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mathematics UK Year 12 Mathematics (AS)

Solved problems for the chapter:

Introduction to Complex Numbers

View the theory »

Complex numbers

1. Complex number

1.1. Imaginary unit

1.2. Imaginary number

1.3. The complex number:

2. Geometric illustration of complex numbers

2.1. Illustration of a complex number in a complex plane

2.2. Illustration of a set of points in the complex plane

3. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:introduction_to_complex_numbers/4511/

Example 1.

Simplify .

Example 2.

Simplify .

Solve the equation .

Calculate:

Simplify .

Expand:

Simplify:

Obtain the real part of the number:

In the set of complex numbers, factorise the expression:

10.

For obtain the real part, and the imaginary part of .

11.

Verify if the value of the expression is a real number.

12.

Show that the number is an imaginary number.

13.

In the set of complex numbers factorise the expression:

14.

In the set of complex numbers factorise the expression:

15.

In the set of complex numbers, factorise the expression:

16.

Write down the imaginary part of the number:

17.

Simplify:

18.

In the set of complex numbers factorise the expression:

Simplify:

Simplify:

Simplify:

For find the value of the expression:

23.

For find the value of the expression:

24.

Simplify:

25.

Find the real numbers x and y for which:

26.

Find the range of real values of x for which the expression below is a real number:

27.

Find the range of values of real numbers x for the expression below to be a complex number:

28.

For which complex number z will be the value of the expression below equal to 0?

29.

In the set of complex numbers, factorize the expression:

30.

In the set of complex numbers factorise the expression:

31.

Simplify:

32.

In the set of complex numbers factorise the expression:

33.

Given that , evaluate .

34.

Given that and , calculate the value of the expression:

35.

Given that and , calculate the value of the expression:

Solve the equation .

Solve the equation .

Solve the equation:

In the set of complex numbers, factorise the expression:

Simplify .

Evaluate:

Simplify .

Find the complex number z for which the expression:

44.

Obtain all real numbers a for which the expression is an imaginary number and find the corresponding imaginary numbers.

Simplify:

Simplify:

Simplify:

Simplify:

In the set of complex numbers factorise the expression:

50.

Evaluate:

51.

Solve the equation:

52.

Obtain the real and imaginary components of the complex number:

53.

Plot the points , , , , , and in the complex plane.

54.

In the complex plane, draw position vectors (directed lines from the origin of the coordinate system to a given point), which represent complex numbers , , , , and .

55.

Obtain the complex numbers from the complex plane represented by the points , , , and .

56.

Simplify the expression:

57.

Obtain the complex number with for which and .

58.

Obtain the complex number for which and .

59.

In the complex plane, draw the set of points for which .

60.

In the complex plane, draw the set of points for which .

61.

In the complex plane, draw the set of points for which .

62.

In the complex plane, draw the set of points for which .

63.

Draw the set of points for which .

64.

In the complex plane, draw the set of points for which is .

65.

In the complex plane, draw the set of points for which .

66.

In the complex plane, draw the set of points for which and .

67.

The graph in the complex plane below shows the set of points A. Describe the set A with inequalities.

68.

Find the complex number of the form , for which holds.

69.

In the complex plane, plot complex numbers in which the imaginary component is opposite to the real component.

70.

The region A is given in the complex plane in the figure below. Obtain the complex numbers z that belong to the given set.

71.

In the complex plane, draw the set of points for which is .

72.

Solve the following complex number problems.

Plot the graph of the complex number in the complex plane.
Compute all real for which the number is a real number.
Given the complex number . Compute all complex numbers for which and .
In the complex plane, draw the set of points corresponding to the condition .

73.

The complex number is given, evaluate .

74.

Obtain the complex number in the form .

75.

In the complex plane, draw the set of points for which . Determine if the number lies in the given set.

76.

Given the following complex numbers and , evaluate:

77.

Given the equation , evaluate .

78.

Solve equation and plot the solution in the complex plane.

79.

Simplify . Leaving your answer in the form .

80.

Locate the complex number in the complex plane.
Find the absolute value of .
Evaluate and .

81.

Plot the number

in the complex plane.

82.

Solve the equation and plot the solution in the complex plane.

83.

Given the following sets in the complex plane:

Draw the set .
Calculate the real numbers a and b such that
Prove that the complex number

is a real number if .
For which real number is ?

84.

In the set of complex numbers, the number is given.

For which complex numbers does and applies?
Given the equation . Calculate the coefficients a, b and c, so that the numbers and are the solutions of this equation.
Draw the set of complex numbers below in the complex plane:

. Calculate the area of the region A. Leave your answer in the exact form.

85.

Draw the sets below in the complex plane:

Select a number from the set A and write it down in the form

The set A:

The set B:

86.

Given that . Calculate the value of the real number x such that .

87.

Find real numbers x and y that satisfies the equation .

88.

Given the statements in the table below, verify the correctness of each of the statements and mark the appropriate truth value (1 or 0). The first one is done as an example.

Declaration	Declaration value (1)	Declaration value (0)
For each ,	1	0 X
The numbers 8 and 15 are co-prime numbers.	1	0
The number is a multiple of 3.	1	0
For all , holds.	1	0
	1	0
For arbitrary numbers holds.	1	0
There are numbers such that .	1	0

89.

The polynomials p and q are Given the polynomials and . Solve the equation in the set of complex numbers. Prove that every solution of the equation has absolute value of 1.

90.

Let and . Calculate and .
Let . Calculate the values of y for which .
In the complex plane, draw the set of all complex numbers for which is an imaginary number.