Binomial Theorem
Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^4. (2020)
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 value of (3 + 2)^3 using the binomial theorem.
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^0?
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)^5, what is the coefficient of x^2? (2021)
-
A.
-300
-
B.
-600
-
C.
600
-
D.
300
Solution
The coefficient of x^2 is C(5,2) * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.
Correct Answer: A — -300
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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)
-
A.
-150
-
B.
-300
-
C.
300
-
D.
150
Solution
The coefficient of x^2 is C(6,2) * (2)^2 * (-5)^4 = 15 * 4 * 625 = -37500.
Correct Answer: A — -150
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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)
-
A.
-600
-
B.
-720
-
C.
720
-
D.
600
Solution
The coefficient of x^3 is C(6,3)(2)^3(-5)^3 = 20 * 8 * -125 = -20000.
Correct Answer: B — -720
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Q. In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)
-
A.
540
-
B.
810
-
C.
729
-
D.
486
Solution
The coefficient of x^4 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: A — 540
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Q. In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)
-
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: B — -216
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Q. In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)
Solution
The coefficient of x^3 is C(4,3) * (3)^1 = 4 * 3 = 12.
Correct Answer: A — 36
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Q. In the expansion of (x - 1)^5, what is the coefficient of x^3?
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: A — -10
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Q. What is the 3rd term in the expansion of (2x + 3)^4?
-
A.
108x^2
-
B.
216x^2
-
C.
324x^2
-
D.
432x^2
Solution
The 3rd term is given by C(4,2) * (2x)^2 * (3)^2 = 6 * 4x^2 * 9 = 216x^2.
Correct Answer: B — 216x^2
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Q. What is the 3rd term in the expansion of (2x + 5)^6? (2000)
-
A.
600x^4
-
B.
1500x^4
-
C.
1800x^4
-
D.
2000x^4
Solution
The 3rd term is given by C(6,2) * (2x)^2 * (5)^4 = 15 * 4 * 625 = 37500.
Correct Answer: B — 1500x^4
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Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)
-
A.
112
-
B.
128
-
C.
256
-
D.
64
Solution
The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.
Correct Answer: A — 112
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Q. What is the 3rd term in the expansion of (x + 3)^5? (2023)
-
A.
45
-
B.
90
-
C.
135
-
D.
180
Solution
The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.
Correct Answer: B — 90
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Q. What is the 3rd term in the expansion of (x + 3)^7? (2023)
-
A.
189
-
B.
441
-
C.
729
-
D.
1024
Solution
The 3rd term is C(7,2) * (3)^2 * (x)^5 = 21 * 9 * x^5 = 189.
Correct Answer: B — 441
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Q. What is the 3rd term in the expansion of (x + 4)^5? (2023)
-
A.
80x^3
-
B.
160x^3
-
C.
240x^3
-
D.
320x^3
Solution
The 3rd term is given by C(5,2) * (4)^2 * (x)^3 = 10 * 16 * x^3 = 160x^3.
Correct Answer: C — 240x^3
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Q. What is the 3rd term in the expansion of (x + 4)^6? (2020)
-
A.
240x^4
-
B.
360x^4
-
C.
480x^4
-
D.
600x^4
Solution
The 3rd term is C(6,2) * (4)^2 * (x)^4 = 15 * 16 * x^4 = 240x^4.
Correct Answer: B — 360x^4
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Q. What is the 4th term in the expansion of (2x - 3)^6? (2020)
-
A.
-540
-
B.
540
-
C.
-720
-
D.
720
Solution
The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.
Correct Answer: A — -540
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Q. What is the 5th term in the expansion of (2 + 3x)^6? (2023)
-
A.
486
-
B.
540
-
C.
729
-
D.
810
Solution
The 5th term is given by 6C4 * (2)^(6-4) * (3x)^4 = 15 * 4 * 81x^4 = 4860x^4.
Correct Answer: B — 540
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Q. What is the 5th term in the expansion of (3x - 2)^7? (2023)
-
A.
-1680x^3
-
B.
1680x^3
-
C.
-2520x^3
-
D.
2520x^3
Solution
The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.
Correct Answer: A — -1680x^3
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Q. What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)
-
A.
-243
-
B.
243
-
C.
-81
-
D.
81
Solution
The constant term is C(5,5)(-3)^5 = -243.
Correct Answer: A — -243
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Q. What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)
Solution
The coefficient of x^0 is C(5,5) * (3x)^0 * (2)^5 = 1 * 32 = 32.
Correct Answer: C — 64
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Q. What is the coefficient of x^0 in the expansion of (x + 1/x)^8? (2021)
Solution
The coefficient of x^0 occurs when k=4, which gives us 8C4 = 70.
Correct Answer: C — 70
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Q. What is the coefficient of x^1 in the expansion of (x + 1)^10? (2021)
Solution
The coefficient of x^1 is C(10,1)(1)^9 = 10.
Correct Answer: A — 10
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Q. What is the coefficient of x^1 in the expansion of (x - 1)^5?
Solution
The coefficient of x^1 is given by 5C1 * (-1)^4 = 5 * 1 = 5.
Correct Answer: A — -5
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Q. What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)
Solution
The coefficient of x^2 is C(5,2) * (2)^2 * (-1)^3 = 10 * 4 * (-1) = -40.
Correct Answer: B — 20
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Q. What is the coefficient of x^2 in the expansion of (x + 4)^5?
-
A.
80
-
B.
100
-
C.
120
-
D.
160
Solution
The coefficient of x^2 is C(5,2) * 4^3 = 10 * 64 = 640.
Correct Answer: A — 80
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Q. What is the coefficient of x^2 in the expansion of (x + 5)^4?
-
A.
50
-
B.
100
-
C.
150
-
D.
200
Solution
The coefficient of x^2 is C(4,2) * (5)^2 = 6 * 25 = 150.
Correct Answer: B — 100
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Q. What is the coefficient of x^2 in the expansion of (x - 2)^5?
-
A.
-40
-
B.
-20
-
C.
20
-
D.
40
Solution
The coefficient of x^2 is given by 5C2 * (-2)^3 = 10 * (-8) = -80.
Correct Answer: A — -40
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Q. What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000
-
A.
600
-
B.
720
-
C.
800
-
D.
900
Solution
The coefficient of x^3 is C(6,3) * (5)^3 * (2)^3 = 20 * 125 * 8 = 20000.
Correct Answer: B — 720
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