Complex Numbers
Q. Find the value of (1 + i)².
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A.
2i
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B.
2
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C.
0
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D.
1 + 2i
Solution
(1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.
Correct Answer: B — 2
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Q. If z = 1 + i, find |z|².
Solution
|z|² = (1)² + (1)² = 1 + 1 = 2.
Correct Answer: B — 2
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Q. If z = 1 + i, what is the value of z^3? (2023)
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A.
-2 + 2i
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B.
2i
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C.
0
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D.
2 + 2i
Solution
z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.
Correct Answer: A — -2 + 2i
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Q. If z = 1 + i√3, find the conjugate of z.
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A.
1 - i√3
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B.
1 + i√3
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C.
1 + √3i
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D.
1 - √3i
Solution
The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.
Correct Answer: A — 1 - i√3
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Q. If z = 1 + i√3, what is the value of z^2? (2023)
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A.
-2 + 2i√3
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B.
4
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C.
1 - 2i√3
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D.
1 + 2i√3
Solution
z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.
Correct Answer: A — -2 + 2i√3
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Q. If z = 2 + 2i, find z².
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A.
0
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B.
8i
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C.
8
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D.
4 + 8i
Solution
z² = (2 + 2i)² = 4 + 8i - 4 = 8i.
Correct Answer: C — 8
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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 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. What is the real part of the complex number z = 4 + 5i? (2022)
Solution
The real part of z = 4 + 5i is 4.
Correct Answer: A — 4
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