Complex Numbers and Functions: Interactive Learning Quizzes

Question 1

Determine the complex conjugate of the number (3 + 4i).

Accepted Answer

(3 - 4i)

Answer

(4 + 3i)

Answer

(4 - 3i)

Answer

(3 - 4i)^2

Question 2

Which of the following operations will produce a complex number as the result?

Accepted Answer

Multiplying complex numbers

Answer

Adding real numbers

Answer

Subtracting a real number from a complex number

Answer

Dividing a complex number by a real number

Question 3

Identify the real part of the complex number (2 - 3i).

Accepted Answer

2

Answer

3

Answer

-2

Answer

-3

Question 4

Which of the following is not a valid way to express a complex number?

Accepted Answer

(2, 3)

Answer

4 + 5i

Answer

2 - 3i

Answer

4e^(iπ/3)

Question 5

Determine the imaginary part of the complex number (-2 + 5i).

Accepted Answer

5

Answer

2

Answer

-2

Answer

-5

Question 6

Which of the following is a contour integral?

Accepted Answer

∫_{C} f(z) dz, where C is a closed path

Answer

∫_{a}^{b} f(x) dx, where a and b are real numbers

Answer

∫_{0}^{2π} f(e^(iθ)) dθ

Answer

∫_{0}^{∞} f(x) dx

Question 7

Which operation is not defined for complex numbers?

Accepted Answer

Logarithm

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 8

Which of the following is Euler's formula?

Accepted Answer

e^(ix) = cos(x) + i sin(x)

Answer

e^(ix) = sin(x) + i cos(x)

Answer

e^(x) = cos(x) + i sin(x)

Answer

e^(x) = sin(x) + i cos(x)

Question 9

Which transformation is a conformal mapping?

Accepted Answer

w = e^z

Answer

w = z^2

Answer

w = z + i

Answer

w = |z|

Question 10

Find the complex conjugate of (3 + 4i).

Accepted Answer

(3 - 4i)

Answer

(4 + 3i)

Answer

(4 - 3i)

Question 11

Express the complex number (1 + √3i) in polar form.

Accepted Answer

(2, π/3)

Answer

(2, 2π/3)

Answer

(2, -π/3)

Answer

(2, -2π/3)

Question 12

Determine the order of the pole of the function f(z) = 1/(z - 2)^3 at z = 2.

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 13

Find the derivative of the complex function $f(z) = z^2 + 2z - 1$ with respect to $z$.

Accepted Answer

$2z + 2$

Answer

$z^2 + 2$

Answer

$2z - 1$

Answer

$1$

Question 14

Which of the following is NOT a property of complex numbers?

Accepted Answer

Commutative property of division

Answer

Closure under addition

Answer

Closure under multiplication

Answer

Distributive property

Question 15

Convert the complex number z = 5 + 5i to polar form.

Accepted Answer

5(cos 45° + i sin 45°)

Answer

5(cos 45° - i sin 45°)

Answer

5(sin 45° + i cos 45°)

Answer

5(sin 45° - i cos 45°)

Question 16

Identify the function that is holomorphic.

Accepted Answer

f(z) = z^2

Answer

f(z) = 1/z

Answer

f(z) = |z|

Answer

f(z) = Re(z)

Question 17

Determine the Taylor series expansion of f(z) = e^z about the point z = 0.

Accepted Answer

1 + z + z^2/2! + z^3/3! + ...

Answer

z + z^2/2! + z^3/3! + ...

Answer

1 + z - z^2/2! + z^3/3! + ...

Answer

z - z^2/2! + z^3/3! + ...

Question 18

Find the complex conjugate of (3 + 4i).

Accepted Answer

(3 - 4i)

Answer

(3 + 4i)

Answer

(4 + 3i)

Answer

(4 - 3i)

Question 19

Find the derivative of the complex function f(z) = z^2 - 3z + 2.

Accepted Answer

2z - 3

Answer

z^2 - 3

Answer

2z + 3

Answer

z^2 + 3

Question 20

Which complex function has a singularity at z = 0?

Accepted Answer

f(z) = 1/z

Answer

f(z) = z^2

Answer

f(z) = e^z

Answer

f(z) = sin(z)

Question 21

State the Cauchy-Riemann condition for complex differentiability.

Accepted Answer

∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x

Answer

∂u/∂x = ∂v/∂y and ∂u/∂y = ∂v/∂x

Answer

∂u/∂x = -∂v/∂y and ∂u/∂y = ∂v/∂x

Answer

∂u/∂x = -∂v/∂y and ∂u/∂y = -∂v/∂x

Question 22

Find the complex conjugate of z = 3 + 4i.

Accepted Answer

z* = 3 - 4i

Answer

z* = -3 + 4i

Answer

z* = 3 + 4i

Answer

z* = -3 - 4i

Question 23

Simplify the complex expression: (3 + 4i)(2 - 5i).

Accepted Answer

22 - 2i

Answer

2 - 22i

Answer

22 + 2i

Answer

2 - 2i

Question 24

Find the value of i^4.

