Q. Differentiate f(x) = 4x^2 + 3x - 5. (2019)
-
A.
8x + 3
-
B.
4x + 3
-
C.
2x + 3
-
D.
8x - 3
Solution
Using the power rule, f'(x) = 8x + 3.
Correct Answer:
A
— 8x + 3
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Q. Differentiate f(x) = 4x^5 - 2x^3 + x. (2022)
-
A.
20x^4 - 6x^2 + 1
-
B.
20x^4 - 6x^2
-
C.
4x^4 - 2x^2 + 1
-
D.
5x^4 - 2x^2
Solution
Using the power rule, f'(x) = 20x^4 - 6x^2 + 1.
Correct Answer:
A
— 20x^4 - 6x^2 + 1
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Q. Differentiate f(x) = ln(x^2 + 1). (2022)
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2x/(x^2 - 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Differentiate f(x) = x^2 * e^x. (2022)
-
A.
x^2 * e^x + 2x * e^x
-
B.
2x * e^x + x^2 * e^x
-
C.
x^2 * e^x + e^x
-
D.
2x * e^x
Solution
Using the product rule, f'(x) = x^2 * e^x + 2x * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Differentiate f(x) = x^2 * ln(x).
-
A.
2x * ln(x) + x
-
B.
x * ln(x) + 2x
-
C.
2x * ln(x)
-
D.
x^2/x
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. Differentiate the function f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
2/(x^2 + 1)
-
C.
1/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Differentiate the function f(x) = x^2 * e^x.
-
A.
x^2 * e^x + 2x * e^x
-
B.
2x * e^x + x^2 * e^x
-
C.
x^2 * e^x + e^x
-
D.
2x * e^x + e^x
Solution
Using the product rule, f'(x) = (x^2)' * e^x + x^2 * (e^x)' = 2x * e^x + x^2 * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Evaluate the integral ∫ (3x^2 - 4) dx.
-
A.
x^3 - 4x + C
-
B.
x^3 - 2x + C
-
C.
3x^3 - 4x + C
-
D.
x^3 - 4x
Solution
The integral evaluates to x^3 - 4x + C, where C is the constant of integration.
Correct Answer:
A
— x^3 - 4x + C
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Q. Evaluate the integral ∫ (4x^3 - 2x) dx.
-
A.
x^4 - x^2 + C
-
B.
x^4 - x^2
-
C.
x^4 - x^2 + 2C
-
D.
4x^4 - x^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 the integral ∫ (5x^4) dx.
-
A.
x^5 + C
-
B.
x^5 + 5C
-
C.
x^5 + 1
-
D.
5x^5 + C
Solution
The integral is (5/5)x^5 + C = x^5 + C.
Correct Answer:
A
— x^5 + C
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Q. Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)
-
A.
1/3
-
B.
1/2
-
C.
2/3
-
D.
1
Solution
∫(0 to 1) (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer:
C
— 2/3
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Q. Evaluate the integral ∫(0 to π) sin(x) dx. (2021)
Solution
∫(0 to π) sin(x) dx = [-cos(x)] from 0 to π = -(-1 - 1) = 2.
Correct Answer:
C
— 2
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Q. Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Solution
∫(1 to 2) (3x^2 - 4) dx = [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Correct Answer:
A
— 1
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Q. Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)
Solution
∫(1 to 3) (3x^2 - 2) dx = [x^3 - 2x] from 1 to 3 = (27 - 6) - (1 - 2) = 20.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(1 to 4) (2x + 1) dx. (2021)
Solution
∫(1 to 4) (2x + 1) dx = [x^2 + x] from 1 to 4 = (16 + 4) - (1 + 1) = 18.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(2 to 3) (x^3 - 3x^2 + 2) dx. (2023)
Solution
∫(2 to 3) (x^3 - 3x^2 + 2) dx = [x^4/4 - x^3 + 2x] from 2 to 3 = (81/4 - 27 + 6) - (16/4 - 8 + 4) = 1.
Correct Answer:
B
— 2
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Q. Evaluate the integral ∫(2x + 3) dx from 1 to 2.
Solution
The integral evaluates to [x^2 + 3x] from 1 to 2, which gives (4 + 6) - (1 + 3) = 8 - 4 = 4.
Correct Answer:
B
— 7
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Q. Evaluate the integral ∫(2x + 3) dx. (2021)
-
A.
x^2 + 3x + C
-
B.
x^2 + 3x
-
C.
2x^2 + 3x + C
-
D.
2x^2 + 3x
Solution
The integral of (2x + 3) is (2x^2/2) + 3x + C = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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Q. Evaluate the integral ∫(sin x)dx. (2022)
-
A.
-cos x + C
-
B.
cos x + C
-
C.
sin x + C
-
D.
-sin x + C
Solution
The integral of sin x is -cos x + C.
Correct Answer:
A
— -cos x + C
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Q. Evaluate the integral ∫(x^2 - 2x + 1) dx. (2022)
-
A.
(1/3)x^3 - x^2 + x + C
-
B.
(1/3)x^3 - x^2 + C
-
C.
(1/3)x^3 - 2x + C
-
D.
(1/3)x^3 - x^2 + x
Solution
The integral of (x^2 - 2x + 1) is (1/3)x^3 - x^2 + x + C.
Correct Answer:
A
— (1/3)x^3 - x^2 + x + C
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
-
A.
5, Continuous
-
B.
0, Not continuous
-
C.
5, Not continuous
-
D.
0, Continuous
Solution
Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5, and the function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.
-
A.
5, Continuous
-
B.
0, Continuous
-
C.
5, Not Continuous
-
D.
0, Not Continuous
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
-
A.
5, Continuous
-
B.
5, Discontinuous
-
C.
0, Continuous
-
D.
0, Discontinuous
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0 if defined as f(0) = 5.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.
-
A.
1, Continuous
-
B.
0, Continuous
-
C.
1, Discontinuous
-
D.
0, Discontinuous
Solution
The limit lim (x -> 0) (sin(x)/x) = 1. Since the limit exists and equals the function value at x = 0, it is continuous.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?
-
A.
1, Continuous
-
B.
0, Continuous
-
C.
1, Discontinuous
-
D.
0, Discontinuous
Solution
The limit is 1, and if we define f(0) = 1, then f(x) is continuous at x = 0.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)
-
A.
0, Yes
-
B.
0, No
-
C.
6, Yes
-
D.
6, No
Solution
lim (x -> 3) (x^2 - 9)/(x - 3) = lim (x -> 3) (x + 3) = 6. The function is not defined at x = 3, hence not continuous.
Correct Answer:
C
— 6, Yes
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Q. Evaluate the limit lim x→2 (x^2 - 4)/(x - 2).
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
Using L'Hôpital's Rule, lim x→2 (x^2 - 4)/(x - 2) = lim x→2 (2x)/(1) = 4.
Correct Answer:
C
— 4
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Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
180 degrees
Solution
cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.
Correct Answer:
A
— 0 degrees
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Q. Find the angle between the vectors A = i + j and B = i - j.
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
135 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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