Q. What is the derivative of f(x) = 5x^2 - 4x + 3?
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A.
10x - 4
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B.
10x + 4
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C.
5x - 4
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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)
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A.
15x^2 - 2
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B.
5x^2 - 2
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C.
15x^3 - 2
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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
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B.
15x^4 - 3
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C.
5x^4 - 3
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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) = e^x * ln(x)? (2022)
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A.
e^x * ln(x)
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B.
e^x/x
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C.
e^x * (1 + ln(x))
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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?
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A.
e^x * (2x + 1)
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B.
e^x * (x^2 + 2)
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C.
e^x * 2x
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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
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B.
x
-
C.
e^x
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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)
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A.
2x * sin(x) + x^2 * cos(x)
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B.
2x * cos(x) + x^2 * sin(x)
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C.
2x * sin(x) - x^2 * cos(x)
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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?
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A.
3x^2 * e^x + x^3 * e^x
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B.
x^3 * e^x
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C.
3x^2 * e^x
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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
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B.
3x^2 * ln(x) + x^3/x
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C.
3x^2 * ln(x) + 3x^2
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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)
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A.
3x^2 - 4
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B.
3x^2 + 4
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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^4 - 6x^2 + 9? (2022)
-
A.
4x^3 - 12x
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B.
4x^3 + 12x
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C.
2x^3 - 6x
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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)
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A.
3x + 5
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B.
6x + 5
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C.
6x - 5
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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)
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A.
ad - bc
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B.
ab + cd
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C.
ac + bd
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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
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B.
ab + cd
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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 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 (3, 7) and (3, 1)?
Solution
Using the distance formula: d = √[(3 - 3)² + (1 - 7)²] = √[0 + 36] = √36 = 6.
Correct Answer: A — 6
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Q. What is the distance between the points (5, 5) and (1, 1)?
Solution
Using the distance formula: d = √[(1 - 5)² + (1 - 5)²] = √[16 + 16] = √32 = 4√2.
Correct Answer: A — 4√2
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Q. What is the equation of a circle with center at (2, -3) and radius 4? (2022)
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A.
(x-2)² + (y+3)² = 16
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B.
(x+2)² + (y-3)² = 16
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C.
(x-2)² + (y-3)² = 16
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D.
(x+2)² + (y+3)² = 16
Solution
The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=4. Thus, (x-2)² + (y+3)² = 16.
Correct Answer: A — (x-2)² + (y+3)² = 16
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Q. What is the equation of a circle with center at (3, -2) and radius 5? (2022)
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A.
(x-3)² + (y+2)² = 25
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B.
(x+3)² + (y-2)² = 25
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C.
(x-3)² + (y-2)² = 25
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D.
(x+3)² + (y+2)² = 25
Solution
The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=5, so (x-3)² + (y+2)² = 25.
Correct Answer: A — (x-3)² + (y+2)² = 25
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Q. What is the equation of the circle with center at (2, -3) and radius 5?
-
A.
(x-2)² + (y+3)² = 25
-
B.
(x+2)² + (y-3)² = 25
-
C.
(x-2)² + (y-3)² = 25
-
D.
(x+2)² + (y+3)² = 25
Solution
Standard form of a circle: (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=5.
Correct Answer: A — (x-2)² + (y+3)² = 25
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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (1, 2)? (2020)
-
A.
y = 3x - 1
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B.
y = 3x + 1
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C.
y = 3x + 2
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D.
y = 3x - 2
Solution
Parallel lines have the same slope. Using point-slope form: y - 2 = 3(x - 1) gives y = 3x - 1.
Correct Answer: A — y = 3x - 1
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Q. What is the equation of the line parallel to y = 3x - 4 that passes through the point (2, 1)? (2020)
-
A.
y = 3x - 5
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B.
y = 3x + 1
-
C.
y = 3x - 1
-
D.
y = 3x + 4
Solution
Since parallel lines have the same slope, the equation is y - 1 = 3(x - 2) which simplifies to y = 3x - 5.
Correct Answer: C — y = 3x - 1
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Q. What is the equation of the line that passes through the origin and has a slope of -4? (2023)
-
A.
y = -4x
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B.
y = 4x
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C.
y = -x/4
-
D.
y = 1/4x
Solution
Using the slope-intercept form y = mx + b, with m = -4 and b = 0, the equation is y = -4x.
Correct Answer: A — y = -4x
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Q. What is the equation of the line that passes through the origin and has a slope of -3? (2022)
-
A.
y = -3x
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B.
y = 3x
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C.
y = -x/3
-
D.
y = 1/3x
Solution
The equation of a line through the origin with slope m is y = mx. Thus, y = -3x.
Correct Answer: A — y = -3x
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Q. What is the expansion of (2x - 3y)²?
-
A.
4x² - 9y²
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B.
4x² - 12xy + 9y²
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C.
4x² + 9y²
-
D.
4x² + 12xy + 9y²
Solution
(2x - 3y)² = 4x² - 12xy + 9y² by applying the square of a binomial.
Correct Answer: B — 4x² - 12xy + 9y²
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