Q. If z = 2 + 2i, find the argument of z.
-
A.
π/4
-
B.
π/2
-
C.
3π/4
-
D.
0
Solution
The argument is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.
Correct Answer: A — π/4
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Q. If z = 2 + 2i, find the conjugate of z.
-
A.
2 - 2i
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B.
2 + 2i
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C.
-2 + 2i
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D.
-2 - 2i
Solution
The conjugate of z = 2 + 2i is z̅ = 2 - 2i.
Correct Answer: A — 2 - 2i
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Q. If z = 2 + 2i, find the modulus of z.
Solution
|z| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Correct Answer: A — 2√2
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Q. If z = 2 + 2i, find the value of z/z*.
Solution
z/z* = (2 + 2i)/(2 - 2i) = (2 + 2i)(2 + 2i)/(4 + 4) = (4 + 8i - 4)/(8) = i.
Correct Answer: C — i
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Q. If z = 2 + 2i, find the value of z^3.
-
A.
-8 + 8i
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B.
0
-
C.
8 + 8i
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D.
8 - 8i
Solution
z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = -8 + 4i.
Correct Answer: A — -8 + 8i
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Q. If z = 2 + 2i, find the value of |z|^2.
Solution
|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.
Correct Answer: B — 8
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Q. If z = 2 + 2i, find z².
-
A.
0
-
B.
8i
-
C.
8
-
D.
4 + 8i
Solution
z² = (2 + 2i)² = 4 + 8i - 4 = 8i.
Correct Answer: C — 8
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Q. If z = 2 + 2i, what is the argument of z? (2023)
-
A.
π/4
-
B.
π/2
-
C.
3π/4
-
D.
0
Solution
The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.
Correct Answer: A — π/4
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Q. If z = 2 + 2i, what is the value of z^2?
Solution
z^2 = (2 + 2i)^2 = 4 + 8i - 4 = 8i.
Correct Answer: C — 8
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Q. If z = 2 + 2i, what is |z|^2? (2021)
Solution
|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.
Correct Answer: A — 8
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Q. If z = 2 + 3i, find the conjugate of z.
-
A.
2 - 3i
-
B.
3 - 2i
-
C.
-2 + 3i
-
D.
-3 - 2i
Solution
The conjugate of z is 2 - 3i.
Correct Answer: A — 2 - 3i
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Q. If z = 2 + 3i, what is the argument of z?
-
A.
arctan(3/2)
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B.
arctan(2/3)
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C.
π/4
-
D.
0
Solution
The argument of z = 2 + 3i is θ = arctan(3/2).
Correct Answer: A — arctan(3/2)
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Q. If z = 2 - 3i, what is the conjugate of z? (2020)
-
A.
2 + 3i
-
B.
2 - 3i
-
C.
-2 + 3i
-
D.
-2 - 3i
Solution
The conjugate of z = 2 - 3i is 2 + 3i.
Correct Answer: A — 2 + 3i
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Q. If z = 2(cos(θ) + i sin(θ)), what is the value of z when θ = π/3?
-
A.
1 + i
-
B.
1 + √3i
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C.
2 + 2i
-
D.
1 + 2i
Solution
z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.
Correct Answer: B — 1 + √3i
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Q. If z = 2(cos(θ) + i sin(θ)), what is the value of |z|?
Solution
|z| = 2, as |z| = r where z = r(cos(θ) + i sin(θ)).
Correct Answer: A — 2
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Q. If z = 2(cos(π/3) + i sin(π/3)), find z in rectangular form.
-
A.
1 + √3i
-
B.
2 + √3i
-
C.
1 + 2i
-
D.
2 + 2i
Solution
z = 2(1/2 + i√3/2) = 1 + √3i.
Correct Answer: A — 1 + √3i
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Q. If z = 2(cos(π/3) + i sin(π/3)), what is z in rectangular form? (2022)
-
A.
1 + √3i
-
B.
2 + √3i
-
C.
1 + 2i
-
D.
2 + 2i
Solution
Using Euler's formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.
Correct Answer: A — 1 + √3i
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Q. If z = 2(cos(π/4) + i sin(π/4)), find the rectangular form of z.
-
A.
√2 + √2i
-
B.
2 + 2i
-
C.
1 + i
-
D.
0 + 0i
Solution
z = 2(cos(π/4) + i sin(π/4)) = 2(√2/2 + i√2/2) = √2 + √2i.
Correct Answer: A — √2 + √2i
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Q. If z = 2(cos(π/4) + i sin(π/4)), find |z|.
Solution
|z| = 2, as |r(cosθ + isinθ)| = r.
Correct Answer: A — 2
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Q. If z = 2e^(iπ/3), find the rectangular form of z.
-
A.
1 + √3i
-
B.
2 + 2i
-
C.
2 + √3i
-
D.
√3 + 1i
Solution
z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.
Correct Answer: A — 1 + √3i
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Q. If z = 2e^(iπ/3), what is the value of z?
-
A.
1 + i√3
-
B.
2 + 0i
-
C.
0 + 2i
-
D.
2 - 2i
Solution
z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3.
Correct Answer: A — 1 + i√3
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Q. If z = 2e^(iπ/4), then z^2 is?
-
A.
4e^(iπ/2)
-
B.
4e^(iπ/4)
-
C.
2e^(iπ/2)
-
D.
2e^(iπ/4)
Solution
z^2 = (2e^(iπ/4))^2 = 4e^(iπ/2).
Correct Answer: A — 4e^(iπ/2)
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Q. If z = 3 + 4i, find |z|.
Solution
The modulus |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. If z = 3 + 4i, then |z| is equal to?
Solution
The modulus |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. If z = 3 + 4i, what is the imaginary part of z? (2022) 2022
Solution
The imaginary part of the complex number z = 3 + 4i is 4.
Correct Answer: B — 4
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Q. If z = 3 + 4i, what is the modulus of z? (2021)
Solution
The modulus of z is given by |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. If z = 3 + 4i, what is the value of z + z* (where z* is the conjugate of z)? (2017)
Solution
z + z* = (3 + 4i) + (3 - 4i) = 6.
Correct Answer: B — 8
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Q. If z = 3 + 4i, what is the value of z + z̅? (2022)
Solution
z + z̅ = (3 + 4i) + (3 - 4i) = 6.
Correct Answer: B — 8
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Q. If z = 3 + 4i, what is the value of |z|^2? (2019)
Solution
|z|^2 = 3^2 + 4^2 = 9 + 16 = 25.
Correct Answer: A — 25
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Q. If z = 3 + 4i, what is |z|?
Solution
The modulus |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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