Q. If z = re^(iθ), what is the value of z^3?
-
A.
r^3 e^(i3θ)
-
B.
r^3 e^(iθ)
-
C.
3re^(iθ)
-
D.
r^3 e^(iθ^3)
Solution
Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).
Correct Answer: A — r^3 e^(i3θ)
Learn More →
Q. If z = re^(iθ), what is the value of |z|?
Solution
The modulus |z| = r, as |re^(iθ)| = r.
Correct Answer: A — r
Learn More →
Q. If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?
-
A.
x^2 + y^2 = 100
-
B.
x^2 + y^2 = 10
-
C.
x^2 + y^2 = 50
-
D.
x^2 + y^2 = 25
Solution
The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.
Correct Answer: A — x^2 + y^2 = 100
Learn More →
Q. If z = x + yi, find the real part of z^3.
-
A.
x^3 - 3xy^2
-
B.
3x^2y - y^3
-
C.
x^3 + 3xy^2
-
D.
3x^2 - y^3
Solution
Using the binomial expansion, z^3 = (x + yi)^3 = x^3 - 3xy^2 + (3x^2y - y^3)i.
Correct Answer: A — x^3 - 3xy^2
Learn More →
Q. If z1 = 1 + i and z2 = 1 - i, find z1 * z2.
Solution
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
Learn More →
Q. If z1 = 1 + i and z2 = 2 - 3i, find z1 * z2.
-
A.
7 - i
-
B.
7 + i
-
C.
5 - i
-
D.
5 + i
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i + 3 = 5 - i.
Correct Answer: A — 7 - i
Learn More →
Q. If z1 = 1 + i and z2 = 2 - 3i, what is z1 * z2?
-
A.
5 - i
-
B.
8 - i
-
C.
7 + i
-
D.
1 + 5i
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i.
Correct Answer: A — 5 - i
Learn More →
Q. If z1 = 1 + i and z2 = 2 - i, find z1 * z2.
-
A.
3 + i
-
B.
3 - i
-
C.
2 + 3i
-
D.
2 - 3i
Solution
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - 1 = 3 + i.
Correct Answer: A — 3 + i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - 5i, then z1 + z2 is?
-
A.
6 - 2i
-
B.
6 + 2i
-
C.
2 - 2i
-
D.
2 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - 5i) = 6 - 2i.
Correct Answer: A — 6 - 2i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - i, find z1 + z2.
-
A.
6 + 2i
-
B.
6 + 4i
-
C.
2 + 6i
-
D.
8 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - i, what is z1 + z2?
-
A.
6 + 2i
-
B.
6 + 4i
-
C.
2 + 4i
-
D.
2 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
Learn More →
Q. The argument of the complex number z = -1 - i is?
-
A.
-3π/4
-
B.
3π/4
-
C.
π/4
-
D.
-π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.
Correct Answer: A — -3π/4
Learn More →
Q. The product of two complex numbers z1 = 1 + i and z2 = 2 - i is?
-
A.
3 + i
-
B.
3 - i
-
C.
2 + 3i
-
D.
2 - 3i
Solution
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - i^2 = 2 + 1 + i = 3 + i.
Correct Answer: A — 3 + i
Learn More →
Q. The real part of the complex number z = 4 - 3i is?
Solution
The real part of z = 4 - 3i is 4.
Correct Answer: A — 4
Learn More →
Q. The value of (1 + i)^2 is?
-
A.
2i
-
B.
2
-
C.
0
-
D.
1 + 2i
Solution
(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2.
Correct Answer: B — 2
Learn More →
Q. What is the argument of the complex number z = -1 + 0i?
Solution
The argument of z = -1 + 0i is arg(z) = π.
Correct Answer: A — π
Learn More →
Q. What is the argument of the complex number z = -1 - i?
-
A.
-3π/4
-
B.
3π/4
-
C.
-π/4
-
D.
π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.
Correct Answer: A — -3π/4
Learn More →
Q. What is the conjugate of the complex number z = 2 + 5i?
-
A.
2 - 5i
-
B.
2 + 5i
-
C.
-2 + 5i
-
D.
-2 - 5i
Solution
The conjugate of z = 2 + 5i is z̅ = 2 - 5i.
Correct Answer: A — 2 - 5i
Learn More →
Q. What is the conjugate of the complex number z = 5 - 2i?
-
A.
5 + 2i
-
B.
5 - 2i
-
C.
2 - 5i
-
D.
2 + 5i
Solution
The conjugate of z = 5 - 2i is 5 + 2i.
Correct Answer: A — 5 + 2i
Learn More →
Q. What is the conjugate of the complex number z = 5 - 6i?
-
A.
5 + 6i
-
B.
5 - 6i
-
C.
6 - 5i
-
D.
6 + 5i
Solution
The conjugate of z = 5 - 6i is 5 + 6i.
Correct Answer: A — 5 + 6i
Learn More →
Q. What is the modulus of the complex number z = -3 + 4i?
Solution
The modulus |z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
Learn More →
Q. What is the product of (1 + i) and (1 - i)?
Solution
(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
Learn More →
Q. What is the product of the complex numbers (1 + i) and (1 - i)?
Solution
(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
Learn More →
Q. What is the product of the complex numbers z1 = 1 + i and z2 = 1 - i?
Solution
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2.
Correct Answer: A — 2
Learn More →
Q. What is the real part of the complex number 5 - 7i?
Solution
The real part of 5 - 7i is 5.
Correct Answer: A — 5
Learn More →
Q. What is the real part of the complex number z = 2e^(iπ/3)?
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
Learn More →
Q. What is the real part of the complex number z = 4e^(iπ/3)?
Solution
The real part is Re(z) = 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: C — 2√3
Learn More →
Q. What is the real part of the complex number z = 5 - 2i?
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
Learn More →
Q. What is the real part of the complex number z = 5 - 4i?
Solution
The real part of z = 5 - 4i is 5.
Correct Answer: A — 5
Learn More →
Q. What is the real part of the complex number z = 5 - 6i?
Solution
The real part of z = 5 - 6i is 5.
Correct Answer: A — 5
Learn More →
Showing 61 to 90 of 101 (4 Pages)
