Q. What is the derivative of f(x) = 3x^4 - 5x^2 + 2? (2021)
-
A.
12x^3 - 10x
-
B.
12x^3 - 5
-
C.
6x^3 - 5x
-
D.
3x^3 - 5
Solution
Using the power rule, f'(x) = 12x^3 - 10x.
Correct Answer:
A
— 12x^3 - 10x
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Q. What is the derivative of f(x) = 3x^4 - 5x^3 + 2x - 7?
-
A.
12x^3 - 15x^2 + 2
-
B.
12x^3 - 15x^2 - 2
-
C.
3x^3 - 5x^2 + 2
-
D.
3x^3 - 5x^2 - 2
Solution
Using the power rule, f'(x) = 12x^3 - 15x^2 + 2.
Correct Answer:
A
— 12x^3 - 15x^2 + 2
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Q. What is the derivative of f(x) = 4/x? (2022)
-
A.
-4/x^2
-
B.
4/x^2
-
C.
-4/x
-
D.
4/x
Solution
Using the power rule, f'(x) = -4/x^2.
Correct Answer:
A
— -4/x^2
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Q. What is the derivative of f(x) = 5x^2 - 4x + 3?
-
A.
10x - 4
-
B.
10x + 4
-
C.
5x - 4
-
D.
5x + 4
Solution
Using the power rule, f'(x) = 10x - 4.
Correct Answer:
A
— 10x - 4
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Q. What is the derivative of f(x) = 5x^3 - 2x + 1? (2023)
-
A.
15x^2 - 2
-
B.
5x^2 - 2
-
C.
15x^3 - 2
-
D.
5x^3 - 2
Solution
The derivative f'(x) = 15x^2 - 2.
Correct Answer:
A
— 15x^2 - 2
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Q. What is the derivative of f(x) = 5x^5 - 3x + 7? (2020)
-
A.
25x^4 - 3
-
B.
15x^4 - 3
-
C.
5x^4 - 3
-
D.
20x^4 - 3
Solution
Using the power rule, f'(x) = 25x^4 - 3.
Correct Answer:
A
— 25x^4 - 3
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Q. What is the derivative of f(x) = 7x^2 + 2x - 5? (2023)
-
A.
14x + 2
-
B.
14x - 2
-
C.
7x + 2
-
D.
2x - 5
Solution
Using the power rule, f'(x) = 14x + 2.
Correct Answer:
A
— 14x + 2
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Q. What is the derivative of f(x) = e^x * ln(x)? (2022)
-
A.
e^x * ln(x)
-
B.
e^x/x
-
C.
e^x * (1 + ln(x))
-
D.
e^x * ln(x)/x
Solution
Using the product rule, f'(x) = e^x * ln(x) + e^x * (1/x) = e^x * (ln(x) + 1/x).
Correct Answer:
C
— e^x * (1 + ln(x))
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Q. What is the derivative of f(x) = e^x * x^2?
-
A.
e^x * (2x + 1)
-
B.
e^x * (x^2 + 2)
-
C.
e^x * 2x
-
D.
2e^x * x
Solution
Using the product rule, f'(x) = e^x * x^2 + e^x * 2x = e^x * (x^2 + 2x).
Correct Answer:
A
— e^x * (2x + 1)
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Q. What is the derivative of f(x) = ln(x)? (2019)
-
A.
1/x
-
B.
x
-
C.
e^x
-
D.
x^2
Solution
The derivative f'(x) = d/dx(ln(x)) = 1/x.
Correct Answer:
A
— 1/x
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Q. What is the derivative of f(x) = x^2 * sin(x)? (2023)
-
A.
2x * sin(x) + x^2 * cos(x)
-
B.
2x * cos(x) + x^2 * sin(x)
-
C.
2x * sin(x) - x^2 * cos(x)
-
D.
x^2 * sin(x) + 2x * cos(x)
Solution
Using the product rule, f'(x) = 2x * sin(x) + x^2 * cos(x).
Correct Answer:
A
— 2x * sin(x) + x^2 * cos(x)
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Q. What is the derivative of f(x) = x^3 * e^x?
-
A.
3x^2 * e^x + x^3 * e^x
-
B.
x^3 * e^x
-
C.
3x^2 * e^x
-
D.
x^3 * e^(x+1)
Solution
Using the product rule, f'(x) = 3x^2 * e^x + x^3 * e^x.
Correct Answer:
A
— 3x^2 * e^x + x^3 * e^x
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Q. What is the derivative of f(x) = x^3 * ln(x)? (2023)
-
A.
