2-2 De Moivre's theorem: practice questions

Correct answer: (A) -4

Explanation: Step 1: (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i Step 2: (1 + i)⁴ = [(1 + i)²]² = (2i)² = 4i² = -4 ∴ (1 + i)⁴ = -4

Correct answer: (D) 1/2 - (√3/2)i

Explanation: Step 1: by De Moivre (cos θ + i sin θ)⁻¹ = cos(-θ) + i sin(-θ) = cos θ - i sin θ Step 2: = cos π/3 - i sin π/3 = 1/2 - (√3/2)i ∴ result = 1/2 - (√3/2)i

Correct answer: (A) 3 sin θ − 4 sin³θ

Explanation: Step 1: (cos θ + i sin θ)³ = cos 3θ + i sin 3θ (De Moivre). Step 2: Expanding by the binomial theorem, the imaginary part is 3cos²θ sin θ − sin³θ. Step 3: So sin 3θ = 3(1 − sin²θ) sin θ − sin³θ. Step 4: sin 3θ = 3 sin θ − 4 sin³θ.

Correct answer: (B) -1 + √3 i

Explanation: Step 1: 8 = 8(cos 0 + i sin 0) , cube roots have modulus ∛8 = 2 Step 2: amplitudes are 0 , 2π/3 , 4π/3 , giving 2 , 2(cos 2π/3 + i sin 2π/3) , 2(cos 4π/3 + i sin 4π/3) Step 3: 2(cos 2π/3 + i sin 2π/3) = 2(-1/2 + (√3/2)i) = -1 + √3 i ∴ one cube root is -1 + √3 i

Correct answer: (C) 1/2 + (√3/2)i

Explanation: Step 1: by De Moivre's theorem (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ Step 2: = cos(3 · π/9) + i sin(3 · π/9) = cos π/3 + i sin π/3 Step 3: cos π/3 = 1/2 , sin π/3 = √3/2 ∴ result = 1/2 + (√3/2)i

Correct answer: (A) ±(4 − i)

Explanation: Step 1: Let (x + yi)² = 15 − 8i, so x² − y² = 15 and 2xy = −8. Step 2: Also x² + y² = |15 − 8i| = √(225 + 64) = 17. Step 3: Adding: 2x² = 32, so x = ±4, and xy = −4 gives y = ∓1. Step 4: The square roots are ±(4 − i).

Correct answer: (C) modulus 8 , amplitude 120°

Explanation: Step 1: by De Moivre z³ = 2³(cos(3·40°) + i sin(3·40°)) Step 2: modulus = 2³ = 8 , amplitude = 3 × 40° = 120° ∴ modulus 8 , amplitude 120°

Correct answer: (B) {1 , 2 + i}

Explanation: Step 1: Discriminant: (3 + i)² − 4(2 + i) = 9 + 6i − 1 − 8 − 4i = 2i. Step 2: Square roots of 2i: (a + bi)² = 2i gives a² − b² = 0, 2ab = 2, so √(2i) = ±(1 + i). Step 3: x = ((3 + i) ± (1 + i))/2, so x = 2 + i or x = 1. Step 4: Check: sum = 3 + i and product = 2 + i. S.S. = {1 , 2 + i}.

Correct answer: (D) −2

Explanation: Step 1: For |z| = 1, z + 1/z = 2cos θ = √3 ⇒ θ = ±π/6. Step 2: By De Moivre, z⁶ + 1/z⁶ = 2cos 6θ = 2cos(±π) = −2. ∴ −2

Correct answer: (C) 1 , i , -1 , -i

Explanation: Step 1: 1 = cos 0 + i sin 0 ; the fourth roots have modulus 1 and amplitudes 0 , π/2 , π , 3π/2 Step 2: cos 0 + i sin 0 = 1 ; cos π/2 + i sin π/2 = i ; cos π + i sin π = -1 ; cos 3π/2 + i sin 3π/2 = -i ∴ the fourth roots are 1 , i , -1 , -i