Q. Determine the coefficient of x^5 in the expansion of (3x - 4)^7.
A.
252
B.
336
C.
672
D.
840
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Solution
The coefficient of x^5 in (3x - 4)^7 is C(7, 5) * (3)^5 * (-4)^2 = 21 * 243 * 16 = 68016.
Correct Answer:
A
— 252
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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 coefficient of x^2 in the expansion of (3x - 2)^5.
A.
-60
B.
-90
C.
90
D.
60
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Solution
The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.
Correct Answer:
B
— -90
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^6.
A.
96
B.
144
C.
216
D.
256
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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. Find the coefficient of x^3 in the expansion of (3x - 4)^5.
A.
-540
B.
-720
C.
720
D.
540
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Solution
The coefficient of x^3 in (3x - 4)^5 is given by 5C3 * (3)^3 * (-4)^2 = 10 * 27 * 16 = -720.
Correct Answer:
B
— -720
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Q. Find the coefficient of x^3 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^3 in (x + 1)^8 is given by 8C3 = 56.
Correct Answer:
C
— 84
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Q. Find the coefficient of x^4 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^4 is C(8, 4) = 70.
Correct Answer:
A
— 70
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Q. Find the coefficient of x^5 in the expansion of (3x + 2)^6.
A.
486
B.
729
C.
729
D.
486
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Solution
The coefficient of x^5 in (3x + 2)^6 is C(6, 5)(3)^5(2)^1 = 6 * 243 * 2 = 2916.
Correct Answer:
A
— 486
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Q. Find the value of (1 + x)^6 when x = 2.
A.
64
B.
128
C.
256
D.
512
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Solution
Using the binomial theorem, (1 + 2)^6 = 3^6 = 729.
Correct Answer:
C
— 256
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Q. Find the value of (a + b)^4 when a = 2 and b = 3.
A.
81
B.
125
C.
625
D.
256
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Solution
Using the binomial theorem, (a + b)^4 = C(4, 0)a^4b^0 + C(4, 1)a^3b^1 + C(4, 2)a^2b^2 + C(4, 3)a^1b^3 + C(4, 4)a^0b^4. Substituting a = 2 and b = 3 gives 16 + 4*6 + 6*9 + 4*27 + 81 = 81.
Correct Answer:
A
— 81
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Q. Find the value of k if the coefficient of x^2 in the expansion of (x + k)^4 is 6.
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Solution
The coefficient of x^2 in (x + k)^4 is C(4, 2) * k^2 = 6. Thus, 6k^2 = 6, giving k^2 = 1, so k = 1 or -1.
Correct Answer:
B
— 2
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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 binomial coefficient C(7, 4).
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Solution
C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7*6*5)/(3*2*1) = 35.
Correct Answer:
B
— 35
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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. If the expansion of (x + y)^n has 15 terms, what is the value of n?
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Solution
The number of terms in the expansion of (x + y)^n is n + 1. Therefore, n + 1 = 15, which gives n = 14.
Correct Answer:
A
— 14
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Q. In the expansion of (2 + 3x)^5, what is the coefficient of x?
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Solution
The coefficient of x is given by C(5, 1) * (2)^4 * (3) = 5 * 16 * 3 = 240.
Correct Answer:
A
— 15
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Q. In the expansion of (2 - x)^5, what is the coefficient of x^3?
A.
-80
B.
-60
C.
60
D.
80
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Solution
The coefficient of x^3 in (2 - x)^5 is given by 5C3 * 2^2 * (-1)^3 = 10 * 4 * (-1) = -40.
Correct Answer:
A
— -80
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Q. In the expansion of (2x + 5)^3, what is the coefficient of x^2?
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Solution
The coefficient of x^2 in (2x + 5)^3 is given by 3C2 * (2x)^2 * 5^1 = 3 * 4 * 5 = 60.
Correct Answer:
B
— 60
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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
Show solution
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 - 2)^4, what is the coefficient of x^1?
A.
-48
B.
-72
C.
72
D.
48
Show solution
Solution
The coefficient of x^1 in (3x - 2)^4 is given by 4C3 * (3x)^1 * (-2)^3 = 4 * 3 * (-8) = -96.
Correct Answer:
B
— -72
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Q. In the expansion of (3x - 2)^5, what is the coefficient of x^3?
A.
-240
B.
-360
C.
360
D.
240
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Solution
The coefficient of x^3 in (3x - 2)^5 is given by 5C2 * (3)^3 * (-2)^2 = 10 * 27 * 4 = -1080.
Correct Answer:
B
— -360
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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
Show solution
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)^6, what is the coefficient of a^2b^4?
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Solution
The coefficient of a^2b^4 in (a + b)^6 is given by 6C2 = 15.
Correct Answer:
B
— 30
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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 + 2)^5, what is the coefficient of x^4?
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Solution
The coefficient of x^4 is given by C(5, 4)(2)^1 = 5 * 2 = 10.
Correct Answer:
B
— 10
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Q. In the expansion of (x + 2)^8, what is the coefficient of x^6?
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Solution
The coefficient of x^6 in (x + 2)^8 is C(8, 6) * (2)^2 = 28 * 4 = 112.
Correct Answer:
C
— 84
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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
Show solution
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
Show solution
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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