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Complex numbers: mixed assessment

Read the idea, work independently, then explain what changed.

TOPIC 01

Complex numbers: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find the imaginary part.

z=7-9i

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

The imaginary part is the real coefficient of i.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$\operatorname{Im}z=-9$
The imaginary part is not the whole term bi.

The requested value is -9.

Checks and common pitfalls: The imaginary part is not the whole term bi.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Simplify the quotient.

\frac{7+7i}{1+i}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

Factor the numerator.

Worked solution

Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$\frac{7(1+i)}{1+i}=7$
The factor 1+i is nonzero, so cancellation is legal.

Checks and common pitfalls: The factor 1+i is nonzero, so cancellation is legal.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

Find the real part using de Moivre’s formula.

[7(\cos(\pi/3)+i\sin(\pi/3))]^3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use modulus and argument, remembering that arguments differ by full turns.

Hint 2

Cube the modulus and triple the angle.

Worked solution

Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$343(\cos\pi+i\sin\pi)=-343$
The angle multiplication must accompany the modulus power.

The requested value is -343.

Checks and common pitfalls: The angle multiplication must accompany the modulus power.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Write the conjugate.

z=8-3i

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

Reflect the complex-plane point across the real axis.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$\overline z=8+3i$
Only the sign of the imaginary term changes.

8+3i.

Checks and common pitfalls: Only the sign of the imaginary term changes.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Solve the quadratic over the complex numbers.

z^2-16z+65=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

Complete the square.

Worked solution

Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$(z-8)^2=-1\Rightarrow z=8±i$
Substituting either root gives zero in the original polynomial.

z=8±i.

Checks and common pitfalls: Substituting either root gives zero in the original polynomial.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Solve 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.

\begin{gathered}\text{B: }i^{42}\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use modulus and argument, remembering that arguments differ by full turns.

Hint 2

B: Expand with i²=−1 and use a conjugate to remove a complex denominator.

Worked solution

Part A reasoning
Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$r(\cos(\theta+2\pi)+i\sin(\theta+2\pi))=r(\cos\theta+i\sin\theta)$
A representation is not unique even though the complex number is fixed.
Part B reasoning
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$i^{42}=(i^4)^{10}i^2=-1$
Reduce the exponent modulo four.

A: No; sine and cosine are unchanged by one full turn. B: The requested value is -1.

Checks and common pitfalls: A representation is not unique even though the complex number is fixed. Reduce the exponent modulo four.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

Find the sum a+b when the complex number is zero.

(a-9)+(b+2)i=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

Both real and imaginary parts must vanish.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$a=9,\quad b=-2\Rightarrow a+b=7$
One zero part alone does not make the complex number zero.

The requested value is 7.

Checks and common pitfalls: One zero part alone does not make the complex number zero.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find the real part of the product.

(9+i)(2-i)

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

The product i(−i) equals +1.

Worked solution

Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$(9+i)(2-i)=19+-7i$
Separate the real terms from the coefficient of i.

The requested value is 19.

Checks and common pitfalls: Separate the real terms from the coefficient of i.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the modulus of the product.

|z_1|=9,\quad|z_2|=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use modulus and argument, remembering that arguments differ by full turns.

Hint 2

Multiplication multiplies moduli.

Worked solution

Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$|z_1z_2|=|z_1||z_2|=27$
Arguments add, but lengths multiply.

The requested value is 27.

Checks and common pitfalls: Arguments add, but lengths multiply.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Find the modulus.

z=30+40i

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

Interpret z as a point in the complex plane.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$|z|=\sqrt{(30)^2+(40)^2}=50$
Modulus is the nonnegative distance from the origin.

The requested value is 50.

Checks and common pitfalls: Modulus is the nonnegative distance from the origin.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Evaluate the power.

i^{42}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

Powers of i repeat every four exponents.

Worked solution

Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$i^{42}=(i^4)^{10}i^2=-1$
Reduce the exponent modulo four.

The requested value is -1.

Checks and common pitfalls: Reduce the exponent modulo four.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

Find the modulus of the quotient.

|z_1|=30,\quad|z_2|=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use modulus and argument, remembering that arguments differ by full turns.

Hint 2

The divisor must be nonzero.

Worked solution

Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$|z_1/z_2|=|z_1|/|z_2|=10$
Arguments subtract under division.

The requested value is 10.

Checks and common pitfalls: Arguments subtract under division.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Solve over the complex numbers.

z^2+121=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

Use i²=−1.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$z^2=-121;\quad (±11i)^2=-121$
The roots 11i and −11i are distinct because 11>0.

z=±11i.

Checks and common pitfalls: The roots 11i and −11i are distinct because 11>0.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

Solve both parts and justify the conditions used. Part A: Simplify the sum and explain the cancellation. Part B: Write the conjugate.

\begin{gathered}\text{A: }1+i+i^2+i^3\\\text{B: }z=8-3i\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Expand with i²=−1 and use a conjugate to remove a complex denominator.

Hint 2

B: Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Worked solution

Part A reasoning
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Calculate or simplify this relation.
$1+i-1-i=0$
The four points cancel in opposite pairs on the unit circle.
Part B reasoning
Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$\overline z=8+3i$
Only the sign of the imaginary term changes.

A: The requested value is 0. B: 8+3i.

Checks and common pitfalls: The four points cancel in opposite pairs on the unit circle. Only the sign of the imaginary term changes.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

Give a polar representation.

z=11i

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use modulus and argument, remembering that arguments differ by full turns.

Hint 2

The point lies on the positive imaginary axis.

Worked solution

Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$|z|=11,\quad\arg z=\pi/2+2n\pi$
A chosen argument is one representative from infinitely many.

11(cos(π/2)+i sin(π/2)).

Checks and common pitfalls: A chosen argument is one representative from infinitely many.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

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Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.