Q. Determine the coefficient of x^2 in the expansion of (3x - 4)^4.
-
A.
144
-
B.
216
-
C.
108
-
D.
96
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. Determine the coefficient of x^2 in the expansion of (3x - 4)^6.
-
A.
540
-
B.
720
-
C.
480
-
D.
360
Solution
The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.
Correct Answer: B — 720
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Q. Determine the coefficient of x^2 in the expansion of (x - 2)^6.
-
A.
-60
-
B.
-30
-
C.
15
-
D.
20
Solution
The coefficient of x^2 is C(6,2)(-2)^4 = 15 * 16 = 240.
Correct Answer: A — -60
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Q. Find the coefficient of x^0 in the expansion of (2x + 3)^4.
Solution
The coefficient of x^0 is C(4, 0) * (2x)^0 * 3^4 = 1 * 81 = 81.
Correct Answer: A — 81
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Q. Find the coefficient of x^1 in the expansion of (x + 2)^5.
Solution
The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.
Correct Answer: B — 20
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Q. Find the coefficient of x^2 in the expansion of (3x - 4)^6.
-
A.
540
-
B.
720
-
C.
480
-
D.
360
Solution
The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.
Correct Answer: B — 720
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^6.
-
A.
-540
-
B.
-720
-
C.
540
-
D.
720
Solution
The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Correct Answer: A — -540
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Q. Find the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution
The coefficient of x^3 is C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Correct Answer: B — 15
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Q. Find the coefficient of x^3 in the expansion of (x + 2)^6.
-
A.
80
-
B.
120
-
C.
160
-
D.
240
Solution
The coefficient of x^3 is C(6,3) * (2)^3 = 20 * 8 = 160.
Correct Answer: B — 120
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^5.
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: A — -10
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^6.
-
A.
-20
-
B.
-15
-
C.
-10
-
D.
-6
Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: A — -20
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Q. Find the coefficient of x^3 in the expansion of (x - 3)^5.
-
A.
-135
-
B.
-90
-
C.
-60
-
D.
-45
Solution
The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.
Correct Answer: A — -135
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Q. Find the coefficient of x^4 in the expansion of (3x - 2)^6.
-
A.
540
-
B.
720
-
C.
810
-
D.
960
Solution
Using the binomial theorem, the coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: C — 810
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^8.
Solution
The coefficient of x^5 is C(8,5) = 56.
Correct Answer: B — 70
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Q. Find the coefficient of x^5 in the expansion of (x + 3)^8.
-
A.
56
-
B.
168
-
C.
336
-
D.
672
Solution
The coefficient of x^5 is C(8,5) * (3)^3 = 56 * 27 = 1512.
Correct Answer: B — 168
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Q. Find the coefficient of x^5 in the expansion of (x - 3)^7.
-
A.
-1890
-
B.
-2187
-
C.
-2401
-
D.
-2430
Solution
The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -1890.
Correct Answer: A — -1890
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Q. Find the value of (1 + 2)^4 using the binomial theorem.
Solution
Using the binomial theorem, (1 + 2)^4 = C(4,0) * 1^4 * 2^0 + C(4,1) * 1^3 * 2^1 + C(4,2) * 1^2 * 2^2 + C(4,3) * 1^1 * 2^3 + C(4,4) * 1^0 * 2^4 = 1 + 8 + 24 + 32 + 16 = 81.
Correct Answer: A — 16
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Q. Find the value of (1 + x)^10 at x = 1. (2048)
-
A.
10
-
B.
11
-
C.
1024
-
D.
2048
Solution
Using the binomial theorem, (1 + 1)^10 = 2^10 = 1024.
Correct Answer: C — 1024
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Q. Find the value of (1 + x)^10 at x = 2.
-
A.
1024
-
B.
2048
-
C.
512
-
D.
256
Solution
Using the binomial theorem, (1 + 2)^10 = 3^10 = 59049.
Correct Answer: B — 2048
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Q. Find the value of k such that the coefficient of x^4 in the expansion of (x + k)^6 is 240.
Solution
The coefficient of x^4 is C(6,4) * k^2 = 15k^2. Setting 15k^2 = 240 gives k^2 = 16, so k = 4 or -4.
Correct Answer: B — 5
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Q. Find the value of the coefficient of x^2 in the expansion of (3x - 4)^4.
-
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: A — -144
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Q. If the expansion of (x + a)^n has a term 15x^3a^2, what is the value of n?
Solution
The term is given by C(n, 3) * a^2 * x^3. Setting C(n, 3) * a^2 = 15 gives n = 6.
Correct Answer: B — 6
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Q. In the expansion of (1 + x)^10, what is the coefficient of x^5?
-
A.
252
-
B.
210
-
C.
120
-
D.
300
Solution
The coefficient of x^5 is C(10,5) = 10! / (5!5!) = 252.
Correct Answer: A — 252
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Q. In the expansion of (2 + 3x)^4, what is the coefficient of x?
Solution
The coefficient of x is C(4,1) * 2^3 * 3 = 4 * 8 * 3 = 96.
Correct Answer: A — 12
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Q. In the expansion of (2 + 3x)^4, what is the coefficient of x^2?
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: B — 54
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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^1?
Solution
The coefficient of x^1 is C(4,1) * (2)^1 * (3)^3 = 4 * 2 * 27 = 216.
Correct Answer: B — 48
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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^2?
-
A.
108
-
B.
72
-
C.
36
-
D.
144
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: B — 72
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Q. In the expansion of (2x + 3)^6, what is the coefficient of x^4?
-
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: B — 720
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Q. In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?
-
A.
108
-
B.
72
-
C.
36
-
D.
144
Solution
The coefficient is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: A — 108
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Q. In the expansion of (2x + 5)^4, what is the coefficient of x^2?
-
A.
300
-
B.
600
-
C.
450
-
D.
500
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (5)^2 = 6 * 4 * 25 = 600.
Correct Answer: A — 300
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