Accepted Answer

1

Answer

2i

Answer

i

Answer

-1

Question 25

Determine the magnitude of the complex number 4 - 3i.

Accepted Answer

5

Answer

7

Answer

1

Question 26

Find the complex conjugate of the complex number z = 2 + 3i.

Accepted Answer

2 - 3i

Answer

-2 + 3i

Answer

3 + 2i

Answer

-2 - 3i

Question 27

Convert the complex number -2 + 2i to its polar form.

Accepted Answer

2√2 * cis(135°)

Answer

√2 * cis(135°)

Answer

√2 * cis(45°)

Answer

2√2 * cis(45°)

Question 28

A voltage source in an electrical circuit is given by the complex number 10∠30°. Determine the magnitude of the voltage.

Accepted Answer

10 volts

Answer

5 volts

Answer

√3 volts

Answer

10√3 volts

Question 29

In a circuit, the impedance is represented by the complex number Z = 5 + 10i ohms. Calculate the reactance of the circuit.

Accepted Answer

10 ohms

Answer

15 ohms

Answer

5 ohms

Question 30

Which of the following is the complex conjugate of (3 + 4i)?

Accepted Answer

3 - 4i

Answer

-3 - 4i

Answer

-3 + 4i

Answer

3 + 4i

Question 31

Which of the following is the polar form of the complex number 3 + 4i?

Accepted Answer

5(cos(53.13°) + i sin(53.13°))

Answer

5(cos(36.87°) + i sin(36.87°))

Answer

3(cos(36.87°) + i sin(36.87°))

Answer

3(cos(53.13°) + i sin(53.13°))

Question 32

Find the complex roots of the equation z² - 4z + 5 = 0.

Accepted Answer

2 ± i

Answer

-2 ± i

Answer

-2 ± 2i

Answer

2 ± 2i

Question 33

Which of the following is the derivative of the complex function f(z) = z² + 2z - 1?

Accepted Answer

2z + 2

Answer

z² + 2

Answer

1

Answer

2z - 1

Question 34

Which of the following is the Taylor series expansion of the complex function f(z) = cos(z) around z = 0?

Accepted Answer

1 - z²/2! + z^4/4! - z^6/6! + ...

Answer

z - z³/3! + z^5/5! - z^7/7! + ...

Answer

cos(1) - z sin(1)/1! + z^2 cos(1)/2! - z^3 sin(1)/3! + ...

Answer

e^z - z²/2! + z^4/4! - z^6/6! + ...

Question 35

Which of the following is the conformal mapping of the complex plane w = z²?

Accepted Answer

A circle maps to an ellipse.

Answer

A parabola maps to a circle.

Answer

An ellipse maps to a straight line.

Answer

A straight line maps to a parabola.

Question 36

Which of the following is the standard form of the complex number (3 + 4i) - (5 - 2i)?

Accepted Answer

-2 + 6i

Answer

-8 - 2i

Answer

8 + 2i

Answer

2 - 6i

Question 37

What is the conjugate of the complex number z = 3 - 4i?

Accepted Answer

3 + 4i

Answer

-3 - 4i

Answer

4 + 3i

Answer

-3 + 4i

Question 38

Find the magnitude of the complex number z = -2 + 3i.

Accepted Answer

√(13)

Answer

-√(13)

Answer

13

Answer

2 + 3i

Question 39

Which of the following is the complex exponential form of the complex number z = 5(cos(π/6) + i sin(π/6))?

Accepted Answer

5e^(iπ/6)

Answer

5e^(-iπ/6)

Answer

-5e^(-iπ/6)

Answer

-5e^(iπ/6)

Question 40

Find the value of i^(12).

Accepted Answer

1

Answer

-i

Answer

i

Answer

-1

Question 41

Which of the following is a polynomial function of degree 2 with complex coefficients?

Accepted Answer

z^2 - 3z + 2

Answer

sin(z) + cos(z)

Answer

e^z + z^2

Answer

z^3 - 2z^2 + z - 1

Question 42

Find the zeros of the complex function f(z) = z^2 + 2z + 5.

Accepted Answer

-1 ± 2i

Answer

1 ± 2i

Answer

1 ± 2

Answer

-1 ± 2

Question 43

Which of the following is not a property of complex functions?

Accepted Answer

They are continuous everywhere.

Answer

They have a unique derivative at each point where they are differentiable.

Answer

They are differentiable everywhere except at isolated points.

Answer

They satisfy the Cauchy-Riemann equations.

Question 44

Find the Taylor series expansion of the complex function f(z) = e^z around z = 0.

Accepted Answer

∑(n=0 to ∞) (z^n / n!)

Answer

∑(n=1 to ∞) (z^(n-1) / n!)

Answer

∑(n=1 to ∞) (z^n / n)

Answer

∑(n=0 to ∞) (z^(n+1) / (n+1)!)

Question 45

Which of the following contour integrals is equal to 2πi?

Accepted Answer

∮(z-i) dz around the unit circle centered at the origin.

Answer

∮(z^2+1) dz around the square with vertices at ±1 ± i.

Answer

∮e^z dz around the circle |z| = 2.