Binomial Theorem for Competitive Exam Success

Binomial theorem

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

A. -9
B. -18
C. -27
D. -6

Solution

The coefficient of x is C(3,1) * (2)^1 * (-3)^2 = 3 * 2 * 9 = -54.

Correct Answer: B — -18

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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^1?

A. -15
B. -30
C. 30
D. 15

Solution

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

Correct Answer: B — -30

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

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

Solution

The coefficient of x^2 is C(5,2) * (2)^2 * (-3)^3 = 10 * 4 * (-27) = -1080.

Correct Answer: B — -120

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

A. -540
B. -720
C. -960
D. -1080

Solution

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

Correct Answer: C — -960

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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^4?

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

Solution

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

Correct Answer: B — -120

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Q. In the expansion of (x + 1/x)^6, what is the coefficient of x^0?

A. 6
B. 15
C. 20
D. 30

Solution

The coefficient of x^0 in (x + 1/x)^6 is given by 6C3 = 20.

Correct Answer: C — 20

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Q. In the expansion of (x + 1/x)^8, what is the coefficient of x^4?

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

Solution

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

Correct Answer: B — 56

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

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

Solution

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

Correct Answer: B — 60

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Q. In the expansion of (x + 2)^6, what is the term containing x^4?

A. 240
B. 360
C. 720
D. 480

Solution

The term containing x^4 is C(6,4) * (2)^2 * (x)^4 = 15 * 4 * x^4 = 60x^4.

Correct Answer: C — 720

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Q. In the expansion of (x + 2)^7, what is the term containing x^5?

A. 672
B. 6720
C. 67200
D. 672000

Solution

The term containing x^5 is given by 7C5 * (2)^2 * (x)^5 = 21 * 4 * x^5 = 84x^5. Thus, the term is 84x^5.

Correct Answer: B — 6720

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

A. 150
B. 200
C. 250
D. 300

Solution

The coefficient of x^3 is C(6, 3) * 5^3 = 20 * 125 = 250.

Correct Answer: B — 200

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

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

Solution

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

Correct Answer: C — 90

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Q. In the expansion of (x - 2)^8, what is the coefficient of x^5?

A. -2240
B. -1280
C. -1120
D. -960

Solution

The coefficient of x^5 is C(8,5) * (-2)^3 = 56 * (-8) = -448.

Correct Answer: A — -2240

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Q. What is the coefficient of x^0 in the expansion of (3x - 4)^5?

A. -1024
B. 243
C. -243
D. 1024

Solution

The coefficient of x^0 is (-4)^5 = -1024.

Correct Answer: A — -1024

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

A. 125
B. 100
C. 75
D. 50

Solution

The coefficient of x^0 is C(3,0) * 5^3 = 1 * 125 = 125.

Correct Answer: A — 125

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

A. 3125
B. 625
C. 125
D. 25

Solution

The coefficient of x^0 is C(5,0) * 5^5 = 1 * 3125 = 3125.

Correct Answer: A — 3125

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

A. 144
B. 216
C. 432
D. 576

Solution

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

Correct Answer: C — 432

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

A. 240
B. 360
C. 480
D. 720

Solution

Using the binomial theorem, the coefficient of x^2 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-4, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-4)^2 = 15 * 16 = 240.

Correct Answer: B — 360

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

A. 90
B. 100
C. 120
D. 150

Solution

Using the binomial theorem, the coefficient of x^3 is given by C(5,3) * (2)^3 * (3)^(5-3) = 10 * 8 * 9 = 720.

Correct Answer: A — 90

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

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

Solution

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

Correct Answer: C — 810

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

A. 240
B. 300
C. 360
D. 420

Solution

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

Correct Answer: C — 360

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Q. What is 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) = 15.

Correct Answer: D — 35

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Q. What is the coefficient of x^5 in the expansion of (x + 1/2)^10?

A. 252
B. 210
C. 120
D. 300

Solution

The coefficient of x^5 is C(10,5) * (1/2)^5 = 252 * 1/32 = 7.875.

Correct Answer: A — 252

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

A. 21
B. 42
C. 35
D. 28

Solution

The coefficient of x^5 is C(7,5)(2)^2 = 21 * 4 = 84.

Correct Answer: C — 35

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Q. What is the coefficient of x^5 in the expansion of (x + 2)^8?

A. 672
B. 1280
C. 960
D. 720

Solution

The coefficient of x^5 is C(8,5) * (2)^3 = 56 * 8 = 448.

Correct Answer: A — 672

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

A. 1680
B. 1440
C. 1260
D. 1080

Solution

Using the binomial theorem, the coefficient of x^5 in (x + 3)^8 is given by 8C5 * (3)^3 = 56 * 27 = 1512.

Correct Answer: A — 1680

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Q. What is the coefficient of x^6 in the expansion of (x + 5)^8?

A. 6720
B. 5600
C. 4480
D. 3360

Solution

The coefficient of x^6 is C(8,6) * (5)^2 = 28 * 25 = 700.

Correct Answer: A — 6720

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

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

Solution

The middle term is T(5) = C(8,4) * (2)^4 = 70 * 16 = 1120.

Correct Answer: A — 128

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

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

Solution

The sum of the coefficients is found by substituting x = 1: (2 + 3*1)^4 = 5^4 = 625.

Correct Answer: A — 81

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

A. 5
B. 6
C. 31
D. 32

Solution

The sum of the coefficients is found by substituting x = 1, so (1 + 1)^5 = 2^5 = 32.

Correct Answer: D — 32

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Showing 31 to 60 of 69 (3 Pages)

Binomial theorem

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