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
-
B.
0
-
C.
8 + 8i
-
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, 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 + 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)
-
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(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/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 |z|?
Solution
The modulus |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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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
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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?
-
A.
cos(4θ) + i sin(4θ)
-
B.
cos(2θ) + i sin(2θ)
-
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)
-
B.
re^(iθ)
-
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)
-
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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