MHT-CET
Q. Calculate the value of 6^2 - 4^2. (2023) 2023
Solution
6^2 = 36 and 4^2 = 16, so 36 - 16 = 20.
Correct Answer: A — 20
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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
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. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(6, 0, 0), and C(0, 8, 0). (2023)
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A.
(2, 2, 0)
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B.
(2, 3, 0)
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C.
(3, 2, 0)
-
D.
(0, 0, 0)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3, (0+0+0)/3) = (2, 2.67, 0).
Correct Answer: A — (2, 2, 0)
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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)
-
A.
(1, 1, 0)
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B.
(2, 1, 0)
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C.
(4/3, 1, 0)
-
D.
(0, 1, 0)
Solution
Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).
Correct Answer: B — (2, 1, 0)
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Q. Determine the critical points of f(x) = e^x - 2x. (2021)
Solution
f'(x) = e^x - 2. Setting f'(x) = 0 gives e^x = 2, so x = ln(2).
Correct Answer: B — 1
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Q. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.
Solution
Using the formula for distance from a point to a line, d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), we find d = |2(3) + 3(4) - 12| / sqrt(2^2 + 3^2) = 3.
Correct Answer: B — 3
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Q. Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)
-
A.
(-∞, 0)
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B.
(0, 2)
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C.
(2, ∞)
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D.
(0, 4)
Solution
f''(x) = -2, which is always negative, indicating concave down everywhere.
Correct Answer: C — (2, ∞)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)
-
A.
(-∞, 0)
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B.
(0, 2)
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C.
(2, ∞)
-
D.
(0, 4)
Solution
f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). f'(x) > 0 for x in (0, 3).
Correct Answer: B — (0, 2)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has local minima. (2020)
-
A.
(0, 2)
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B.
(1, 3)
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C.
(2, 4)
-
D.
(0, 1)
Solution
f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).
Correct Answer: A — (0, 2)
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Q. Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)
-
A.
(0, 1)
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B.
(1, 3)
-
C.
(2, 5)
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D.
(3, 4)
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.
Correct Answer: B — (1, 3)
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Q. Determine the local minima of f(x) = x^4 - 4x^2. (2021)
Solution
f'(x) = 4x^3 - 8x. Setting f'(x) = 0 gives x = 0, ±2. f(0) = 0.
Correct Answer: B — 0
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Q. Determine the maximum area of a triangle with a base of 10 units and height as a function of x. (2020)
Solution
Area = 1/2 * base * height = 5h. Max area occurs when h is maximized, thus Area = 50 when h = 10.
Correct Answer: B — 50
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Q. Determine the maximum height of the function f(x) = -x^2 + 6x + 5. (2020) 2020
Solution
The vertex occurs at x = 3. f(3) = -3^2 + 6*3 + 5 = 8.
Correct Answer: A — 8
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Q. Determine the minimum value of f(x) = x^2 - 4x + 5. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.
Correct Answer: A — 1
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Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)
-
A.
(1, 3)
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B.
(2, 2)
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C.
(0, 6)
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D.
(3, 0)
Solution
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.
Correct Answer: A — (1, 3)
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Q. Determine the product of the roots of the equation x² + 6x + 9 = 0. (2021)
Solution
The product of the roots is c/a = 9/1 = 9.
Correct Answer: A — 9
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Q. Determine the roots of the equation x² + 2x - 8 = 0. (2023)
-
A.
-4 and 2
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B.
4 and -2
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C.
2 and -4
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D.
0 and 8
Solution
Factoring gives (x + 4)(x - 2) = 0, hence the roots are -4 and 2.
Correct Answer: A — -4 and 2
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Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)
-
A.
x^3 + x^2 + C
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B.
x^3 + x^2 + 2C
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C.
x^3 + x^2 + 1
-
D.
x^3 + 2x + C
Solution
The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.
Correct Answer: A — x^3 + x^2 + C
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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)
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A.
x^3 + 2x + C
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B.
x^3 + 2x^2 + C
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C.
x^3 + 2x^3 + C
-
D.
3x^3 + 2x + C
Solution
Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Correct Answer: A — x^3 + 2x + C
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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)
-
A.
0
-
B.
1
-
C.
∞
-
D.
Undefined
Solution
Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.
Correct Answer: B — 1
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Q. Evaluate ∫ (2x + 3) dx. (2022)
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A.
x^2 + 3x + C
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B.
x^2 + 3 + C
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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)
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A.
x^4 - x^2 + C
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B.
x^4 - x^2 + 2C
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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)
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A.
5x - (3/2)x^2 + C
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B.
5x - (3/3)x^2 + C
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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 ∫(5x^4)dx. (2020)
-
A.
(5/5)x^5 + C
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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. 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^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^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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