Pure Mathematics — First: Algebra · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 51 questions on this lesson, each with an explanation.
If Z = 4 e^((−π/3)i), then Z in the algebraic form x + yi is ..........
Correct answer: (C) 2 − 2√3 i
Explanation: Step 1: r = 4, θ = −π/3, so Z = 4(cos(−π/3) + i sin(−π/3)). Step 2: cos(−π/3) = 1/2 and sin(−π/3) = −√3/2. Step 3: Z = 4(1/2 − (√3/2)i) = 2 − 2√3 i. ∴ Z = 2 − 2√3 i (2√3 − 2i swaps cos and sin).
Find the modulus of z = 8 + 6i .
Correct answer: (B) 10
Explanation: Step 1: |z| = √(8² + 6²) Step 2: = √(64 + 36) = √100 ∴ |z| = 10
What is the modulus of the complex number z = 3 + 4i ?
Correct answer: (D) 5
Explanation: Step 1: The modulus |z| = √(a² + b²) with a = 3 , b = 4 Step 2: |z| = √(3² + 4²) = √(9 + 16) = √25 ∴ |z| = 5
What is the principal amplitude of z = √3 + i ?
Correct answer: (C) π/6
Explanation: Step 1: z is in the first quadrant Step 2: tan θ = b/a = 1/√3 Step 3: θ = tan⁻¹(1/√3) = 30° = π/6 ∴ principal amplitude = π/6
What is the modulus of z = cos π/6 + i sin π/6 ?
Correct answer: (D) 1
Explanation: Step 1: this is the trigonometric form r(cos θ + i sin θ) with r = 1 Step 2: for any angle θ , |cos θ + i sin θ| = √(cos²θ + sin²θ) = √1 = 1 ∴ |z| = 1
Write z = -1 + i in trigonometric form.
Correct answer: (D) √2(cos 3π/4 + i sin 3π/4)
Explanation: Step 1: r = √((-1)² + 1²) = √2 Step 2: a = -1 (negative) , b = 1 (positive) so z is in the second quadrant Step 3: the reference angle is tan⁻¹(1/1) = π/4 , so θ = π - π/4 = 3π/4 ∴ z = √2(cos 3π/4 + i sin 3π/4)
If z1 = 8(cos 5π/6 + i sin 5π/6) and z2 = 2(cos π/6 + i sin π/6), find z1 ÷ z2 .
Correct answer: (A) 4(cos 2π/3 + i sin 2π/3)
Explanation: Step 1: divide the moduli and subtract the amplitudes: r = 8 ÷ 2 = 4 , θ = 5π/6 - π/6 = 4π/6 = 2π/3 Step 2: z1 ÷ z2 = 4(cos 2π/3 + i sin 2π/3) ∴ z1 ÷ z2 = 4(cos 2π/3 + i sin 2π/3)
The exponential form of the complex number Z = −√3 − i is ..........
Correct answer: (B) 2 e^((−5π/6)i)
Explanation: Step 1: r = √((−√3)² + (−1)²) = √4 = 2. Step 2: x < 0, y < 0, so Z lies in the third quadrant: θ = −(π − π/6) = −5π/6. Step 3: Z = r e^(θi) = 2 e^((−5π/6)i). ∴ 2 e^((−5π/6)i) (π/6 ignores the quadrant; 5π/6 is the amplitude of −√3 + i; 4 is r² not r).
If Z₁ = 6 e^((5π/6)i) and Z₂ = 3 e^((π/3)i), then Z₁ ÷ Z₂ = ..........
Correct answer: (A) 2i
Explanation: Step 1: Z₁/Z₂ = (r₁/r₂) e^((θ₁ − θ₂)i) = (6/3) e^((5π/6 − π/3)i). Step 2: 5π/6 − π/3 = π/2, so Z₁/Z₂ = 2 e^((π/2)i). Step 3: 2(cos π/2 + i sin π/2) = 2(0 + i) = 2i. ∴ Z₁ ÷ Z₂ = 2i (−2i subtracts the amplitudes the wrong way; −√3 − i adds them).
What is the principal amplitude of the complex number z = −2(cos 20° − i sin 20°)?
Correct answer: (D) 160°
Explanation: Step 1: z = −2 cos 20° + 2i sin 20°, so x = −2 cos 20° is negative and y = 2 sin 20° is positive: z is in the second quadrant. Step 2: −cos 20° = cos(180° − 20°) and sin 20° = sin(180° − 20°). Step 3: z = 2(cos 160° + i sin 160°), with |z| = 2. Step 4: The principal amplitude is 160°.
The first 10 questions in every subject are free, each with an explanation.