigonometric ratios and function graphs · FoundationWrite 60° as cπ radians. Find c.Original teaching material · Answer check Trigonometric ratios and function graphs · FoundationEvaluate exactly.Original teaching material · Answer check Trigonometric ratios and function graphs · StandardWrite the least positive period as cπ. Find c.Original teaching material · Answer check Trigonometric ratios and function graphs · StandardA student claims sin(x+π)=sin x for all real x. Give a counterexample and state the correct identity.Original teaching material · Reasoning checklist