1. The value of i⁹ is equal to:
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Answer: A. i
Explanation: i⁹ = (i⁴)² × i = (1)² × i = i.
2. The value of i⁻³⁹ is equal to:
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Answer: A. i
Explanation: i⁻³⁹ = 1 / i³⁹ = 1 / (i³⁶ × i³) = 1 / (1 × -i) = 1 / -i = i / (-i²) = i / 1 = i.
3. Express (5i)(-3/5 i) in the form a + ib:
A. 3 + 0i
B. -3 + 0i
C. 0 + 3i
D. 0 - 3i
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Answer: A. 3 + 0i
Explanation: (5i)(-3/5 i) = 5 × (-3/5) × i² = -3 × (-1) = 3 = 3 + 0i.
4. The modulus of the complex number z = 3 + 4i is:
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Answer: A. 5
Explanation: |z| = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5.
5. The conjugate of the complex number z = 4 - 3i is:
A. 4 + 3i
B. -4 - 3i
C. -4 + 3i
D. 3 - 4i
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Answer: A. 4 + 3i
Explanation: The conjugate of z = a + ib is z̄ = a - ib. Thus, conjugate of 4 - 3i is 4 + 3i.
6. The multiplicative inverse of 4 - 3i is:
A. (4/25) + (3/25)i
B. (4/25) - (3/25)i
C. (3/25) + (4/25)i
D. 4 + 3i
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Answer: A. (4/25) + (3/25)i
Explanation: Multiplicative inverse = z̄ / |z|² = (4 + 3i) / (4² + (-3)²) = (4 + 3i) / 25 = (4/25) + (3/25)i.
7. The value of (1 + i)⁴ is:
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Answer: A. -4
Explanation: (1 + i)² = 1 + 2i + i² = 1 + 2i - 1 = 2i. Therefore, (1 + i)⁴ = (2i)² = 4i² = -4.
8. The multiplicative inverse of √5 + 3i is:
A. (√5/14) - (3/14)i
B. (√5/14) + (3/14)i
C. (√5/2) - (3/2)i
D. √5 - 3i
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Answer: A. (√5/14) - (3/14)i
Explanation: Inverse = z̄ / |z|² = (√5 - 3i) / ((√5)² + 3²) = (√5 - 3i) / (5 + 9) = (√5/14) - (3/14)i.
9. The solutions of the quadratic equation x² + 3 = 0 are:
A. ± √3 i
B. ± 3i
C. ± √3
D. ± 9i
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Answer: A. ± √3 i
Explanation: x² = -3 ⇒ x = ± √(-3) = ± √3 i.
10. The solutions of the quadratic equation x² + x + 1 = 0 are:
A. (-1 ± √3 i) / 2
B. (1 ± √3 i) / 2
C. (-1 ± 3i) / 2
D. -1 ± √3 i
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Answer: A. (-1 ± √3 i) / 2
Explanation: Using quadratic formula x = [-b ± √(b² - 4ac)] / 2a = [-1 ± √(1 - 4)] / 2 = (-1 ± √3 i) / 2.
11. The value of 1 + i² + i⁴ + i⁶ + ... + i²ⁿ (when n is odd) is:
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Answer: A. 0
Explanation: Sequence is 1 - 1 + 1 - 1 ... with an even number of terms (n + 1 terms), which sums up to 0.
12. If z = a + ib, then real part Re(z) is given by:
A. (z + z̄) / 2
B. (z - z̄) / 2
C. (z + z̄) / 2i
D. (z - z̄) / 2i
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Answer: A. (z + z̄) / 2
Explanation: z + z̄ = (a + ib) + (a - ib) = 2a ⇒ a = (z + z̄) / 2.
13. If z = a + ib, then imaginary part Im(z) is given by:
A. (z - z̄) / 2i
B. (z + z̄) / 2i
C. (z - z̄) / 2
D. (z + z̄) / 2
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Answer: A. (z - z̄) / 2i
Explanation: z - z̄ = (a + ib) - (a - ib) = 2ib ⇒ b = (z - z̄) / 2i.
14. The real value of x for which (x - 1) + i(y + 2) = 2 + 3i is:
A. x = 3, y = 1
B. x = 1, y = 3
C. x = 2, y = 2
D. x = 3, y = 5
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Answer: A. x = 3, y = 1
Explanation: Equating real and imaginary parts: x - 1 = 2 ⇒ x = 3, and y + 2 = 3 ⇒ y = 1.
