Multiple Choice Questions
Numerical Analysis and Computational Mathematics
Topic: Complex Numbers
Grade: 12
Question 1:
Which of the following complex numbers is a solution to the equation z^2 + 2z + 2 = 0?
a) -1
b) i
c) -i
d) 1
Answer: c) -i
Explanation: To solve the equation, we can use the quadratic formula. Applying the formula, we get z = (-2 ± √(-4))/2. Simplifying further, we have z = -1 ± i. Therefore, the solutions to the equation are -1 + i and -1 – i. Among the answer choices, only -i matches one of the solutions. For example, if we substitute -i into the equation, we get (-i)^2 + 2(-i) + 2 = -1 – 2i + 2 = -1 – 2i + 2 = 0.
Example 1: Solve the equation z^2 – 2z + 2 = 0.
Example 2: Find the two complex solutions to the equation z^2 – 5z + 6 = 0.
Question 2:
What is the square of the complex number (2 + 3i)?
a) -5 + 12i
b) -5 – 12i
c) 13 + 12i
d) 13 – 12i
Answer: d) 13 – 12i
Explanation: To square a complex number, we multiply it by itself. Applying the FOIL method, we have (2 + 3i)(2 + 3i) = 4 + 6i + 6i + 9i^2 = 4 + 12i + 9(-1) = 4 + 12i – 9 = -5 + 12i. Therefore, the square of (2 + 3i) is -5 + 12i.
Example 1: Find the square of the complex number (1 – i).
Example 2: Calculate the square of the complex number (-3 + 4i).
Question 3:
Which of the following complex numbers is a solution to the equation z^3 + 3z^2 + 2z + 8 = 0?
a) 2
b) -2
c) -4
d) 4
Answer: a) 2
Explanation: To solve the equation, we can use either synthetic division or factor theorem. By trying different values, we find that z = 2 is a solution. Dividing the equation by (z – 2), we obtain z^2 + 5z + 4. Factoring the quadratic, we have (z + 1)(z + 4) = 0. Therefore, the solutions to the equation are z = 2, z = -1, and z = -4.
Example 1: Solve the equation z^3 – 2z^2 + z – 2 = 0.
Example 2: Find the solutions to the equation z^3 + 8z^2 + 19z + 12 = 0.
Question 4:
What is the modulus of the complex number 3 – 4i?
a) 7
b) 5
c) 3
d) 4
Answer: b) 5
Explanation: The modulus (or absolute value) of a complex number is calculated as the square root of the sum of the squares of its real and imaginary parts. For the given complex number, the modulus is √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.
Example 1: Find the modulus of the complex number -2i.
Example 2: Calculate the modulus of the complex number 1 + 2i.