Q. Find the coefficient of x^2 in the expansion of (2x - 3)^4.
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Solution
Using the binomial theorem, the coefficient of x^2 in (2x - 3)^4 is given by 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.
Correct Answer: C — 54
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Q. Find the value of k in the expansion of (x + 2)^6 such that the term containing x^4 is 240.
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Solution
The term containing x^4 is C(6,4) * (2)^2 * x^4 = 15 * 4 * x^4 = 60x^4. Setting 60 = 240 gives k = 4.
Correct Answer: A — 4
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Q. Find the value of the coefficient of x^4 in the expansion of (x - 2)^6.
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Solution
Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.
Correct Answer: C — 30
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Q. In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)
A.
-108
B.
-72
C.
72
D.
108
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Solution
The coefficient of x^3 in (2x - 3)^4 is given by 4C1 * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96. The coefficient is -108.
Correct Answer: A — -108
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Q. In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?
A.
-720
B.
-540
C.
540
D.
720
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Solution
The coefficient is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = 720.
Correct Answer: C — 540
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Q. In the expansion of (3x - 4)^7, what is the coefficient of x^5? (1920)
A.
1260
B.
1440
C.
1680
D.
1920
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Solution
Using the binomial theorem, the coefficient of x^5 in (3x - 4)^7 is given by 7C5 * (3)^5 * (-4)^2 = 21 * 243 * 16 = 21 * 3888 = 81588.
Correct Answer: A — 1260
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Q. In the expansion of (a + b)^n, if the coefficient of a^3b^2 is 60, what is the value of n?
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Solution
C(n,3) * b^2 = 60. For n = 5, C(5,3) = 10, which does not satisfy. For n = 6, C(6,3) = 20, which does not satisfy. For n = 7, C(7,3) = 35, which does not satisfy. For n = 8, C(8,3) = 56, which satisfies.
Correct Answer: B — 6
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Q. In the expansion of (x + 3)^5, what is the coefficient of x^3?
A.
60
B.
90
C.
100
D.
120
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Solution
Using the binomial theorem, the coefficient of x^3 in (x + 3)^5 is given by 5C3 * (3)^2 = 10 * 9 = 90.
Correct Answer: B — 90
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Q. In the expansion of (x + 3)^6, what is the coefficient of x^4?
A.
540
B.
720
C.
810
D.
900
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Solution
Using the binomial theorem, the coefficient of x^4 in (x + 3)^6 is given by 6C4 * (3)^2 = 15 * 9 = 135.
Correct Answer: B — 720
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Q. In the expansion of (x - 1)^8, what is the coefficient of x^5?
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Solution
The coefficient of x^5 in (x - 1)^8 is C(8,5) * (-1)^3 = 56 * (-1) = -56.
Correct Answer: A — -56
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Q. What is the 4th term in the expansion of (3x + 2)^6?
A.
540x^4
B.
540x^3
C.
720x^4
D.
720x^3
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Solution
The 4th term is given by C(6,3) * (3x)^3 * (2)^3 = 20 * 27x^3 * 8 = 4320x^3.
Correct Answer: A — 540x^4
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Q. What is the coefficient of x^2 in the expansion of (2x + 5)^4?
A.
60
B.
80
C.
100
D.
120
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Solution
Using the binomial theorem, the coefficient of x^2 in (2x + 5)^4 is given by 4C2 * (2)^2 * (5)^2 = 6 * 4 * 25 = 600.
Correct Answer: A — 60
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Q. What is the coefficient of x^2 in the expansion of (2x - 5)^5? (2019)
A.
-300
B.
-600
C.
600
D.
300
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Solution
The coefficient of x^2 in (2x - 5)^5 is given by 5C2 * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.
Correct Answer: B — -600
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Q. What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)
A.
90
B.
180
C.
270
D.
360
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Solution
The coefficient of x^3 in (3x + 2)^5 is given by 5C3 * (3)^3 * (2)^2 = 10 * 27 * 4 = 1080.
Correct Answer: B — 180
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Q. What is the sum of the coefficients in the expansion of (2x - 3)^4?
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Solution
To find the sum of the coefficients, substitute x = 1: (2*1 - 3)^4 = (-1)^4 = 1. The sum of coefficients is 81.
Correct Answer: B — 81
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Q. What is the sum of the coefficients in the expansion of (x + 1)^8?
A.
256
B.
512
C.
128
D.
64
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Solution
The sum of the coefficients in the expansion of (x + 1)^n is given by (1 + 1)^n = 2^n. Here, n=8, so the sum is 2^8 = 256.
Correct Answer: B — 512
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Q. What is the value of the 5th term in the expansion of (x + 2)^7?
A.
672
B.
672x^4
C.
672x^3
D.
672x^2
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Solution
The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.
Correct Answer: C — 672x^3
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