Multiple Choice Questions
Advanced Topics in Applied Mathematics
Topic: Complex Numbers
Grade: 12
Question 1:
Which of the following complex numbers is a solution to the equation z^2 + 2z + 1 = 0?
a) 1
b) -1
c) i
d) -i
Answer: b) -1
Explanation: To solve the equation, we can use the quadratic formula. Substituting the values of a, b, and c from the equation, we get z = (-2 ± √(2^2 – 4(1)(1))) / (2(1)). Simplifying further, we have z = (-2 ± √0) / 2. Since the discriminant is zero, there is only one solution, which is -1. For example, if we substitute z = -1 back into the equation, we get (-1)^2 + 2(-1) + 1 = 0, which is true.
Question 2:
What is the conjugate of the complex number 3 + 4i?
a) 3 – 4i
b) -3 + 4i
c) -3 – 4i
d) 3 + 4i
Answer: a) 3 – 4i
Explanation: The conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part. Therefore, the conjugate of 3 + 4i is 3 – 4i. For example, if we multiply the complex number 3 + 4i by its conjugate, we get (3 + 4i)(3 – 4i) = 9 – 12i + 12i – 16i^2 = 9 + 16 = 25.
Question 3:
What is the magnitude of the complex number 2 – 3i?
a) 2
b) 3
c) √13
d) 13
Answer: c) √13
Explanation: The magnitude (or absolute value) of a complex number a + bi is given by |a + bi| = √(a^2 + b^2). Therefore, the magnitude of 2 – 3i is √(2^2 + (-3)^2) = √(4 + 9) = √13. For example, if we calculate the magnitude of 2 – 3i using the formula, we get |2 – 3i| = √(2^2 + (-3)^2) = √(4 + 9) = √13.
Question 4:
What is the principal argument of the complex number -2 + 2i?
a) π/2
b) -Ï€/4
c) π/4
d) -Ï€/2
Answer: c) π/4
Explanation: The principal argument of a complex number a + bi is the angle θ such that tan(θ) = b/a. Therefore, the principal argument of -2 + 2i is tan(θ) = 2/(-2) = -1, which gives θ = π/4. For example, if we plot the complex number -2 + 2i on the complex plane, we can see that it lies in the first quadrant with an angle of π/4 with the positive real axis.
Question 5:
What is the product of the complex numbers (2 + i)(3 – 2i)?
a) 7 + 2i
b) 7 – 4i
c) 6 + 7i
d) 8 – i
Answer: b) 7 – 4i
Explanation: To find the product of two complex numbers, we can use the distributive property and the fact that i^2 = -1. Therefore, (2 + i)(3 – 2i) = 6 – 4i + 3i – 2i^2 = 6 – i – 2(-1) = 6 – i + 2 = 8 – i. For example, if we multiply (2 + i)(3 – 2i) using the distributive property, we get (2)(3) + (2)(-2i) + (i)(3) + (i)(-2i) = 6 – 4i + 3i – 2i^2 = 6 – i – 2(-1) = 6 – i + 2 = 8 – i.