3x^2 * ln(x) + x^2
-
B.
3x^2 * ln(x) + x^3/x
-
C.
3x^2 * ln(x) + 3x^2
-
D.
3x^2 * ln(x) + 1
Solution
Using the product rule, f'(x) = 3x^2 * ln(x) + x^2.
Correct Answer:
A
— 3x^2 * ln(x) + x^2
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Q. What is the derivative of f(x) = x^3 - 4x + 7? (2021)
-
A.
3x^2 - 4
-
B.
3x^2 + 4
-
C.
x^2 - 4
-
D.
3x^2 - 7
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
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Q. What is the derivative of f(x) = x^3 - 6x^2 + 9x?
-
A.
3x^2 - 12x + 9
-
B.
3x^2 - 6x + 9
-
C.
6x - 12
-
D.
3x^2 - 9
Solution
The derivative f'(x) = 3x^2 - 12x + 9.
Correct Answer:
A
— 3x^2 - 12x + 9
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Q. What is the derivative of f(x) = x^4 - 6x^2 + 9? (2022)
-
A.
4x^3 - 12x
-
B.
4x^3 + 12x
-
C.
2x^3 - 6x
-
D.
2x^3 + 6x
Solution
Using the power rule, f'(x) = 4x^3 - 12x.
Correct Answer:
A
— 4x^3 - 12x
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Q. What is the derivative of the function f(x) = 3x^2 + 5x - 7? (2021)
-
A.
3x + 5
-
B.
6x + 5
-
C.
6x - 5
-
D.
3x^2 + 5
Solution
The derivative f'(x) = d/dx(3x^2 + 5x - 7) = 6x + 5.
Correct Answer:
B
— 6x + 5
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Q. What is the determinant of a 1x1 matrix [[5]]? (2021)
Solution
The determinant of a 1x1 matrix is simply the value of the single element. Therefore, the determinant of [[5]] is 5.
Correct Answer:
B
— 5
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Q. What is the determinant of a 2x2 matrix A = [[a, b], [c, d]]? (2021)
-
A.
ad - bc
-
B.
ab + cd
-
C.
ac + bd
-
D.
ad + bc
Solution
The determinant of a 2x2 matrix is calculated as ad - bc.
Correct Answer:
A
— ad - bc
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Q. What is the determinant of a 2x2 matrix [[a, b], [c, d]]? (2020)
-
A.
ad - bc
-
B.
ab + cd
-
C.
ac - bd
-
D.
bc - ad
Solution
The determinant of a 2x2 matrix is calculated as ad - bc.
Correct Answer:
A
— ad - bc
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Q. What is the determinant of the matrix E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
Solution
The determinant of E is 0 because the rows are linearly dependent.
Correct Answer:
A
— 0
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Q. What is the determinant of the matrix H = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]?
Solution
The determinant can be calculated using the formula for 3x3 matrices. Here, the first column is the same, leading to a determinant of 0.
Correct Answer:
A
— 0
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Q. What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \)? (2023)
Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer:
A
— 0
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Q. What is the determinant of the matrix \( J = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \)? (2021)
Solution
Det(J) = (5*8) - (6*7) = 40 - 42 = -2.
Correct Answer:
A
— -2
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Q. What is the directrix of the parabola defined by the equation y^2 = 20x?
-
A.
x = -5
-
B.
x = 5
-
C.
y = 5
-
D.
y = -5
Solution
For the equation y^2 = 4px, p = 5. The directrix is given by x = -p, which is x = -5.
Correct Answer:
A
— x = -5
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Q. What is the distance between the points (0, 0) and (3, 4)?
Solution
Using the distance formula: d = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. What is the distance between the points (0, 0) and (8, 6)?
Solution
Using the distance formula: d = √((8 - 0)² + (6 - 0)²) = √(64 + 36) = √100 = 10.
Correct Answer:
A
— 10
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Q. What is the distance between the points (0, 0) and (x, y) where x = 3 and y = 4? (2022)
Solution
Using the distance formula: d = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. What is the distance between the points (2, 3) and (6, 7)?
Solution
Using the distance formula: d = √[(6 - 2)² + (7 - 3)²] = √[16 + 16] = √32 = 4√2 ≈ 5.66.
Correct Answer:
A
— 5
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Q. What is the distance between the points (3, 2) and (3, -2)?
Solution
Using the distance formula: d = √[(3 - 3)² + (-2 - 2)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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