Complex Numbers Study Guide for Competitive Exams

Complex numbers

Q. If z = re^(iθ), what is the value of z^3?

A. r^3 e^(i3θ)
B. r^3 e^(iθ)
C. 3re^(iθ)
D. r^3 e^(iθ^3)

Solution

Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).

Correct Answer: A — r^3 e^(i3θ)

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Q. If z = re^(iθ), what is the value of |z|?

A. r
B. θ
C. re
D. 1

Solution

The modulus |z| = r, as |re^(iθ)| = r.

Correct Answer: A — r

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Q. If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?

A. x^2 + y^2 = 100
B. x^2 + y^2 = 10
C. x^2 + y^2 = 50
D. x^2 + y^2 = 25

Solution

The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.

Correct Answer: A — x^2 + y^2 = 100

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Q. If z = x + yi, find the real part of z^3.

A. x^3 - 3xy^2
B. 3x^2y - y^3
C. x^3 + 3xy^2
D. 3x^2 - y^3

Solution

Using the binomial expansion, z^3 = (x + yi)^3 = x^3 - 3xy^2 + (3x^2y - y^3)i.

Correct Answer: A — x^3 - 3xy^2

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Q. If z1 = 1 + i and z2 = 1 - i, find z1 * z2.

A. 2
B. 0
C. 1
D. -1

Solution

z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.

Correct Answer: A — 2

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Q. If z1 = 1 + i and z2 = 2 - 3i, find z1 * z2.

A. 7 - i
B. 7 + i
C. 5 - i
D. 5 + i

Solution

z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i + 3 = 5 - i.

Correct Answer: A — 7 - i

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Q. If z1 = 1 + i and z2 = 2 - 3i, what is z1 * z2?

A. 5 - i
B. 8 - i
C. 7 + i
D. 1 + 5i

Solution

z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i.

Correct Answer: A — 5 - i

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Q. If z1 = 1 + i and z2 = 2 - i, find z1 * z2.

A. 3 + i
B. 3 - i
C. 2 + 3i
D. 2 - 3i

Solution

z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - 1 = 3 + i.

Correct Answer: A — 3 + i

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Q. If z1 = 2 + 3i and z2 = 4 - 5i, then z1 + z2 is?

A. 6 - 2i
B. 6 + 2i
C. 2 - 2i
D. 2 + 2i

Solution

z1 + z2 = (2 + 3i) + (4 - 5i) = 6 - 2i.

Correct Answer: A — 6 - 2i

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Q. If z1 = 2 + 3i and z2 = 4 - i, find z1 + z2.

A. 6 + 2i
B. 6 + 4i
C. 2 + 6i
D. 8 + 2i

Solution

z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.

Correct Answer: A — 6 + 2i

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Q. If z1 = 2 + 3i and z2 = 4 - i, what is z1 + z2?

A. 6 + 2i
B. 6 + 4i
C. 2 + 4i
D. 2 + 2i

Solution

z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.

Correct Answer: A — 6 + 2i

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Q. The argument of the complex number z = -1 - i is?

A. -3π/4
B. 3π/4
C. π/4
D. -π/4

Solution

The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.

Correct Answer: A — -3π/4

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Q. The product of two complex numbers z1 = 1 + i and z2 = 2 - i is?

A. 3 + i
B. 3 - i
C. 2 + 3i
D. 2 - 3i

Solution

z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - i^2 = 2 + 1 + i = 3 + i.

Correct Answer: A — 3 + i

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Q. The real part of the complex number z = 4 - 3i is?

A. 4
B. -3
C. 3
D. 0

Solution

The real part of z = 4 - 3i is 4.

Correct Answer: A — 4

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Q. The value of (1 + i)^2 is?

A. 2i
B. 2
C. 0
D. 1 + 2i

Solution

(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2.

Correct Answer: B — 2

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Q. What is the argument of the complex number z = -1 + 0i?

A. π
B. 0
C. π/2
D. -π

Solution

The argument of z = -1 + 0i is arg(z) = π.

Correct Answer: A — π

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Q. What is the argument of the complex number z = -1 - i?

A. -3π/4
B. 3π/4
C. -π/4
D. π/4

Solution

The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.

Correct Answer: A — -3π/4

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Q. What is the conjugate of the complex number z = 2 + 5i?

A. 2 - 5i
B. 2 + 5i
C. -2 + 5i
D. -2 - 5i

Solution

The conjugate of z = 2 + 5i is z̅ = 2 - 5i.

Correct Answer: A — 2 - 5i

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Q. What is the conjugate of the complex number z = 5 - 2i?

A. 5 + 2i
B. 5 - 2i
C. 2 - 5i
D. 2 + 5i

Solution

The conjugate of z = 5 - 2i is 5 + 2i.

Correct Answer: A — 5 + 2i

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Q. What is the conjugate of the complex number z = 5 - 6i?

A. 5 + 6i
B. 5 - 6i
C. 6 - 5i
D. 6 + 5i

Solution

The conjugate of z = 5 - 6i is 5 + 6i.

Correct Answer: A — 5 + 6i

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Q. What is the modulus of the complex number z = -3 + 4i?

A. 5
B. 7
C. 3
D. 4

Solution

The modulus |z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. What is the product of (1 + i) and (1 - i)?

A. 2
B. 0
C. 1
D. 1 + 0i

Solution

(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.

Correct Answer: A — 2

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Q. What is the product of the complex numbers (1 + i) and (1 - i)?

A. 2
B. 0
C. 1
D. 1 + 0i

Solution

(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.

Correct Answer: A — 2

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Q. What is the product of the complex numbers z1 = 1 + i and z2 = 1 - i?

A. 2
B. 0
C. 1
D. 1 + 0i

Solution

z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2.

Correct Answer: A — 2

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Q. What is the real part of the complex number 5 - 7i?

A. 5
B. -7
C. 0
D. 7

Solution

The real part of 5 - 7i is 5.

Correct Answer: A — 5

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Q. What is the real part of the complex number z = 2e^(iπ/3)?

A. 1
B. 2
C. 3
D. 0

Solution

The real part is 2 * cos(π/3) = 2 * 1/2 = 1.

Correct Answer: B — 2

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Q. What is the real part of the complex number z = 4e^(iπ/3)?

A. 2
B. 4
C. 2√3
D. 4√3

Solution

The real part is Re(z) = 4 * cos(π/3) = 4 * 1/2 = 2.

Correct Answer: C — 2√3

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Q. What is the real part of the complex number z = 5 - 2i?

A. 5
B. -2
C. 2
D. 0

Solution

The real part of z = 5 - 2i is 5.

Correct Answer: A — 5

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Q. What is the real part of the complex number z = 5 - 4i?

A. 5
B. -4
C. 4
D. 0

Solution

The real part of z = 5 - 4i is 5.

Correct Answer: A — 5

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Q. What is the real part of the complex number z = 5 - 6i?

A. 5
B. -6
C. 11
D. 1

Solution

The real part of z = 5 - 6i is 5.

Correct Answer: A — 5

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Complex numbers

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