15. The value of (1 + i) / (1 - i) is equal to:
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Answer: A. i
Explanation: Multiplying numerator and denominator by (1 + i): (1 + i)² / (1² - i²) = 2i / 2 = i.
16. The value of (1 - i) / (1 + i) is equal to:
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Answer: A. -i
Explanation: Multiplying numerator and denominator by (1 - i): (1 - i)² / (1² - i²) = -2i / 2 = -i.
17. If |z| = 1, then 1 / z is equal to:
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Answer: A. z̄
Explanation: Property: z · z̄ = |z|². Given |z| = 1, z · z̄ = 1 ⇒ 1/z = z̄.
18. The argument of purely real negative complex number is:
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Answer: A. π
Explanation: Purely real negative numbers lie on the negative real axis, making an angle of π with the positive real axis.
19. The argument of purely imaginary positive complex number is:
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Answer: A. π/2
Explanation: Purely positive imaginary numbers lie on the positive imaginary axis, so the principal argument is π/2.
20. If z₁ = 2 + 3i and z₂ = 1 - 2i, then z₁ + z₂ is:
A. 3 + i
B. 3 - i
C. 1 + 5i
D. 3 + 5i
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Answer: A. 3 + i
Explanation: z₁ + z₂ = (2 + 1) + (3 - 2)i = 3 + i.
21. If z₁ = 3 + 5i and z₂ = 2 - 3i, then z₁ - z₂ is:
A. 1 + 8i
B. 1 + 2i
C. 5 + 2i
D. 1 - 8i
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Answer: A. 1 + 8i
Explanation: z₁ - z₂ = (3 - 2) + (5 - (-3))i = 1 + 8i.
22. The product of (2 + 3i) and (2 - 3i) is:
A. 13
B. -5
C. 4 - 9i
D. 13i
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Answer: A. 13
Explanation: (a + ib)(a - ib) = a² + b² = 2² + 3² = 4 + 9 = 13.
23. The value of i + i² + i³ + i⁴ is equal to:
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Answer: A. 0
Explanation: i + (-1) + (-i) + 1 = 0. The sum of four consecutive powers of i is always 0.
24. The solutions of 2x² + x + 1 = 0 are:
A. (-1 ± √7 i) / 4
B. (1 ± √7 i) / 4
C. (-1 ± 7i) / 4
D. (-1 ± √7 i) / 2
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Answer: A. (-1 ± √7 i) / 4
Explanation: x = [-1 ± √(1 - 4(2)(1))] / (2 × 2) = [-1 ± √(-7)] / 4 = (-1 ± √7 i) / 4.
25. The smallest positive integer n for which ((1 + i)/(1 - i))ⁿ = 1 is:
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Answer: A. 4
Explanation: Since (1 + i)/(1 - i) = i, we need iⁿ = 1. The smallest positive integer is n = 4.
26. If z = x + iy lies in the second quadrant, then:
A. x < 0, y > 0
B. x > 0, y > 0
C. x < 0, y < 0
D. x > 0, y < 0
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Answer: A. x < 0, y > 0
Explanation: In Quadrant II, the real axis x is negative (x < 0) and the imaginary axis y is positive (y > 0).
27. Value of √( -25 ) × √( -9 ) is:
A. -15
B. 15
C. 15i
D. -15i
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Answer: A. -15
Explanation: √(-25) = 5i and √(-9) = 3i. Their product is (5i)(3i) = 15i² = -15.
28. Modulus of (1 + i)/(1 - i) is equal to:
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Answer: A. 1
Explanation: |(1 + i)/(1 - i)| = |1 + i| / |1 - i| = √2 / √2 = 1 (or since (1 + i)/(1 - i) = i, |i| = 1).
29. If z = 0, then argument of z is:
A. Undefined
B. 0
C. π
D. π/2
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Answer: A. Undefined
Explanation: The origin (0, 0) in the complex plane does not form a unique angle with the positive real axis, so its argument is undefined.
30. The discriminant of x² + x + 1 = 0 is:
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Answer: A. -3
Explanation: Discriminant D = b² - 4ac = 1² - 4(1)(1) = 1 - 4 = -3.
