Binomial Theorem for Competitive Exam Success

Binomial Theorem

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

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

Solution

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

Correct Answer: B — 48

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

A. 15/4
B. 45/8
C. 15/8
D. 5/4

Solution

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

Correct Answer: B — 45/8

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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/2)^8. (2021)

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

Solution

The coefficient of x^2 is C(8,2) * (1/2)^6 = 28 * 1/64 = 28/64 = 7/16.

Correct Answer: C — 70

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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

A. 15
B. 30
C. 20
D. 10

Solution

The coefficient of x^2 is given by 6C2 * (1)^4 = 15.

Correct Answer: A — 15

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

A. 96
B. 144
C. 192
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. Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.

A. 20
B. 30
C. 40
D. 50

Solution

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

Correct Answer: A — 20

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

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

Solution

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

Correct Answer: B — 10

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

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

Solution

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

Correct Answer: A — 540

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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)

A. 15/8
B. 45/8
C. 5/8
D. 1/8

Solution

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

Correct Answer: B — 45/8

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Q. Calculate 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 C(6,4) * 2^2 = 15 * 4 = 60.

Correct Answer: C — 90

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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

A. 54
B. 81
C. 108
D. 729

Solution

The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.

Correct Answer: C — 108

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Q. Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

A. 150
B. 600
C. 750
D. 1000

Solution

The coefficient of x^4 is C(6,4) * 5^2 = 15 * 25 = 375.

Correct Answer: C — 750

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Q. Calculate the coefficient of x^5 in the expansion of (x + 2)^7.

A. 21
B. 42
C. 63
D. 84

Solution

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

Correct Answer: C — 63

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Q. Calculate the coefficient of x^5 in the expansion of (x - 3)^7. (2021)

A. -189
B. -243
C. -126
D. -21

Solution

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

Correct Answer: A — -189

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Q. Calculate the term containing x^3 in the expansion of (2x + 5)^6. (2000)

A. 1500
B. 1800
C. 2000
D. 2500

Solution

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

Correct Answer: B — 1800

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Q. Calculate the term containing x^3 in the expansion of (x + 2)^7.

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

Solution

The term containing x^3 is C(7,3) * (2)^4 = 35 * 16 = 560.

Correct Answer: B — 84

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Q. Calculate the term independent of x in the expansion of (2x - 3)^5.

A. -243
B. 0
C. 243
D. 81

Solution

The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.

Correct Answer: A — -243

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.

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

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.

Correct Answer: B — 24

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^5.

A. 80
B. 120
C. 160
D. 200

Solution

The term independent of x occurs when the powers of x cancel out. This can be calculated using the binomial expansion.

Correct Answer: C — 160

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x)^4.

A. -81
B. 108
C. -108
D. 81

Solution

The term independent of x occurs when 2k = 4, k = 2. The term is C(4,2) * (2x^2)^2 * (-3x)^2 = 6 * 4 * 9 = -216.

Correct Answer: C — -108

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^6.

A. 729
B. 729/64
C. 729/32
D. 729/16

Solution

The term independent of x occurs when k = 3, which gives C(6,3) * (x/2)^3 * (-3)^3 = 20 * (1/8) * (-27) = -67.5.

Correct Answer: B — 729/64

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^8.

A. -3
B. -8
C. 0
D. 256

Solution

The term independent of x occurs when k = 4. Thus, the term is C(8,4) * (x/2)^4 * (-3)^4 = 70 * (1/16) * 81 = 256.

Correct Answer: D — 256

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Q. Calculate the term independent of x in the expansion of (x^2 - 3x + 2)^4.

A. 8
B. 12
C. 16
D. 20

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(4,2) * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.

Correct Answer: B — 12

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Q. Calculate the value of (1 + 3)^5 using the binomial theorem.

A. 81
B. 243
C. 125
D. 256

Solution

(1 + 3)^5 = 4^5 = 1024.

Correct Answer: B — 243

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Q. Determine 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 given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.

Correct Answer: B — 720

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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 + 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 (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 - 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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Binomial Theorem

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