Q. If z = 1 + i, what is the argument of z?
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Solution
The argument of a complex number z = a + bi is given by θ = tan⁻¹(b/a). Here, θ = tan⁻¹(1/1) = π/4.
Correct Answer:
A
— π/4
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Q. If z = 1 + i, what is the value of z^3? (2023)
A.
-2 + 2i
B.
2i
C.
0
D.
2 + 2i
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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.
A.
1 - i√3
B.
1 + i√3
C.
1 + √3i
D.
1 - √3i
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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)
A.
-2 + 2i√3
B.
4
C.
1 - 2i√3
D.
1 + 2i√3
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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 = 1 + √3i, what is the square of the modulus of z?
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Solution
The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 4.
Correct Answer:
A
— 4
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Q. If z = 2 + 2i, find z².
A.
0
B.
8i
C.
8
D.
4 + 8i
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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
Show solution
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^3? (2022)
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Solution
z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = 8.
Correct Answer:
C
— 8
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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
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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 = 3 + 4i, what is the modulus of z? (2021)
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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 = 4 - 3i, find the conjugate of z. (2023)
A.
4 + 3i
B.
4 - 3i
C.
-4 + 3i
D.
-4 - 3i
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Solution
The conjugate of z = 4 - 3i is z* = 4 + 3i.
Correct Answer:
A
— 4 + 3i
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Q. If z = 4e^(iπ/3), find the rectangular form of z.
A.
2 + 2√3i
B.
4 + 0i
C.
0 + 4i
D.
4 + 4i
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Solution
z = 4(cos(π/3) + i sin(π/3)) = 4(1/2 + i√3/2) = 2 + 2√3i.
Correct Answer:
A
— 2 + 2√3i
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Q. If z = 5 + 12i, find the argument of z. (2020)
A.
arctan(12/5)
B.
arctan(5/12)
C.
π/2
D.
0
Show solution
Solution
The argument of z = 5 + 12i is θ = arctan(12/5).
Correct Answer:
A
— arctan(12/5)
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Q. If z = 5(cos(θ) + i sin(θ)), what is the value of z when θ = π/3? (2023)
A.
5 + 5i
B.
5 + 2.5i
C.
2.5 + 5i
D.
2.5 + 2.5i
Show solution
Solution
z = 5(cos(π/3) + i sin(π/3)) = 5(1/2 + i√3/2) = 2.5 + 5i√3/2.
Correct Answer:
B
— 5 + 2.5i
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Q. If z1 = 2 + 2i and z2 = 1 - i, find z1 + z2.
A.
3 + i
B.
1 + i
C.
3 - i
D.
1 - i
Show solution
Solution
z1 + z2 = (2 + 2i) + (1 - i) = 3 + i.
Correct Answer:
A
— 3 + i
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Q. If z1 = 2 + 2i and z2 = 2 - 2i, what is the value of z1 * z2? (2019)
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Solution
z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.
Correct Answer:
A
— 8
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Q. If z1 = 2 + 2i and z2 = 3 - i, find z1 + z2. (2023)
A.
5 + i
B.
5 + 3i
C.
1 + i
D.
1 - i
Show solution
Solution
z1 + z2 = (2 + 2i) + (3 - i) = 5 + i.
Correct Answer:
A
— 5 + i
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Q. In the equation x^2 - 4x + 4 = 0, what is the nature of the roots? (2021)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
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Solution
The discriminant is 0, which indicates that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. In the expansion of (2 + 3x)^5, what is the coefficient of x?
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Solution
The coefficient of x is given by C(5, 1) * (2)^4 * (3) = 5 * 16 * 3 = 240.
Correct Answer:
A
— 15
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Q. In the expansion of (2 - x)^5, what is the coefficient of x^3?
A.
-80
B.
-60
C.
60
D.
80
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Solution
The coefficient of x^3 in (2 - x)^5 is given by 5C3 * 2^2 * (-1)^3 = 10 * 4 * (-1) = -40.
Correct Answer:
A
— -80
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Q. In the expansion of (2x + 5)^3, what is the coefficient of x^2?
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Solution
The coefficient of x^2 in (2x + 5)^3 is given by 3C2 * (2x)^2 * 5^1 = 3 * 4 * 5 = 60.
Correct Answer:
B
— 60
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Q. In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)
A.
-108
B.
-72
C.
72
D.
108
Show solution
Solution
The coefficient of x^3 in (2x - 3)^4 is given by 4C1 * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96. The coefficient is -108.
Correct Answer:
A
— -108
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Q. In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?
A.
-720
B.
-540
C.
540
D.
720
Show solution
Solution
The coefficient is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = 720.
Correct Answer:
C
— 540
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Q. In the expansion of (3x - 2)^4, what is the coefficient of x^1?
A.
-48
B.
-72
C.
72
D.
48
Show solution
Solution
The coefficient of x^1 in (3x - 2)^4 is given by 4C3 * (3x)^1 * (-2)^3 = 4 * 3 * (-8) = -96.
Correct Answer:
B
— -72
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Q. In the expansion of (3x - 2)^5, what is the coefficient of x^3?
A.
-240
B.
-360
C.
360
D.
240
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Solution
The coefficient of x^3 in (3x - 2)^5 is given by 5C2 * (3)^3 * (-2)^2 = 10 * 27 * 4 = -1080.
Correct Answer:
B
— -360
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Q. In the expansion of (3x - 4)^7, what is the coefficient of x^5? (1920)
A.
1260
B.
1440
C.
1680
D.
1920
Show solution
Solution
Using the binomial theorem, the coefficient of x^5 in (3x - 4)^7 is given by 7C5 * (3)^5 * (-4)^2 = 21 * 243 * 16 = 21 * 3888 = 81588.
Correct Answer:
A
— 1260
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Q. In the expansion of (a + b)^6, what is the coefficient of a^2b^4?
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Solution
The coefficient of a^2b^4 in (a + b)^6 is given by 6C2 = 15.
Correct Answer:
B
— 30
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Q. In the expansion of (a + b)^n, if the coefficient of a^3b^2 is 60, what is the value of n?
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Solution
C(n,3) * b^2 = 60. For n = 5, C(5,3) = 10, which does not satisfy. For n = 6, C(6,3) = 20, which does not satisfy. For n = 7, C(7,3) = 35, which does not satisfy. For n = 8, C(8,3) = 56, which satisfies.
Correct Answer:
B
— 6
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Q. In the expansion of (x + 2)^5, what is the coefficient of x^4?
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Solution
The coefficient of x^4 is given by C(5, 4)(2)^1 = 5 * 2 = 10.
Correct Answer:
B
— 10
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Q. In the expansion of (x + 2)^8, what is the coefficient of x^6?
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Solution
The coefficient of x^6 in (x + 2)^8 is C(8, 6) * (2)^2 = 28 * 4 = 112.
Correct Answer:
C
— 84
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