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Q. Find the arithmetic mean of the first five prime numbers.

A. 5
B. 6
C. 7
D. 8

Solution

First five primes: 2, 3, 5, 7, 11. Mean = (2 + 3 + 5 + 7 + 11) / 5 = 28 / 5 = 5.6.

Correct Answer: C — 7

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Q. Find the arithmetic mean of the numbers 12, 15, 18, 21, and 24.

A. 18
B. 19
C. 20
D. 21

Solution

Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18.

Correct Answer: A — 18

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Q. Find the coefficient of x^0 in the expansion of (2x + 3)^4.

A. 81
B. 64
C. 27
D. 16

Solution

The coefficient of x^0 is C(4, 0) * (2x)^0 * 3^4 = 1 * 81 = 81.

Correct Answer: A — 81

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Q. Find the coefficient of x^0 in the expansion of (2x - 3)^3.

A. -27
B. -24
C. -18
D. -12

Solution

The coefficient of x^0 is (-3)^3 = -27.

Correct Answer: A — -27

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Q. Find the coefficient of x^1 in the expansion of (x + 2)^5.

A. 10
B. 20
C. 30
D. 40

Solution

The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.

Correct Answer: B — 20

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Q. Find the coefficient of x^1 in the expansion of (x + 4)^3.

A. 12
B. 48
C. 36
D. 24

Solution

The coefficient of x^1 is C(3,1) * (4)^2 = 3 * 16 = 48.

Correct Answer: A — 12

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Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.

A. 540
B. 720
C. 810
D. 960

Solution

The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.

Correct Answer: A — 540

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Q. Find the coefficient of x^2 in the expansion of (2x - 3)^4.

A. 36
B. 48
C. 54
D. 72

Solution

Using the binomial theorem, the coefficient of x^2 in (2x - 3)^4 is given by 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.

Correct Answer: C — 54

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Q. Find the coefficient of x^2 in the expansion of (3x - 2)^5.

A. -60
B. -90
C. 90
D. 60

Solution

The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.

Correct Answer: B — -90

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Q. Find the coefficient of x^2 in the expansion of (3x - 4)^6.

A. 540
B. 720
C. 480
D. 360

Solution

The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.

Correct Answer: B — 720

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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)

A. 80
B. 100
C. 120
D. 160

Solution

The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.

Correct Answer: A — 80

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Q. Find the coefficient of x^2 in the expansion of (x + 4)^6.

A. 96
B. 144
C. 216
D. 256

Solution

The coefficient of x^2 is given by C(6, 2)(4)^4 = 15 * 256 = 3840.

Correct Answer: A — 96

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Q. Find the coefficient of x^2 in the expansion of (x - 5)^5.

A. 100
B. 150
C. 200
D. 250

Solution

The coefficient of x^2 is C(5,2) * (-5)^3 = 10 * (-125) = -1250.

Correct Answer: A — 100

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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)

A. -54
B. -108
C. 108
D. 54

Solution

The coefficient of x^3 is C(4,3) * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96.

Correct Answer: B — -108

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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^6.

A. -540
B. -720
C. 540
D. 720

Solution

The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.

Correct Answer: A — -540

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Q. Find the coefficient of x^3 in the expansion of (3x - 4)^5.

A. -540
B. -720
C. 720
D. 540

Solution

The coefficient of x^3 in (3x - 4)^5 is given by 5C3 * (3)^3 * (-4)^2 = 10 * 27 * 16 = -720.

Correct Answer: B — -720

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Q. Find the coefficient of x^3 in the expansion of (x + 1)^8.

A. 56
B. 70
C. 84
D. 120

Solution

The coefficient of x^3 in (x + 1)^8 is given by 8C3 = 56.

Correct Answer: C — 84

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Q. Find the coefficient of x^3 in the expansion of (x + 1/2)^6.

A. 20
B. 15
C. 10
D. 5

Solution

The coefficient of x^3 is C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.

Correct Answer: B — 15

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Q. Find the coefficient of x^3 in the expansion of (x + 2)^6.

A. 80
B. 120
C. 160
D. 240

Solution

The coefficient of x^3 is C(6,3) * (2)^3 = 20 * 8 = 160.

Correct Answer: B — 120

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Q. Find the coefficient of x^3 in the expansion of (x - 1)^5.

A. -10
B. -5
C. 5
D. 10

Solution

The coefficient of x^3 is C(5,3) * (-1)^2 = 10.

Correct Answer: A — -10

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Q. Find the coefficient of x^3 in the expansion of (x - 1)^6.

A. -20
B. -15
C. -10
D. -6

Solution

The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.

Correct Answer: A — -20

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Q. Find the coefficient of x^3 in the expansion of (x - 1)^7.

A. -35
B. -21
C. -7
D. -49

Solution

The coefficient of x^3 is given by 7C3 * (-1)^4 = 35.

Correct Answer: A — -35

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Q. Find the coefficient of x^3 in the expansion of (x - 2)^5.

A. -40
B. -80
C. -60
D. -100

Solution

The coefficient of x^3 is C(5,3) * (-2)^2 = 10 * 4 = 40.

Correct Answer: A — -40

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Q. Find the coefficient of x^3 in the expansion of (x - 3)^5.

A. -135
B. -90
C. -60
D. -45

Solution

The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.

Correct Answer: A — -135

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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^5.

A. -240
B. -360
C. -480
D. -600

Solution

The coefficient of x^4 is C(5,4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.

Correct Answer: A — -240

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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^6.

A. 540
B. 720
C. 810
D. 900

Solution

The coefficient of x^4 is C(6,4) * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.

Correct Answer: A — 540

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Q. Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)

A. 240
B. 360
C. 480
D. 600

Solution

The coefficient of x^4 is C(5,4)(3)^4(2)^1 = 5 * 81 * 2 = 810.

Correct Answer: B — 360

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Q. Find the coefficient of x^4 in the expansion of (3x - 2)^6.

A. 540
B. 720
C. 810
D. 960

Solution

Using the binomial theorem, the coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: C — 810

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Q. Find the coefficient of x^4 in the expansion of (x + 1)^6.

A. 15
B. 20
C. 30
D. 35

Solution

The coefficient of x^4 is C(6,4) * 1^2 = 15.

Correct Answer: B — 20

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Q. Find the coefficient of x^4 in the expansion of (x + 1)^8.

A. 70
B. 80
C. 90
D. 100

Solution

The coefficient of x^4 is C(8, 4) = 70.

Correct Answer: A — 70

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Major Competitive Exams

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