Sets, Relations, Functions, Trigonometry & Complex Numbers — Practice

Question 1

In a survey of 100 people, 60 read newspaper A, 40 read newspaper B, and 20 read both. Find

Accepted Answer

A

Question 2

Prove that: (sin 3A − sin A)/(cos A − cos 3A) = cot 2A.

Accepted Answer

Use sum-to-product formulas. Numerator: sin 3A − sin A = 2 cos[(3A+A)/2] · sin[(3A−A)/2] = 2 cos 2A · sin A. Denominator: cos A − cos 3A = −2 sin[(A+3A)/2] · sin[(A−3A)/2] = −2 sin 2A · sin(−A) = −2 sin 2A · (−sin A) = 2 sin 2A · sin A. Therefore: (sin 3A − sin A)/(cos A − cos 3A) = (2 cos 2A · sin A)/(2 sin 2A · sin A) = cos 2A/sin 2A = cot 2A. ∎

Question 3

If 1, ω, ω² are the cube roots of unity, prove that (1 − ω + ω²)(1 + ω − ω²) = 4.

Accepted Answer

We know: 1 + ω + ω² = 0, so ω + ω² = −1, and ω³ = 1. Compute (1 − ω + ω²): use ω² = −1 − ω (from 1 + ω + ω² = 0). So 1 − ω + ω² = 1 − ω + (−1 − ω) = −2ω. Compute (1 + ω − ω²): 1 + ω − ω² = 1 + ω − (−1−ω) = 1 + ω + 1 + ω = 2 + 2ω = 2(1 + ω) = 2(−ω²) = −2ω². Therefore: (1 − ω + ω²)(1 + ω − ω²) = (−2ω)(−2ω²) = 4ω³ = 4 × 1 = 4. ∎

Practice — Sets, Relations, Functions, Trigonometry & Complex Numbers

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