Binomial Theorem for Competitive Exam Success

Binomial Theorem

Q. What is the coefficient of x^2 in the expansion of (2x + 1)^6?

A. 15
B. 30
C. 45
D. 60

Solution

The coefficient of x^2 is C(6,2) * (2)^2 * (1)^4 = 15 * 4 = 60.

Correct Answer: B — 30

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Q. What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

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

Solution

The coefficient of x^2 is C(5,2) * (2)^2 * (-1)^3 = 10 * 4 * (-1) = -40.

Correct Answer: B — 20

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

A. 540
B. 720
C. 1080
D. 1440

Solution

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

Correct Answer: C — 1080

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Q. What is the coefficient of x^2 in the expansion of (x + 1/2)^8?

A. 28
B. 56
C. 70
D. 84

Solution

The coefficient of x^2 is C(8,2) * (1/2)^2 = 28 * 1/4 = 7.

Correct Answer: C — 70

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Q. What is the coefficient of x^2 in the expansion of (x + 3)^5?

A. 45
B. 60
C. 75
D. 90

Solution

The coefficient of x^2 is given by 5C2 * (3)^3 = 10 * 27 = 270.

Correct Answer: A — 45

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Q. What is the coefficient of x^2 in the expansion of (x + 4)^5?

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. What is the coefficient of x^2 in the expansion of (x + 5)^4?

A. 50
B. 100
C. 150
D. 200

Solution

The coefficient of x^2 is C(4,2) * (5)^2 = 6 * 25 = 150.

Correct Answer: B — 100

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Q. What is the coefficient of x^2 in the expansion of (x + 5)^7? (2021)

A. 21
B. 35
C. 105
D. 175

Solution

The coefficient of x^2 is given by 7C2 * 5^5 = 21 * 3125 = 65625.

Correct Answer: B — 35

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Q. What is the coefficient of x^2 in the expansion of (x - 1)^5?

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

Solution

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

Correct Answer: A — -10

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Q. What is the coefficient of x^2 in the expansion of (x - 2)^5?

A. -40
B. -20
C. 20
D. 40

Solution

The coefficient of x^2 is given by 5C2 * (-2)^3 = 10 * (-8) = -80.

Correct Answer: A — -40

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Q. What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

A. 600
B. 720
C. 800
D. 900

Solution

The coefficient of x^3 is C(6,3) * (5)^3 * (2)^3 = 20 * 125 * 8 = 20000.

Correct Answer: B — 720

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Q. What is the coefficient of x^3 in the expansion of (x + 5)^8?

A. 2800
B. 3360
C. 4000
D. 4200

Solution

The coefficient of x^3 is given by 8C3 * (5)^5 = 56 * 3125 = 175000.

Correct Answer: B — 3360

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Q. What is the coefficient of x^3 in the expansion of (x - 1)^5? (2022)

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

Solution

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

Correct Answer: A — -10

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Q. What is the coefficient of x^3 in the expansion of (x - 2)^5?

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

Solution

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

Correct Answer: A — 10

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Q. What is the coefficient of x^3 in the expansion of (x - 2)^6?

A. -80
B. -120
C. -240
D. -720

Solution

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

Correct Answer: B — -120

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Q. What is the coefficient of x^4 in the expansion of (2x + 5)^6? (2022)

A. 150
B. 300
C. 600
D. 450

Solution

The coefficient of x^4 is C(6,4) * (2)^4 * (5)^2 = 15 * 16 * 25 = 600.

Correct Answer: D — 450

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Q. What is the coefficient of x^4 in the expansion of (3x + 2)^6?

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

Solution

The coefficient of x^4 is C(6,4) * (3)^4 * (2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: B — 720

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Q. What is the coefficient of x^4 in the expansion of (3x - 2)^5? (2021)

A. 90
B. 120
C. 150
D. 180

Solution

The coefficient of x^4 is given by 5C4 * (3)^4 * (-2)^1 = 5 * 81 * (-2) = -810.

Correct Answer: C — 150

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Q. What is the coefficient of x^4 in the expansion of (x + 1)^8? (2020)

A. 70
B. 56
C. 28
D. 8

Solution

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

Correct Answer: B — 56

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Q. What is the coefficient of x^4 in the expansion of (x + 2)^6?

A. 15
B. 60
C. 90
D. 120

Solution

The coefficient of x^4 is given by 6C4 * (2)^2 = 15 * 4 = 60.

Correct Answer: C — 90

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Q. What is the coefficient of x^5 in the expansion of (2x - 1)^6?

A. 32
B. 48
C. 64
D. 80

Solution

The coefficient of x^5 is C(6,5) * (2)^5 * (-1)^1 = 6 * 32 * (-1) = -192.

Correct Answer: D — 80

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Q. What is the coefficient of x^5 in the expansion of (x + 4)^7? (2020)

A. 672
B. 6720
C. 1680
D. 840

Solution

The coefficient of x^5 is C(7,5) * (4)^2 = 21 * 16 = 336.

Correct Answer: C — 1680

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Q. What is the coefficient of x^5 in the expansion of (x - 3)^7?

A. -21
B. -63
C. -126
D. -189

Solution

The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -189.

Correct Answer: B — -63

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Q. What is the coefficient of x^6 in the expansion of (2x - 1)^8? (2022)

A. -112
B. 112
C. 64
D. 128

Solution

The coefficient of x^6 is C(8,6) * (2)^6 * (-1)^2 = 28 * 64 = 1792.

Correct Answer: A — -112

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Q. What is the coefficient of x^6 in the expansion of (x + 3)^8? (2020)

A. 56
B. 84
C. 112
D. 128

Solution

The coefficient of x^6 is C(8,6) * (3)^2 = 28 * 9 = 252.

Correct Answer: B — 84

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Q. What is the coefficient of x^6 in the expansion of (x + 3)^9? (2020)

A. 84
B. 126
C. 36
D. 54

Solution

The coefficient of x^6 is C(9,6)(3)^3 = 84 * 27 = 2268.

Correct Answer: B — 126

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Q. What is the coefficient of x^7 in the expansion of (2x - 3)^8? (2022)

A. -6720
B. 6720
C. -3360
D. 3360

Solution

The coefficient of x^7 is C(8,7)(2)^7(-3)^1 = 8 * 128 * -3 = -3072.

Correct Answer: A — -6720

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Q. What is the constant term in the expansion of (2x^2 - 3/x)^5? (2020)

A. -243
B. 0
C. 32
D. 64

Solution

The constant term occurs when 2k - 5 = 0, k = 2. The term is C(5,2)(-3)^3(2^2) = 10 * -27 * 4 = -1080.

Correct Answer: A — -243

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Q. What is the sum of the coefficients in the expansion of (2 + 3)^4? (2022)

A. 81
B. 64
C. 100
D. 125

Solution

The sum of the coefficients is (2 + 3)^4 = 5^4 = 625.

Correct Answer: A — 81

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Q. What is the sum of the coefficients in the expansion of (3x - 2)^4? (2023)

A. 1
B. 81
C. 64
D. 256

Solution

To find the sum of the coefficients, substitute x = 1. Thus, (3(1) - 2)^4 = (3 - 2)^4 = 1^4 = 1.

Correct Answer: B — 81

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Showing 91 to 120 of 139 (5 Pages)

Binomial Theorem

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