Binomial Theorem for Competitive Exam Success

Binomial Theorem

Q. In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

A. -300
B. -600
C. 600
D. 300

Solution

The coefficient of x^2 is C(5,2) * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.

Correct Answer: A — -300

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Q. In the expansion of (2x - 5)^5, what is the coefficient of x^3? (2021)

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

Solution

The coefficient of x^3 is C(5,3) * (2)^3 * (-5)^2 = 10 * 8 * 25 = 2000.

Correct Answer: C — 450

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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

A. -150
B. -300
C. 300
D. 150

Solution

The coefficient of x^2 is C(6,2) * (2)^2 * (-5)^4 = 15 * 4 * 625 = -37500.

Correct Answer: A — -150

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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

A. -600
B. -720
C. 720
D. 600

Solution

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

Correct Answer: B — -720

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Q. In the expansion of (3 + 2x)^4, what is the coefficient of x^2? (2023)

A. 36
B. 48
C. 54
D. 24

Solution

The coefficient of x^2 is C(4,2) * (3)^2 * (2)^2 = 6 * 9 * 4 = 216.

Correct Answer: B — 48

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Q. In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

A. 540
B. 810
C. 729
D. 486

Solution

The coefficient of x^4 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: A — 540

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Q. In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

A. -144
B. -216
C. 216
D. 144

Solution

The coefficient of x^2 is C(4,2)(3)^2(-4)^2 = 6 * 9 * 16 = -864.

Correct Answer: B — -216

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Q. In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

A. 36
B. 48
C. 54
D. 64

Solution

The coefficient of x^3 is C(4,3) * (3)^1 = 4 * 3 = 12.

Correct Answer: A — 36

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Q. In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

A. 240
B. 320
C. 80
D. 160

Solution

The coefficient of x^3 is C(5,3) * (4)^2 = 10 * 16 = 160.

Correct Answer: A — 240

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Q. In the expansion of (x - 1)^5, what is the coefficient of x^3?

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

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 3rd term in the expansion of (2x + 3)^4?

A. 108x^2
B. 216x^2
C. 324x^2
D. 432x^2

Solution

The 3rd term is given by C(4,2) * (2x)^2 * (3)^2 = 6 * 4x^2 * 9 = 216x^2.

Correct Answer: B — 216x^2

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

A. 600x^4
B. 1500x^4
C. 1800x^4
D. 2000x^4

Solution

The 3rd term is given by C(6,2) * (2x)^2 * (5)^4 = 15 * 4 * 625 = 37500.

Correct Answer: B — 1500x^4

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Q. What is the 3rd term in the expansion of (2x - 3)^5? (2022)

A. -90x^3
B. 90x^3
C. -60x^3
D. 60x^3

Solution

The 3rd term is C(5,2) * (2x)^3 * (-3)^2 = 10 * 8x^3 * 9 = -720x^3.

Correct Answer: A — -90x^3

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Q. What is the 3rd term in the expansion of (x + 2)^5? (2021)

A. 40
B. 80
C. 60
D. 20

Solution

The 3rd term is given by C(5,2) * (x)^3 * (2)^2 = 10 * x^3 * 4 = 40.

Correct Answer: C — 60

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Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)

A. 112
B. 128
C. 256
D. 64

Solution

The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.

Correct Answer: A — 112

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Q. What is the 3rd term in the expansion of (x + 3)^4? (2023)

A. 36x^2
B. 54x^2
C. 72x^2
D. 108x^2

Solution

The 3rd term is C(4,2) * (3)^2 * (x)^2 = 6 * 9 * x^2 = 54x^2.

Correct Answer: B — 54x^2

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Q. What is the 3rd term in the expansion of (x + 3)^5? (2023)

A. 45
B. 90
C. 135
D. 180

Solution

The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.

Correct Answer: B — 90

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Q. What is the 3rd term in the expansion of (x + 3)^7? (2023)

A. 189
B. 441
C. 729
D. 1024

Solution

The 3rd term is C(7,2) * (3)^2 * (x)^5 = 21 * 9 * x^5 = 189.

Correct Answer: B — 441

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Q. What is the 3rd term in the expansion of (x + 4)^5? (2023)

A. 80x^3
B. 160x^3
C. 240x^3
D. 320x^3

Solution

The 3rd term is given by C(5,2) * (4)^2 * (x)^3 = 10 * 16 * x^3 = 160x^3.

Correct Answer: C — 240x^3

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Q. What is the 3rd term in the expansion of (x + 4)^6? (2020)

A. 240x^4
B. 360x^4
C. 480x^4
D. 600x^4

Solution

The 3rd term is C(6,2) * (4)^2 * (x)^4 = 15 * 16 * x^4 = 240x^4.

Correct Answer: B — 360x^4

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Q. What is the 4th term in the expansion of (2x - 3)^6? (2020)

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

Solution

The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.

Correct Answer: A — -540

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Q. What is the 5th term in the expansion of (2 + 3x)^6? (2023)

A. 486
B. 540
C. 729
D. 810

Solution

The 5th term is given by 6C4 * (2)^(6-4) * (3x)^4 = 15 * 4 * 81x^4 = 4860x^4.

Correct Answer: B — 540

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Q. What is the 5th term in the expansion of (3x - 2)^7? (2023)

A. -1680x^3
B. 1680x^3
C. -2520x^3
D. 2520x^3

Solution

The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.

Correct Answer: A — -1680x^3

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Q. What is the 5th term in the expansion of (x - 1)^8? (2022)

A. -56x^4
B. 56x^4
C. -70x^4
D. 70x^4

Solution

The 5th term corresponds to k=4. Using the binomial theorem, it is given by 8C4 * (x)^4 * (-1)^4 = 70x^4.

Correct Answer: A — -56x^4

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Q. What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

A. -243
B. 243
C. -81
D. 81

Solution

The constant term is C(5,5)(-3)^5 = -243.

Correct Answer: A — -243

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Q. What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

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

Solution

The coefficient of x^0 is C(5,5) * (3x)^0 * (2)^5 = 1 * 32 = 32.

Correct Answer: C — 64

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Q. What is the coefficient of x^0 in the expansion of (x + 1/x)^8? (2021)

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

Solution

The coefficient of x^0 occurs when k=4, which gives us 8C4 = 70.

Correct Answer: C — 70

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Q. What is the coefficient of x^1 in the expansion of (x + 1)^10? (2021)

A. 10
B. 90
C. 100
D. 1

Solution

The coefficient of x^1 is C(10,1)(1)^9 = 10.

Correct Answer: A — 10

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

A. 30
B. 60
C. 90
D. 120

Solution

The coefficient of x^1 is C(6,1) * 5^5 = 6 * 3125 = 18750.

Correct Answer: C — 90

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

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

Solution

The coefficient of x^1 is given by 5C1 * (-1)^4 = 5 * 1 = 5.

Correct Answer: A — -5

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Showing 61 to 90 of 139 (5 Pages)

Binomial Theorem

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