Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6 units. (2022)
-
A.
√49
-
B.
√45
-
C.
√36
-
D.
√50
Solution
Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 units.
Correct Answer:
A
— √49
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Q. Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2/sin(x)) = lim (x -> 0) (x^2/x) = lim (x -> 0) x = 0.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2023)
-
A.
0
-
B.
1
-
C.
4
-
D.
Undefined
Solution
Factoring gives ((x - 1)(x^3 + x^2 + x + 1))/(x - 1). For x ≠ 1, this simplifies to x^3 + x^2 + x + 1. Thus, lim (x -> 1) = 4.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> 2) (x^2 + 3x - 10)/(x - 2) (2021)
Solution
Factoring gives (x - 2)(x + 5)/(x - 2). For x ≠ 2, this simplifies to x + 5. Evaluating at x = 2 gives 7.
Correct Answer:
D
— 7
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Q. Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)
-
A.
1
-
B.
2
-
C.
0
-
D.
Undefined
Solution
The expression is undefined at x=2. The limit does not exist as the function approaches infinity.
Correct Answer:
D
— Undefined
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Q. Find the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2023)
Solution
The expression can be factored as ((x - 3)(x + 3))/(x - 3). For x ≠ 3, this simplifies to x + 3. Thus, lim (x -> 3) (x + 3) = 6.
Correct Answer:
A
— 0
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Q. Find the local maxima of f(x) = -x^2 + 4x + 1. (2020)
Solution
The maximum occurs at x = -b/(2a) = -4/(2*-1) = 2. f(2) = -2^2 + 4(2) + 1 = 5.
Correct Answer:
B
— 5
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Q. Find the local maxima of f(x) = -x^3 + 3x^2 + 1. (2020)
-
A.
(0, 1)
-
B.
(1, 3)
-
C.
(2, 5)
-
D.
(3, 1)
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. f(2) = 5.
Correct Answer:
B
— (1, 3)
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Q. Find the local maximum of f(x) = -x^3 + 3x^2 + 4. (2020)
Solution
Set f'(x) = 0 to find critical points. The local maximum occurs at x = 2. f(2) = 5.
Correct Answer:
B
— 5
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Q. Find the magnitude of the vector A = 3i - 4j. (2020)
Solution
|A| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the maximum area of a triangle with a base of 10 m and height varying. (2020)
Solution
Area = 1/2 * base * height. Max area occurs when height is maximized, thus Area = 1/2 * 10 * 10 = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a base of 10 units and height as a function of the base. (2021)
Solution
Area = 1/2 * base * height. Max area occurs when height is maximized at 10 units, giving Area = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a base of 10 units and height as a function of x. (2022)
Solution
Area = 1/2 * base * height. Max area occurs when height is maximized, which is 10 units, giving Area = 50.
Correct Answer:
B
— 50
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Q. Find the maximum area of a triangle with a fixed perimeter of 30 cm. (2022)
-
A.
75 cm²
-
B.
100 cm²
-
C.
50 cm²
-
D.
60 cm²
Solution
For maximum area, the triangle should be equilateral. Area = (sqrt(3)/4) * (10)^2 = 75 cm².
Correct Answer:
A
— 75 cm²
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Q. Find the maximum height of the projectile modeled by h(t) = -16t^2 + 32t + 48. (2020)
Solution
The maximum occurs at t = -b/(2a) = -32/(2*-16) = 1. h(1) = 64.
Correct Answer:
A
— 48
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Q. Find the maximum height of the projectile modeled by h(t) = -16t^2 + 64t + 48. (2020)
Solution
The maximum occurs at t = -b/(2a) = 64/(2*16) = 2. h(2) = -16(2^2) + 64(2) + 48 = 80.
Correct Answer:
B
— 64
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Q. Find the maximum value of the function f(x) = -2x^2 + 8x - 3. (2021) 2021
Solution
The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = -8/(2*-2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 8.
Correct Answer:
B
— 8
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Q. Find the midpoint of the line segment joining the points (2, 3) and (4, 7). (2022) 2022
-
A.
(3, 5)
-
B.
(2, 5)
-
C.
(4, 5)
-
D.
(3, 4)
Solution
Midpoint = ((2+4)/2, (3+7)/2) = (3, 5).
Correct Answer:
A
— (3, 5)
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Q. Find the minimum value of f(x) = 4x^2 - 16x + 20. (2022)
Solution
The vertex gives the minimum at x = 2. f(2) = 4(2^2) - 16(2) + 20 = 4.
Correct Answer:
A
— 4
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Q. Find the minimum value of f(x) = x^2 - 4x + 6. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2.
Correct Answer:
A
— 2
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Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.
Correct Answer:
A
— 3
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Q. Find the minimum value of f(x) = x^2 - 4x + 7. (2021) 2021
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4*2 + 7 = 3.
Correct Answer:
A
— 3
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Q. Find the minimum value of the function f(x) = 2x^2 - 8x + 10. (2022)
Solution
The minimum occurs at x = 2. f(2) = 2(2^2) - 8(2) + 10 = 6.
Correct Answer:
B
— 4
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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(x)
-
D.
y = 2e^(x)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(2x)
-
D.
y = 2e^(x/4)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)
-
A.
(1, 4)
-
B.
(2, 3)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. f(2) = 3.
Correct Answer:
C
— (3, 0)
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Q. Find the point of intersection of the lines y = x + 2 and y = -x + 4. (2023)
-
A.
(1, 3)
-
B.
(2, 4)
-
C.
(3, 5)
-
D.
(0, 2)
Solution
Setting x + 2 = -x + 4 gives 2x = 2, so x = 1. Substituting x back gives y = 3. Thus, the point is (1, 3).
Correct Answer:
A
— (1, 3)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 that has a horizontal tangent. (2023)
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 2)
-
D.
(3, 4)
Solution
To find horizontal tangents, set the derivative y' = 3x^2 - 6x = 0. This gives x = 0 and x = 2. The point (1, 2) has a horizontal tangent.
Correct Answer:
B
— (1, 2)
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Q. Find the point on the curve y = x^3 - 3x^2 + 4 where the tangent is horizontal. (2023)
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 2)
-
D.
(3, 4)
Solution
To find horizontal tangents, set dy/dx = 0. dy/dx = 3x^2 - 6x = 0 gives x = 0 and x = 2. At x = 1, y = 2.
Correct Answer:
B
— (1, 2)
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Q. Find the real part of the complex number 4 + 5i. (2023)
Solution
The real part of the complex number 4 + 5i is 4.
Correct Answer:
A
— 4
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