Q. If D = [[2, 1], [1, 2]], what is the trace of D?
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Solution
The trace of a matrix is the sum of its diagonal elements. Trace(D) = 2 + 2 = 4.
Correct Answer:
C
— 3
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Q. If D = [[4, 2], [1, 3]], find the inverse of D. (2022)
A.
[[3, -2], [-1, 4]]
B.
[[3, 2], [-1, 4]]
C.
[[3, -2], [1, 4]]
D.
[[4, -2], [-1, 3]]
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Solution
The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10, and adj(D) = [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Correct Answer:
A
— [[3, -2], [-1, 4]]
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Q. If D = [[4, 2], [1, 3]], what is the inverse of D?
A.
[[3, -2], [-1, 4]]
B.
[[3, 2], [-1, 4]]
C.
[[4, -2], [-1, 3]]
D.
[[3, -4], [1, 2]]
Show solution
Solution
The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Correct Answer:
A
— [[3, -2], [-1, 4]]
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Q. If E = [[1, 2], [2, 4]], what can be said about the matrix E? (2023)
A.
Invertible
B.
Singular
C.
Non-square
D.
Diagonal
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Solution
Matrix E is singular because its determinant is 0 (1*4 - 2*2 = 0).
Correct Answer:
B
— Singular
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Q. If F = [[1, 0], [0, 1]], what is F^(-1)?
A.
[[1, 0], [0, 1]]
B.
[[0, 1], [1, 0]]
C.
[[1, 1], [1, 1]]
D.
[[0, 0], [0, 0]]
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Solution
The inverse of the identity matrix F is itself, so F^(-1) = F.
Correct Answer:
A
— [[1, 0], [0, 1]]
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Q. If F = [[1, 2], [2, 4]], what is the determinant of F? (2021)
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Solution
The determinant of F is calculated as (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer:
A
— 0
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Q. If F = [[1, 2], [2, 4]], what is the rank of F?
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Solution
The second row is a multiple of the first row, so there is only one linearly independent row. Therefore, the rank of F is 1.
Correct Answer:
A
— 1
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Q. If F = [[1, 2], [3, 5]], what is the trace of F? (2020)
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Solution
The trace of F is the sum of the diagonal elements: 1 + 5 = 6.
Correct Answer:
D
— 8
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Q. If f(x) = 3x + 2, what is the value of f(1) and is it continuous?
A.
5, Continuous
B.
5, Not Continuous
C.
3, Continuous
D.
3, Not Continuous
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Solution
f(1) = 3(1) + 2 = 5. Since f(x) is a linear function, it is continuous everywhere.
Correct Answer:
A
— 5, Continuous
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Q. If f(x) = 3x + 2, what is the value of f(2) and is it continuous?
A.
8, Continuous
B.
8, Discontinuous
C.
7, Continuous
D.
7, Discontinuous
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Solution
f(2) = 3(2) + 2 = 8. Since f(x) is a polynomial, it is continuous everywhere.
Correct Answer:
A
— 8, Continuous
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Q. If f(x) = 3x^2 + 2x, what is f'(2)? (2023)
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Solution
First, find f'(x) = 6x + 2. Then, f'(2) = 6(2) + 2 = 12 + 2 = 14.
Correct Answer:
A
— 10
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Q. If f(x) = 4x^3 - 2x^2 + x, what is f''(x)?
A.
24x - 4
B.
12x - 2
C.
12x - 4
D.
24x - 2
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Solution
First, find f'(x) = 12x^2 - 4x + 1, then differentiate again to get f''(x) = 24x - 4.
Correct Answer:
A
— 24x - 4
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Q. If f(x) = 5x^2 + 3x - 1, what is f''(x)? (2020)
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Solution
The first derivative f'(x) = 10x + 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = 5x^2 + 3x - 1, what is f'(2)? (2020)
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Solution
First, find f'(x) = 10x + 3. Then, f'(2) = 10(2) + 3 = 20 + 3 = 23.
Correct Answer:
A
— 27
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Q. If f(x) = 5x^2 - 3x + 7, what is f''(x)? (2020)
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Solution
The first derivative f'(x) = 10x - 3, and the second derivative f''(x) = 10.
