Q. Find the argument of the complex number z = -1 - i.
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A.
-3π/4
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B.
3π/4
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C.
π/4
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D.
-π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.
Correct Answer: A — -3π/4
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Q. Find the conjugate of the complex number z = 5 - 6i.
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A.
5 + 6i
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B.
5 - 6i
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C.
-5 + 6i
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D.
-5 - 6i
Solution
The conjugate of z = 5 - 6i is z̅ = 5 + 6i.
Correct Answer: A — 5 + 6i
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Q. Find the real part of the complex number z = 2 + 3i.
Solution
The real part of z = 2 + 3i is 2.
Correct Answer: A — 2
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Q. Find 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
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Q. Find the real part of the complex number z = 3 + 4i.
Solution
The real part of z is 3.
Correct Answer: A — 3
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Q. Find the real part of the complex number z = 4 + 3i.
Solution
The real part of z = 4 + 3i is 4.
Correct Answer: A — 4
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Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).
Solution
The real part is 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: A — 2
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Q. Find the real part of the complex number z = 5 - 2i.
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
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Q. Find the value of (1 + i)^2.
Solution
(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: B — 2
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Q. Find the value of (1 + i)^4.
Solution
(1 + i)^4 = (√2 e^(iπ/4))^4 = 4 e^(iπ) = 4(-1) = -4.
Correct Answer: C — 8
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Q. Find the value of i^4.
Solution
i^4 = (i^2)^2 = (-1)^2 = 1.
Correct Answer: A — 1
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Q. Find the value of z if z^2 + 4z + 8 = 0.
-
A.
-2 + 2i
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B.
-2 - 2i
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C.
-4 + 0i
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D.
-4 - 0i
Solution
Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.
Correct Answer: A — -2 + 2i
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Q. Find the value of z if z^2 = -16.
Solution
Taking square root, z = ±√(-16) = ±4i.
Correct Answer: A — 4i
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Q. Find the value of z^2 if z = 1 + i.
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: B — 2
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Q. If z = 1 + i, find the conjugate of z.
-
A.
1 - i
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B.
1 + i
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C.
-1 + i
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D.
-1 - i
Solution
The conjugate of z = 1 + i is z̅ = 1 - i.
Correct Answer: A — 1 - i
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Q. If z = 1 + i, find the value of z^3.
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A.
-2 + 2i
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B.
2 + 2i
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C.
0
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D.
1 + 3i
Solution
z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.
Correct Answer: A — -2 + 2i
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Q. If z = 1 + i, find the value of z^4.
Solution
z^4 = (1 + i)^4 = (2e^(iπ/4))^4 = 16e^(iπ) = -16.
Correct Answer: A — -4
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Q. If z = 1 + i, find z^2.
-
A.
2i
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B.
2
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C.
1 + 2i
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D.
0
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: C — 1 + 2i
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Q. If z = 1 + i, find z^3.
-
A.
-2 + 2i
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B.
2 + 2i
-
C.
0
-
D.
1 + i
Solution
z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.
Correct Answer: A — -2 + 2i
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Q. If z = 1 + i, find z^4.
Solution
z^4 = (1 + i)^4 = 4i.
Correct Answer: A — -4
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Q. If z = 1 + i, what is z^2?
-
A.
2i
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B.
2
-
C.
1 + 2i
-
D.
0
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: C — 1 + 2i
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Q. If z = 1 + i√3, find the argument of z.
-
A.
π/3
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B.
2π/3
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C.
π/6
-
D.
5π/6
Solution
The argument θ = tan^(-1)(√3/1) = π/3.
Correct Answer: A — π/3
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Q. If z = 1 + i√3, find the modulus of z.
Solution
|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.
Correct Answer: A — 2
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Q. If z = 1 + i√3, find the value of |z|^2.
Solution
|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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Q. If z = 1 + i√3, find |z|.
Solution
|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.
Correct Answer: A — 2
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Q. If z = 1 + i√3, find |z|^2.
Solution
|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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Q. If z = 1 + i√3, then the argument of z is?
-
A.
π/3
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B.
π/6
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C.
2π/3
-
D.
5π/6
Solution
The argument θ = tan^(-1)(√3/1) = π/3.
Correct Answer: A — π/3
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Q. If z = 1 + i√3, what is |z|^2?
Solution
|z|^2 = 1^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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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
-
D.
-2 - 2i
Solution
The conjugate of z = 2 + 2i is z̅ = 2 - 2i.
Correct Answer: A — 2 - 2i
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