Q. In the expansion of (2x - 3)^3, what is the coefficient of x?
A.
-9
B.
-18
C.
-27
D.
-6
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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
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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
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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
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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
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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?
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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?
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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?
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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
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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
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
Show solution
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
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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?
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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
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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?
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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
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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
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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
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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
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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
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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?
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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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