Binomial Theorem for Competitive Exam Success

Binomial Theorem

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 (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^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 (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 + 2)^6.

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

Solution

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

Correct Answer: B — 30

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

A. 81
B. 162
C. 243
D. 729

Solution

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

Correct Answer: C — 243

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Q. Find the coefficient of x^4 in the expansion of (x + 5)^7.

A. 210
B. 1260
C. 1750
D. 2450

Solution

The coefficient of x^4 is given by C(7,4) * 5^3 = 35 * 125 = 4375.

Correct Answer: B — 1260

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Q. Find the coefficient of x^4 in the expansion of (x - 1)^5.

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

Solution

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

Correct Answer: C — -10

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Q. Find the coefficient of x^4 in the expansion of (x - 5)^6.

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

Solution

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

Correct Answer: B — 200

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)

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

Solution

The coefficient of x^5 is C(6,5) * (2)^5 * (-3)^1 = 6 * 32 * (-3) = -576.

Correct Answer: A — -540

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)

A. 168
B. 252
C. 336
D. 504

Solution

The coefficient of x^5 is C(7,5) * (2)^5 * (-3)^2 = 21 * 32 * 9 = 6048.

Correct Answer: B — 252

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^8.

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

Solution

The coefficient of x^5 is C(8,5) * (2)^5 * (-3)^3 = 56 * 32 * (-27) = -6720.

Correct Answer: A — -6720

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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.

A. 7
B. 21
C. 35
D. 42

Solution

The coefficient of x^5 is C(7,5) = 21.

Correct Answer: C — 35

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

A. 21
B. 42
C. 56
D. 70

Solution

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

Correct Answer: C — 56

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Q. Find the coefficient of x^6 in the expansion of (x + 1)^10. (2023)

A. 10
B. 45
C. 120
D. 210

Solution

The coefficient of x^6 is given by 10C6 = 210.

Correct Answer: D — 210

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Q. Find the constant term in the expansion of (3x - 4/x)^5.

A. -64
B. 0
C. 32
D. 48

Solution

The constant term occurs when the power of x is zero. The term is given by 5C2 * (3x)^2 * (-4/x)^3 = 10 * 9 * (-64) = -5760.

Correct Answer: A — -64

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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)

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

Solution

The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.

Correct Answer: A — -64

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

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

Solution

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

Correct Answer: A — 150

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

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

Solution

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

Correct Answer: C — -10

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

A. 81
B. 108
C. 54
D. 27

Solution

The term independent of x occurs when the powers of x cancel out. The term is 81.

Correct Answer: A — 81

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Q. Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)

A. -15
B. 10
C. 5
D. 0

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.

Correct Answer: A — -15

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Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^4. (2020)

A. 16
B. 64
C. 256
D. 0

Solution

The expression can be rewritten as (x - 2)^4. The term independent of x occurs when k = 4, which gives us (-2)^4 = 16.

Correct Answer: C — 256

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Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^6. (2020)

A. 6
B. 12
C. 24
D. 36

Solution

The expression can be rewritten as (x - 2)^6. The term independent of x occurs when k = 3, which gives us 6C3 * (-2)^3 = 20 * (-8) = -160. The term independent of x is 24.

Correct Answer: C — 24

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Q. Find the value of (3 + 2)^3 using the binomial theorem.

A. 25
B. 27
C. 30
D. 32

Solution

Using the binomial theorem, (3 + 2)^3 = C(3,0) * 3^3 * 2^0 + C(3,1) * 3^2 * 2^1 + C(3,2) * 3^1 * 2^2 + C(3,3) * 3^0 * 2^3 = 27 + 54 + 36 + 8 = 125.

Correct Answer: B — 27

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Q. In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

A. 12
B. 36
C. 48
D. 72

Solution

The coefficient of x is given by 4C1 * (2)^1 * (3)^3 = 4 * 2 * 27 = 216.

Correct Answer: B — 36

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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^0?

A. 81
B. 64
C. 36
D. 16

Solution

The coefficient of x^0 is given by 4C4 * (2x)^0 * (3)^4 = 1 * 81 = 81.

Correct Answer: A — 81

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

A. 30
B. 45
C. 60
D. 75

Solution

The coefficient of x is given by 3C1 * (2)^1 * (5)^2 = 3 * 2 * 25 = 150.

Correct Answer: B — 45

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Showing 31 to 60 of 139 (5 Pages)

Binomial Theorem

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