Q. If z = 4 + 3i, what is the real part of z? (2023)
Solution
The real part of the complex number z = 4 + 3i is 4.
Correct Answer: A — 4
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Q. If z = 4 - 3i, find the conjugate of z. (2023)
-
A.
4 + 3i
-
B.
4 - 3i
-
C.
-4 + 3i
-
D.
-4 - 3i
Solution
The conjugate of z = 4 - 3i is z* = 4 + 3i.
Correct Answer: A — 4 + 3i
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Q. If z = 5 + 12i, find the argument of z. (2020)
-
A.
arctan(12/5)
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B.
arctan(5/12)
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C.
π/2
-
D.
0
Solution
The argument of z = 5 + 12i is θ = arctan(12/5).
Correct Answer: A — arctan(12/5)
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Q. If z = 5 + 12i, what is the argument of z? (2022)
-
A.
arctan(12/5)
-
B.
arctan(5/12)
-
C.
π/2
-
D.
0
Solution
The argument of z = 5 + 12i is given by arctan(12/5).
Correct Answer: A — arctan(12/5)
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Q. If z = 5 + 12i, what is the imaginary part of z? (2022)
Solution
The imaginary part of the complex number z = 5 + 12i is 12.
Correct Answer: B — 12
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Q. If z = a + bi is a complex number such that |z| = 10, what is the equation relating a and b?
-
A.
a^2 + b^2 = 100
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B.
a^2 + b^2 = 10
-
C.
a^2 - b^2 = 100
-
D.
a^2 + b = 10
Solution
The modulus |z| = √(a^2 + b^2) = 10 implies a^2 + b^2 = 100.
Correct Answer: A — a^2 + b^2 = 100
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Q. If z = a + bi, what is the conjugate of z?
-
A.
a - bi
-
B.
a + bi
-
C.
-a + bi
-
D.
-a - bi
Solution
The conjugate of z = a + bi is z̅ = a - bi.
Correct Answer: A — a - bi
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Q. If z = a + bi, where a and b are real numbers, then the conjugate of z is?
-
A.
a + bi
-
B.
a - bi
-
C.
-a + bi
-
D.
-a - bi
Solution
The conjugate of z = a + bi is z̅ = a - bi.
Correct Answer: B — a - bi
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Q. If z = a + bi, where a and b are real numbers, what is the conjugate of z?
-
A.
a - bi
-
B.
a + bi
-
C.
-a + bi
-
D.
-a - bi
Solution
The conjugate of z = a + bi is z̅ = a - bi.
Correct Answer: A — a - bi
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Q. If z = cos(θ) + i sin(θ), what is z^4?
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A.
cos(4θ) + i sin(4θ)
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B.
cos(2θ) + i sin(2θ)
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C.
cos(3θ) + i sin(3θ)
-
D.
cos(θ) + i sin(θ)
Solution
Using De Moivre's theorem, z^4 = (cos(θ) + i sin(θ))^4 = cos(4θ) + i sin(4θ).
Correct Answer: A — cos(4θ) + i sin(4θ)
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Q. If z = e^(iπ/4), find the value of z^8.
Solution
z^8 = (e^(iπ/4))^8 = e^(i2π) = 1.
Correct Answer: A — 1
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Q. If z = re^(iθ), then the value of |z| is?
Solution
The modulus |z| = r in the polar form z = re^(iθ).
Correct Answer: A — r
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Q. If z = re^(iθ), what is the value of r if z = 1 + i?
Solution
r = |z| = √(1^2 + 1^2) = √2.
Correct Answer: A — √2
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Q. If z = re^(iθ), what is the value of r if z = 3 + 4i?
Solution
r = |z| = √(3^2 + 4^2) = √25 = 5.
Correct Answer: A — 5
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Q. If z = re^(iθ), what is the value of r if z = 4 + 3i?
Solution
r = |z| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.
Correct Answer: A — 5
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Q. If z = re^(iθ), what is the value of r if z = 4 + 4i?
Solution
r = |z| = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2.
Correct Answer: A — 4√2
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Q. If z = re^(iθ), what is the value of z^2?
-
A.
r^2e^(i2θ)
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B.
re^(iθ)
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C.
2re^(iθ)
-
D.
r^2e^(iθ)
Solution
Using the property of exponents, z^2 = (re^(iθ))^2 = r^2e^(i2θ).
Correct Answer: A — r^2e^(i2θ)
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Q. If z = re^(iθ), what is the value of z^3?
-
A.
r^3 e^(i3θ)
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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)
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Q. If z = re^(iθ), what is the value of |z|?
Solution
The modulus |z| = r, as |re^(iθ)| = r.
Correct Answer: A — r
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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
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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
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Q. If z = x + yi, find the real part of z^3.
-
A.
x^3 - 3xy^2
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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
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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
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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
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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
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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
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Q. If z1 = 2 + 2i and z2 = 2 - 2i, what is the value of z1 * z2? (2019)
Solution
z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.
Correct Answer: A — 8
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Q. If z1 = 2 + 2i and z2 = 3 - i, find z1 + z2. (2023)
-
A.
5 + i
-
B.
5 + 3i
-
C.
1 + i
-
D.
1 - i
Solution
z1 + z2 = (2 + 2i) + (3 - i) = 5 + i.
Correct Answer: A — 5 + i
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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
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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
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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
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