Q. Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)
-
A.
0
-
B.
1/6
-
C.
1/3
-
D.
1/2
Solution
Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.
Correct Answer: B — 1/6
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Q. Evaluate the limit: lim (x -> 0) (x^3)/(sin(x)) (2022)
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.
Correct Answer: A — 0
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Q. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)
-
A.
3
-
B.
6
-
C.
9
-
D.
Undefined
Solution
Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.
Correct Answer: B — 6
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Q. Evaluate ∫ (2x + 3) dx. (2022)
-
A.
x^2 + 3x + C
-
B.
x^2 + 3 + C
-
C.
x^2 + 3x + 1
-
D.
2x^2 + 3 + C
Solution
The integral is (2/2)x^2 + 3x + C = x^2 + 3x + C.
Correct Answer: A — x^2 + 3x + C
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Q. Evaluate ∫ (4x^3 - 2x) dx. (2019)
-
A.
x^4 - x^2 + C
-
B.
x^4 - x^2 + 2C
-
C.
x^4 - x + C
-
D.
4x^4 - 2x^2 + C
Solution
The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.
Correct Answer: A — x^4 - x^2 + C
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Q. Evaluate ∫ (5 - 3x) dx. (2022)
-
A.
5x - (3/2)x^2 + C
-
B.
5x - (3/3)x^2 + C
-
C.
5x - (3/4)x^2 + C
-
D.
5x - (3/5)x^2 + C
Solution
The integral is 5x - (3/2)x^2 + C.
Correct Answer: A — 5x - (3/2)x^2 + C
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Q. Evaluate ∫(2x^2 + 3x + 1)dx. (2021)
-
A.
(2/3)x^3 + (3/2)x^2 + x + C
-
B.
(2/3)x^3 + (3/2)x + C
-
C.
(2/3)x^3 + (3/2)x^2 + C
-
D.
(2/3)x^3 + 3x + C
Solution
Integrating term by term: ∫2x^2dx = (2/3)x^3, ∫3xdx = (3/2)x^2, and ∫1dx = x. Thus, ∫(2x^2 + 3x + 1)dx = (2/3)x^3 + (3/2)x^2 + x + C.
Correct Answer: A — (2/3)x^3 + (3/2)x^2 + x + C
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Q. Evaluate ∫(5x^4)dx. (2020)
-
A.
(5/5)x^5 + C
-
B.
(1/5)x^5 + C
-
C.
(5/4)x^4 + C
-
D.
(1/4)x^4 + C
Solution
The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.
Correct Answer: A — (5/5)x^5 + C
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Q. Evaluate ∫(6x^2 + 3)dx. (2022)
-
A.
2x^3 + 3x + C
-
B.
2x^3 + 3 + C
-
C.
2x^3 + 3x^2 + C
-
D.
2x^3 + 3x^3 + C
Solution
Integrating term by term: ∫6x^2dx = 2x^3 and ∫3dx = 3x. Thus, ∫(6x^2 + 3)dx = 2x^3 + 3x + C.
Correct Answer: A — 2x^3 + 3x + C
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Q. Find the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)
Solution
Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.
Correct Answer: A — 0.5
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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: A — 6
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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)
Solution
The points are collinear, hence the area = 0.
Correct Answer: A — 0
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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
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
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
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
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^3 in the expansion of (x - 1)^7.
-
A.
-35
-
B.
-21
-
C.
-7
-
D.
-49
Solution
The coefficient of x^3 is given by 7C3 * (-1)^4 = 35.
Correct Answer: A — -35
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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^5.
-
A.
-240
-
B.
-360
-
C.
-480
-
D.
-600
Solution
The coefficient of x^4 is C(5,4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Correct Answer: A — -240
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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
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
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
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^4 in the expansion of (x + 5)^7.
-
A.
210
-
B.
1260
-
C.
1750
-
D.
2450
Solution
The coefficient of x^4 is given by C(7,4) * 5^3 = 35 * 125 = 4375.
Correct Answer: B — 1260
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Q. Find the coefficient of x^4 in the expansion of (x - 1)^5.
Solution
The coefficient of x^4 is C(5,4) * (-1)^1 = 5 * (-1) = -5.
Correct Answer: C — -10
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Q. Find the coefficient of x^4 in the expansion of (x - 5)^6.
-
A.
150
-
B.
200
-
C.
250
-
D.
300
Solution
The coefficient of x^4 is given by C(6,4) * (-5)^2 = 15 * 25 = 375.
Correct Answer: B — 200
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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
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
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
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.
Solution
The coefficient of x^5 is C(7,5) = 21.
Correct Answer: C — 35
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Q. Find the coefficient of x^5 in the expansion of (x + 2)^7.
Solution
The coefficient of x^5 is C(7,5) * 2^2 = 21 * 4 = 84.
Correct Answer: C — 56
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Q. Find the constant term in the expansion of (3x - 4/x)^5.
Solution
The constant term occurs when the power of x is zero. The term is given by 5C2 * (3x)^2 * (-4/x)^3 = 10 * 9 * (-64) = -5760.
Correct Answer: A — -64
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