Multiple Choice Questions
Advanced Topics in Algebra and Geometry
Topic: Complex Numbers
Grade: 12
Question 1:
Which of the following is equivalent to (2 + 3i)(4 – 5i)?
a) -2 – 23i
b) 22 – 7i
c) 23 + 22i
d) 23 + 18i
Answer: b) 22 – 7i
Explanation: To multiply complex numbers, we use the distributive property. First, we multiply the real parts: 2 * 4 = 8. Then, we multiply the imaginary parts: 3i * -5i = -15i^2 = 15. Finally, we combine the real and imaginary parts to get the answer: 8 – 15 = 22 – 7i.
Example 1: (1 + 2i)(3 – 4i) = 3 + 6i – 4i – 8i^2 = 3 + 2i – 8(-1) = 11 + 2i
Example 2: (5 – 2i)(-1 + 3i) = -5 + 15i + 2i – 6i^2 = -5 + 17i + 6 = 1 + 17i
Question 2:
Which of the following is the conjugate of 4 + 3i?
a) 4 – 3i
b) -4 + 3i
c) -4 – 3i
d) 3 + 4i
Answer: a) 4 – 3i
Explanation: The conjugate of a complex number is obtained by changing the sign of its imaginary part. Therefore, the conjugate of 4 + 3i is 4 – 3i.
Example 1: The conjugate of 2 – 7i is 2 + 7i.
Example 2: The conjugate of -6 + 9i is -6 – 9i.
Question 3:
What is the absolute value of -2 + 5i?
a) 7
b) 3
c) 5
d) 2
Answer: c) 5
Explanation: The absolute value of a complex number is the distance between the origin and the point representing the complex number in the complex plane. It can be found using the Pythagorean theorem. The absolute value of -2 + 5i is sqrt((-2)^2 + 5^2) = sqrt(29) ≈ 5.
Example 1: The absolute value of 3 – 4i is sqrt(3^2 + (-4)^2) = 5.
Example 2: The absolute value of -1 + i is sqrt((-1)^2 + 1^2) = sqrt(2).
Question 4:
Which of the following is a solution to the equation z^2 – 6z + 13 = 0?
a) 2 + 3i
b) 3 + 2i
c) -3 + 2i
d) 2 – 3i
Answer: c) -3 + 2i
Explanation: To find the solutions to a quadratic equation with complex coefficients, we can use the quadratic formula. For the equation z^2 – 6z + 13 = 0, the solutions are given by z = (-(-6) ± sqrt((-6)^2 – 4(1)(13))) / (2(1)). Simplifying this expression, we get z = (6 ± sqrt(-20)) / 2. Since the discriminant is negative, the solutions are complex. Therefore, the solution is -3 + 2i.
Example 1: The solutions to the equation z^2 – 4z + 5 = 0 are 2 + i and 2 – i.
Example 2: The solutions to the equation z^2 + 2z + 10 = 0 are -1 + 3i and -1 – 3i.
Question 5:
Which of the following is the polar form of the complex number 2 – 2i?
a) 2√2(cos(π/4) + i sin(π/4))
b) 2√2(cos(π/2) + i sin(π/2))
c) 2√2(cos(π) + i sin(π))
d) 2√2(cos(3π/4) + i sin(3π/4))
Answer: a) 2√2(cos(π/4) + i sin(π/4))
Explanation: The polar form of a complex number is given by r(cos θ + i sin θ), where r is the magnitude and θ is the argument. To find the magnitude, we use the absolute value formula: |2 – 2i| = √((2)^2 + (-2)^2) = √8 = 2√2. To find the argument, we use the inverse tangent function: θ = arctan(-2/2) = arctan(-1) = Ï€/4. Therefore, the polar form is 2√2(cos(Ï€/4) + i sin(Ï€/4)).
Example 1: The polar form of the complex number -3 + 3i is 3√2(cos(π/4) + i sin(π/4)).
Example 2: The polar form of the complex number 1 – i is √2(cos(-Ï€/4) + i sin(-Ï€/4)).