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mathematics UK Year 13 Mathematics (A Levels)

Solved problems for the chapter:

Complex numbers

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Fundamental counting principle for KS4

1. Product rule or basic theorem of combination

2. Addition rule

3. Forum

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Probability of compound events

1. Probability of an impossible event

2. Probability of the opposite event

3. The probability of an event that is the mode of another event

4. Probability of event difference

5. Probability of the sum of events

6. The Maximum of Probability of an Event

7. Forum

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Complex conjugate

1. Properties of the conjugate complex number:

2. Forum

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Division of complex numbers

1. Inverse of a complex number

2. Forum

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Absolute value of a complex number

1. Geometric value of a complex number

2. Absolute value properties

3. Equation of a circle in a complex plane

4. Forum

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Operations with complex numbers

1. Powers of the imaginary unit i:

2. Addition and subtraction of complex numbers:

3. Multiplication of complex numbers:

4. Powers of complex numbers:

5. Addition, subtraction, multiplication and powers of complex numbers:

6. Equality of complex numbers

7. Forum

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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 [7] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:complex_numbers/5509/

Example 1.

In the set of complex numbers factorise the expression:

Example 2.

Write down the complex numbers which are subject to:

Each number in the first column of the spreadsheet is equal to one number in the second column. The terms in the expressions below are labeled with the letters A to K.

In the table there is a space provided for this, enter the letter of the expression that is the same as the expression in the first column of the table (the first line is already filled in correctly).

	D

Solve the following complex number problems.

The complex numbers and are the vertices of the quadrilateral in the complex plane. Calculate the volume of the body produced by rotating this figure by around its side. Calculate the volume of the body produced by rotating this figure by around its diagonal.
Calculate
For each natural number n calculate
We randomly select a natural number . Calculate the probability of the event A being true

Obtain the complex conjugate to the given number below and draw both numbers in the complex plane:

Find the inverse of the complex number .

Simplify and compute its conjugate.

Find the inverse of the complex number .

Calculate the absolute value of the number

10.

For which complex number z does

11.

Write down all complex numbers for which:

12.

For a complex number , show by calculation that .

13.

Calculate the absolute value of the number .

14.

Given that z = 2 - 3, calculate the value of the expression .

15.

For the complex numbers and , calculate the value of the expression:

16.

Calculate if .

17.

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

18.

Solve the equation:

19.

Write all real numbers a for which the complex number is real.

20.

For and . Calculate .

21.

Solve the equation .

22.

Evaluate:

23.

Given the complex number:

Calculate , , and .

24.

Obtain the complex number w, if is and plot its graph.

25.

Calculate .

26.

Simplify .

27.

Given and . The numbers , , and are three of the four vertices of a rectangle. Obtain the fourth vertex of the rectangle and calculate its area.

28.

Find the complex number z for which .

29.

In the complex plane, draw and name the set of all complex numbers z = x + yi, given that:

30.

The complex numbers and are given. Evaluate .

31.

Given the complex number z = (a - i) (- 2 - ai) calculate the value of a such that .

32.

In the set of complex numbers factorise the expression:

33.

Obtain the real and imaginary components of the complex number:

34.

Given:

Evaluate the complex numbers , , . Which of the numbers , and are opposite and which are conjugates?

35.

Obtain the real part of the number:

36.

In the set of complex numbers factorise the expression:

37.

In the set of complex numbers, factorise the expression:

38.

Obtain the complex number with for which and .

39.

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

40.

Obtain the complex number for which and .

41.

In the set of complex numbers factorise the expression:

42.

Show that the number is an imaginary number.

43.

Draw the set of points for which .

44.

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

45.

Write down the imaginary part of the number:

46.

In the set of complex numbers, factorize the expression:

47.

In the set of complex numbers factorise the expression:

48.

In the set of complex numbers factorise the expression:

49.

In the set of complex numbers, factorise the expression:

50.

The complex number is given, evaluate .

51.

In the set of complex numbers factorise the expression:

52.

Obtain the complex number in the form .

53.

Given the following complex numbers and , evaluate:

54.

Simplify . Leaving your answer in the form .

55.

Simplify:

56.

For find the value of the expression:

57.

For find the value of the expression:

58.

In the set of complex numbers, solve the equation . Write down the real and imaginary part of the solution.

59.

Simplify:

60.

Find the real numbers x and y for which:

61.

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

62.

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

63.

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.