Binomial Theorem
Q. Calculate the coefficient of x^2 in the expansion of (x + 1/2)^6.
A.
15/4
B.
45/8
C.
15/8
D.
5/4
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Solution
The coefficient of x^2 is C(6,2) * (1/2)^2 = 15 * 1/4 = 15/4.
Correct Answer: B — 45/8
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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/2)^8. (2021)
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Solution
The coefficient of x^2 is C(8,2) * (1/2)^6 = 28 * 1/64 = 28/64 = 7/16.
Correct Answer: C — 70
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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)
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Solution
The coefficient of x^2 is given by 6C2 * (1)^4 = 15.
Correct Answer: A — 15
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Q. Calculate the coefficient of x^4 in the expansion of (3x - 2)^6.
A.
540
B.
720
C.
810
D.
960
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Solution
The coefficient of x^4 is C(6,4) * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: A — 540
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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)
A.
15/8
B.
45/8
C.
5/8
D.
1/8
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Solution
The coefficient of x^4 is C(6,4)(1/2)^2 = 15 * 1/4 = 15/4.
Correct Answer: B — 45/8
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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)
A.
54
B.
81
C.
108
D.
729
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Solution
The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.
Correct Answer: C — 108
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Q. Calculate the term containing x^3 in the expansion of (x + 2)^7.
A.
56
B.
84
C.
112
D.
128
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Solution
The term containing x^3 is C(7,3) * (2)^4 = 35 * 16 = 560.
Correct Answer: B — 84
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Q. Calculate the term independent of x in the expansion of (2x - 3)^5.
A.
-243
B.
0
C.
243
D.
81
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Solution
The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.
Correct Answer: A — -243
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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.
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Solution
The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.
Correct Answer: B — 24
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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^6.
A.
729
B.
729/64
C.
729/32
D.
729/16
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Solution
The term independent of x occurs when k = 3, which gives C(6,3) * (x/2)^3 * (-3)^3 = 20 * (1/8) * (-27) = -67.5.
Correct Answer: B — 729/64
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Q. Determine the coefficient of x^4 in the expansion of (2x - 3)^6.
A.
540
B.
720
C.
810
D.
960
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Solution
The coefficient of x^4 is given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: B — 720
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Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.
A.
540
B.
720
C.
810
D.
960
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Solution
The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.
Correct Answer: A — 540
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)
A.
80
B.
100
C.
120
D.
160
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Solution
The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.
Correct Answer: A — 80
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Q. Find the coefficient of x^2 in the expansion of (x - 5)^5.
A.
100
B.
150
C.
200
D.
250
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Solution
The coefficient of x^2 is C(5,2) * (-5)^3 = 10 * (-125) = -1250.
Correct Answer: A — 100
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)
A.
-54
B.
-108
C.
108
D.
54
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Solution
The coefficient of x^3 is C(4,3) * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96.
Correct Answer: B — -108
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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^6.
A.
540
B.
720
C.
810
D.
900
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Solution
The coefficient of x^4 is C(6,4) * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: A — 540
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Q. Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)
A.
240
B.
360
C.
480
D.
600
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Solution
The coefficient of x^4 is C(5,4)(3)^4(2)^1 = 5 * 81 * 2 = 810.
Correct Answer: B — 360
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Q. Find the coefficient of x^4 in the expansion of (x + 3)^6.
A.
81
B.
162
C.
243
D.
729
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Solution
The coefficient of x^4 is C(6,4) * (3)^2 = 15 * 9 = 135.
Correct Answer: C — 243
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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)
A.
-540
B.
540
C.
-720
D.
720
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Solution
The coefficient of x^5 is C(6,5) * (2)^5 * (-3)^1 = 6 * 32 * (-3) = -576.
Correct Answer: A — -540
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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)
A.
168
B.
252
C.
336
D.
504
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Solution
The coefficient of x^5 is C(7,5) * (2)^5 * (-3)^2 = 21 * 32 * 9 = 6048.
Correct Answer: B — 252
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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^8.
A.
-6720
B.
6720
C.
-3360
D.
3360
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Solution
The coefficient of x^5 is C(8,5) * (2)^5 * (-3)^3 = 56 * 32 * (-27) = -6720.
Correct Answer: A — -6720
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.
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Solution
The coefficient of x^5 is C(7,5) = 21.
Correct Answer: C — 35
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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)
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Solution
The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.
Correct Answer: A — -64
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Q. Find the term containing x^3 in the expansion of (x - 1)^5.
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Solution
The term containing x^3 is C(5,3) * x^3 * (-1)^2 = 10 * x^3 * 1 = 10.
Correct Answer: C — -10
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Q. Find the term independent of x in the expansion of (x^2 - 2x + 3)^4. (2022)
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Solution
The term independent of x occurs when the powers of x cancel out. The term is 81.
Correct Answer: A — 81
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Q. Find the term independent of x in the expansion of (x^2 - 3x + 1)^5. (2023)
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Solution
The term independent of x occurs when the powers of x cancel out. The term is C(5,2)(-3)^2(1)^3 = 45.
Correct Answer: A — -15
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Q. Find the term independent of x in the expansion of (x^2 - 4x + 4)^4. (2020)
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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.
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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?
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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
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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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