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(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 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^0 in the expansion of (2x - 3)^3.
-
A.
-27
-
B.
-24
-
C.
-18
-
D.
-12
Solution
The coefficient of x^0 is (-3)^3 = -27.
Correct Answer:
A
— -27
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Q. Find the coefficient of x^1 in the expansion of (x + 4)^3.
Solution
The coefficient of x^1 is C(3,1) * (4)^2 = 3 * 16 = 48.
Correct Answer:
A
— 12
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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^3 in the expansion of (x - 2)^5.
-
A.
-40
-
B.
-80
-
C.
-60
-
D.
-100
Solution
The coefficient of x^3 is C(5,3) * (-2)^2 = 10 * 4 = 40.
Correct Answer:
A
— -40
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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 + 1)^6.
Solution
The coefficient of x^4 is C(6,4) * 1^2 = 15.
Correct Answer:
B
— 20
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Q. Find the coefficient of x^4 in the expansion of (x + 2)^6.
Solution
The coefficient of x^4 is C(6,4) * (2)^2 = 15 * 4 = 60.
Correct Answer:
B
— 30
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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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