Q. Find the derivative of f(x) = sin(x) + cos(x) at x = π/4.
Solution
f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.
Correct Answer: C — √2
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Q. Find the derivative of f(x) = sin(x) at x = π/2.
-
A.
0
-
B.
1
-
C.
-1
-
D.
undefined
Solution
f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = tan(x) at x = 0.
-
A.
0
-
B.
1
-
C.
undefined
-
D.
1/2
Solution
f'(x) = sec^2(x); f'(0) = sec^2(0) = 1.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = tan(x) at x = π/4.
Solution
f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.
Correct Answer: A — 1
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Q. Find the derivative of f(x) = tan(x).
-
A.
sec^2(x)
-
B.
csc^2(x)
-
C.
sin^2(x)
-
D.
cos^2(x)
Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer: A — sec^2(x)
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Q. Find the derivative of f(x) = x^2 * e^x.
-
A.
e^x(x^2 + 2x)
-
B.
e^x(x^2 - 2x)
-
C.
2xe^x
-
D.
x^2e^x
Solution
Using the product rule: f'(x) = x^2 * e^x + 2x * e^x = e^x(x^2 + 2x).
Correct Answer: A — e^x(x^2 + 2x)
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Q. Find the derivative of f(x) = x^2 sin(1/x) at x = 0.
-
A.
0
-
B.
1
-
C.
undefined
-
D.
does not exist
Solution
Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.
Correct Answer: A — 0
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.
Solution
f'(x) = 3x^2 - 6x. At x = 2, f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.
Correct Answer: B — 8
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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4x - 5.
-
A.
3x^2 - 6x + 4
-
B.
3x^2 - 3x + 4
-
C.
3x^2 - 6x + 5
-
D.
3x^2 + 6x - 4
Solution
Using the power rule, f'(x) = 3x^2 - 6x + 4.
Correct Answer: A — 3x^2 - 6x + 4
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Q. Find the derivative of f(x) = x^3 - 4x^2 + 6x.
-
A.
3x^2 - 8x + 6
-
B.
3x^2 - 4x + 6
-
C.
3x^2 - 8x
-
D.
x^2 - 4x + 6
Solution
Using the power rule, f'(x) = 3x^2 - 8x + 6.
Correct Answer: A — 3x^2 - 8x + 6
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Q. Find the determinant of the matrix \( D = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements, which is 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).
Solution
The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer: A — 10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of the identity matrix is always 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2*4 - 1*3 = 8 - 3 = 5 \).
Correct Answer: A — 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
Solution
The determinant evaluates to 0.
Correct Answer: A — -1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
Solution
Using the determinant formula, we find it equals 10.
Correct Answer: A — -10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
Solution
The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer: A — 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
-
A.
ad - bc
-
B.
bc - ad
-
C.
a + b + c + d
-
D.
a^2 + b^2
Solution
The determinant is given by the formula \( ad - bc \).
Correct Answer: A — ad - bc
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Q. Find the determinant of the matrix | 1 0 0 | | 0 1 0 | | 0 0 1 |.
Solution
This is the identity matrix, and its determinant is 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.
Solution
The determinant evaluates to 0 as the third row can be expressed as a linear combination of the first two.
Correct Answer: A — -12
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Q. Find the determinant of the matrix: | 1 2 | | 3 5 |.
Solution
det = (1*5) - (2*3) = 5 - 6 = -1.
Correct Answer: A — -1
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Q. Find the directrix of the parabola y^2 = -8x.
-
A.
x = 2
-
B.
x = -2
-
C.
x = 4
-
D.
x = -4
Solution
For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.
Correct Answer: B — x = -2
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Q. Find the distance between the points (3, 4) and (7, 1).
Solution
Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 3.6, closest option is 4.
Correct Answer: A — 5
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Q. Find the distance between the points A(2, 3) and B(5, 7).
Solution
Distance = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer: C — 5
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Q. Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.
Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |3(1) + 4(2) - 12| / sqrt(3^2 + 4^2) = |3 + 8 - 12| / 5 = 1.
Correct Answer: A — 2
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Q. Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.
Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).
Correct Answer: B — 3
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Q. Find the eigenvalues of the matrix A = [[2, 1], [1, 2]].
-
A.
1, 3
-
B.
2, 2
-
C.
3, 1
-
D.
0, 4
Solution
The characteristic polynomial is det(A - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0, giving eigenvalues 1 and 3.
Correct Answer: A — 1, 3
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Q. Find the equation of the circle with center (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
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 5².
Correct Answer: A — (x-2)² + (y+3)² = 25
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Q. Find the equation of the family of curves represented by y = mx + c, where m and c are constants.
-
A.
y = mx + c
-
B.
y = mx^2 + c
-
C.
y = c/x + m
-
D.
y = m^2x + c
Solution
The equation y = mx + c represents a family of straight lines where m is the slope and c is the y-intercept.
Correct Answer: A — y = mx + c
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Q. Find the equation of the line passing through the points (1, 2) and (3, 4).
-
A.
y = x + 1
-
B.
y = 2x
-
C.
y = x + 3
-
D.
y = 2x - 1
Solution
The slope m = (4-2)/(3-1) = 1. Using point-slope form: y - 2 = 1(x - 1) gives y = x + 1.
Correct Answer: A — y = x + 1
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