Kenneth ChengMathematical modeling
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Which set is empty? Variables are real.

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TOPIC 01

2022 JM01

Distinguish ∅ from {∅}.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

Which set is empty? Variables are real.

Official paper · jm01-2022 · I.1 · PDF 2

Official original and suggested answers ↗ · Suggested answer PDF page 5

Skills and prerequisite lessons
  1. Option A{0}\{0\}
  2. Option B{x:sin⁡x+cos⁡x=3}\{x:\sin x+\cos x=3\}
  3. Option C{x:x2−1=0}\{x:x^2-1=0\}
  4. Option D{∅}\{\varnothing\}
  5. Option E{(x,y):x2+y2=0}\{(x,y):x^2+y^2=0\}

Working and explanation

BUILD THE REASONING

Hint 1
Bound sin x+cos x.
Hint 2
A set containing the empty set is not itself empty.
Worked solution

  1. The trigonometric sum cannot attain 3.

    ∣sin⁡x+cos⁡x∣=∣2sin⁡(x+π/4)∣≤2<3|\sin x+\cos x|=|\sqrt2\sin(x+\pi/4)|\le\sqrt2<3

  2. The other sets contain 0, ±1, ∅, or (0,0), respectively.

B.

Checks and common pitfalls: Distinguish ∅ from {∅}.

Reasoning checklist · self / teacher assessment
  • Teaching assessment checklist, independently authored. Use the original paper for official marks.
  • The trigonometric sum cannot attain 3.
    ∣sin⁡x+cos⁡x∣=∣2sin⁡(x+π/4)∣≤2<3|\sin x+\cos x|=|\sqrt2\sin(x+\pi/4)|\le\sqrt2<3
  • The other sets contain 0, ±1, ∅, or (0,0), respectively.

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

Focus on one question

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Curriculum and source notes ↗