FREE Pre-Calculus Complex Numbers Questions and Answers

Question 1

Evaluate (4 - 2i)*(1 - 5i)

Accepted Answer

-6 - 22i

Answer

To multiply complex numbers, use the distributive property (FOIL method) similar to multiplying binomials. Multiply (4 - 2i) by (1 - 5i), remembering that i² = -1. This yields 4 - 20i - 2i + 10i², which simplifies to 4 - 22i - 10, resulting in -6 - 22i.

Question 2

What is the value of the expression (x + yi)(x - yi) if (x + yi) / i = 7 + 9i, where x and y are real numbers?

Accepted Answer

130

Answer

First, solve the given equation (x + yi) / i = 7 + 9i for (x + yi) by multiplying both sides by i. This gives x + yi = i(7 + 9i) = 7i + 9i² = -9 + 7i. The expression to find, (x + yi)(x - yi), is the product of a complex number and its conjugate, which simplifies to x² + y². Substituting x = -9 and y = 7, we get (-9)² + (7)² = 81 + 49 = 130.

Question 3

Find all complex numbers z that satisfy the equation z + 3z' = 5 - 6i, where z' is the complex conjugate of z.

Accepted Answer

z = 5/4 + 3i

Answer

Let the complex number z be represented as a + bi, where z' (its conjugate) is a - bi. Substitute these into the given equation: (a + bi) + 3(a - bi) = 5 - 6i. Simplify the left side to get 4a - 2bi = 5 - 6i. By equating the real and imaginary parts on both sides, we find 4a = 5 (so a = 5/4) and -2b = -6 (so b = 3). Therefore, z = 5/4 + 3i.

Question 4

Define Complex number.

Accepted Answer

A number of the form a + bi, where a and b are real numbers.

Answer

A complex number is formally defined as a number that can be written in the standard form a + bi, where 'a' and 'b' are real numbers. In this expression, 'a' is the real part and 'b' is the imaginary part, with 'i' representing the imaginary unit (√-1).

Question 5

Explain the real and imaginary components of a complex number.

Accepted Answer

The real part is the constant term, and the imaginary part is the coefficient of i.

Answer

In a complex number expressed in the standard form a + bi, the 'a' term is known as the real part, representing the constant numerical value. The 'b' term, which is the coefficient multiplied by the imaginary unit 'i', is defined as the imaginary part of the complex number.

PRE CALCULUS Practice Test

PRE CALCULUS Practice Test

FREE Pre-Calculus Complex Numbers Questions and Answers