Correct Answer:
A
— 10
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Q. If f(x) = e^x + x^2, what is f'(0)? (2021)
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Solution
f'(x) = e^x + 2x. Thus, f'(0) = e^0 + 2(0) = 1 + 0 = 1.
Correct Answer:
A
— 1
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Q. If f(x) = e^x, what is f''(x)? (2020)
A.
e^x
B.
xe^x
C.
2e^x
D.
0
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Solution
The second derivative f''(x) = d^2/dx^2(e^x) = e^x.
Correct Answer:
A
— e^x
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Q. If f(x) = e^x, what is the value of f''(0)? (2021)
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Solution
f'(x) = e^x and f''(x) = e^x. Therefore, f''(0) = e^0 = 1.
Correct Answer:
A
— 1
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Q. If f(x) = ln(x), what is f'(1)? (2020)
A.
1
B.
0
C.
undefined
D.
ln(1)
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Solution
f'(x) = 1/x. Therefore, f'(1) = 1/1 = 1.
Correct Answer:
A
— 1
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Q. If f(x) = ln(x), what is f'(e)?
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Solution
f'(x) = 1/x. Therefore, f'(e) = 1/e.
Correct Answer:
A
— 1
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Q. If f(x) = ln(x^2 + 1), find f'(1). (2022)
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Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 1.
Correct Answer:
B
— 1
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Q. If f(x) = ln(x^2 + 1), what is f'(x)?
A.
2x/(x^2 + 1)
B.
1/(x^2 + 1)
C.
2/(x^2 + 1)
D.
x/(x^2 + 1)
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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. If f(x) = sin(x) + cos(x), what is f'(x)?
A.
cos(x) - sin(x)
B.
-sin(x) + cos(x)
C.
sin(x) + cos(x)
D.
-cos(x) - sin(x)
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Solution
Using the derivative rules, f'(x) = cos(x) - sin(x).
Correct Answer:
B
— -sin(x) + cos(x)
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Q. If f(x) = sin(x) + cos(x), what is f'(π/4)?
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Solution
f'(x) = cos(x) - sin(x). At x = π/4, f'(π/4) = cos(π/4) - sin(π/4) = √2/2 - √2/2 = 0.
Correct Answer:
C
— 1
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Q. If f(x) = x^2 * e^x, find f'(x). (2019)
A.
e^x(x^2 + 2x)
B.
e^x(x^2 - 2x)
C.
x^2 * e^x
D.
2x * e^x
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Solution
Using the product rule, f'(x) = e^x(x^2 + 2x).
Correct Answer:
A
— e^x(x^2 + 2x)
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Q. If f(x) = x^2 * e^x, what is f'(x)? (2019)
A.
e^x(x^2 + 2x)
B.
e^x(x^2 - 2x)
C.
2xe^x
D.
x^2e^x
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Solution
Using the product rule, f'(x) = e^x(x^2 + 2x).
Correct Answer:
A
— e^x(x^2 + 2x)
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Q. If f(x) = x^2 * ln(x), what is f'(x)? (2022)
A.
2x * ln(x) + x
B.
x * ln(x) + 2x
C.
2x * ln(x) - x
D.
x * ln(x) - 2x
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Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. If f(x) = x^2 + 2x + 1, what is f''(x)? (2023)
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Solution
First derivative f'(x) = 2x + 2. Second derivative f''(x) = 2.
Correct Answer:
A
— 2
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Q. If f(x) = x^2 + 2x + 1, what is f(-1)? Is f(x) continuous at x = -1? (2019)
A.
0, Yes
B.
0, No
C.
1, Yes
D.
1, No
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Solution
f(-1) = (-1)^2 + 2*(-1) + 1 = 0. The function is a polynomial and is continuous everywhere, including at x = -1.
Correct Answer:
C
— 1, Yes
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Q. If f(x) = x^2 + 3x + 2, what is f(1) and is it continuous?
A.
6, Continuous
B.
6, Discontinuous
C.
5, Continuous
D.
5, Discontinuous
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Solution
f(1) = 1^2 + 3(1) + 2 = 6. Since f(x) is a polynomial function, it is continuous everywhere.
Correct Answer:
A
— 6, Continuous
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