MHT-CET
Q. Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)
A.
5x^4 - 9x^2
B.
5x^4 + 9x^2
C.
3x^2 - 9x
D.
5x^4 - 3x^2
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Solution
The derivative f'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.
Correct Answer: A — 5x^4 - 9x^2
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Q. Find the determinant of E = [[3, 2], [1, 4]]. (2022)
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Solution
Det(E) = (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Find the determinant of E = [[4, 2], [1, 3]]. (2023)
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Solution
Det(E) = (4*3) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Find the determinant of F = [[4, 5], [6, 7]]. (2020)
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Solution
Det(F) = (4*7) - (5*6) = 28 - 30 = -2.
Correct Answer: A — -2
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Q. Find the determinant of G = [[1, 2], [2, 4]]. (2020)
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Solution
Determinant of G = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. Find the determinant of H = [[3, 1], [2, 5]]. (2021)
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Solution
Determinant of H = (3*5) - (1*2) = 15 - 2 = 13.
Correct Answer: A — 7
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Q. Find the determinant of J = [[5, 2], [1, 3]]. (2020)
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Solution
The determinant of J is calculated as (5*3) - (2*1) = 15 - 2 = 13.
Correct Answer: A — 10
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Q. Find the determinant of the matrix D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)
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Solution
The determinant of D can be calculated using the rule of Sarrus or cofactor expansion, which results in 0.
Correct Answer: A — 0
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Q. Find the determinant of the matrix D = [[4, 2], [3, 1]]. (2023)
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Solution
The determinant of D is calculated as (4*1) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Find the determinant of the matrix \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2021)
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Solution
The determinant is \( 3*4 - 2*1 = 12 - 2 = 10 \).
Correct Answer: A — 10
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Q. Find the dimensions of a rectangle with a fixed area of 50 m^2 that minimizes the perimeter. (2021)
A.
5, 10
B.
7, 7.14
C.
8, 6.25
D.
10, 5
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Solution
For a fixed area, the minimum perimeter occurs when the rectangle is a square. Thus, dimensions are approximately 7 m by 7.14 m.
Correct Answer: B — 7, 7.14
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Q. Find the dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2022) 2022
A.
5, 10
B.
7, 7.14
C.
10, 5
D.
8, 6.25
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Solution
For minimum perimeter, the rectangle should be a square. Thus, side = sqrt(50) ≈ 7.07.
Correct Answer: B — 7, 7.14
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Q. Find the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10. (2023)
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Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.
Correct Answer: A — 2
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Q. Find the equation of the line parallel to y = 3x + 2 and passing through (4, 5).
A.
y = 3x - 7
B.
y = 3x + 5
C.
y = 3x + 2
D.
y = 3x - 2
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Solution
Since the line is parallel, it has the same slope. Using point-slope form: y - 5 = 3(x - 4) gives y = 3x - 7.
Correct Answer: A — y = 3x - 7
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Q. Find the equation of the line parallel to y = 3x + 2 that passes through the point (4, 1).
A.
y = 3x - 11
B.
y = 3x + 1
C.
y = 3x + 2
D.
y = 3x - 2
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Solution
Since the line is parallel, it has the same slope (3). Using point-slope form: y - 1 = 3(x - 4) gives y = 3x - 11.
Correct Answer: A — y = 3x - 11
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Q. Find the equation of the line parallel to y = 5x - 2 that passes through the point (2, 3).
A.
y = 5x - 7
B.
y = 5x + 2
C.
y = 5x - 5
D.
y = 5x + 1
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Solution
Since the slope is the same (5), using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
Correct Answer: A — y = 5x - 7
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Q. Find the equation of the line parallel to y = 5x - 3 that passes through the point (2, 1).
A.
y = 5x - 9
B.
y = 5x + 1
C.
y = 5x - 7
D.
y = 5x + 3
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Solution
Since the slope is the same (5), using point-slope form: y - 1 = 5(x - 2) gives y = 5x - 9.
Correct Answer: A — y = 5x - 9
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Q. Find the equation of the line that passes through (0, 0) and has a slope of 5.
A.
y = 5x
B.
y = x/5
C.
y = 5/x
D.
y = 1/5x
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Solution
Using the slope-intercept form y = mx + b, with m = 5 and b = 0, we get y = 5x.
Correct Answer: A — y = 5x
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Q. Find the equation of the line that passes through the origin and has a slope of -3.
A.
y = -3x
B.
y = 3x
C.
y = -x/3
D.
y = 1/3x
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Solution
Using the slope-intercept form, the equation is y = -3x.
Correct Answer: A — y = -3x
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Q. Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.
A.
y = -1/3x + 5/3
B.
y = 3x - 13
C.
y = -3x + 11
D.
y = 1/3x - 5/3
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Solution
The slope of the given line is 3, so the slope of the perpendicular line is -1/3. Using point-slope form, we get y + 1 = -1/3(x - 4), which simplifies to y = -1/3x + 11/3.
Correct Answer: C — y = -3x + 11
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Q. Find the equation of the line that passes through the point (4, 5) and is perpendicular to the line y = 1/3x + 2.
A.
y = -3x + 17
B.
y = 3x - 7
C.
y = -3x + 5
D.
y = 1/3x + 5
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Solution
The slope of the given line is 1/3, so the slope of the perpendicular line is -3. Using point-slope form, we get y - 5 = -3(x - 4), which simplifies to y = -3x + 17.
Correct Answer: A — y = -3x + 17
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Q. Find the equation of the line that passes through the points (2, 3) and (4, 7).
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 3x - 3
D.
y = x + 1
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Solution
The slope m = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) gives y = 2x + 1.
Correct Answer: B — y = 2x + 1
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Q. Find the general solution of dy/dx = 3x^2. (2020)
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x + C
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Solution
Integrating 3x^2 gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. Find the general solution of the equation y' = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = C/x^3
D.
y = Cx^3
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Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer: A — y = Ce^(x^3)
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Q. Find the integral of (2x + 1)^3 dx. (2019)
A.
(1/4)(2x + 1)^4 + C
B.
(1/3)(2x + 1)^4 + C
C.
(1/5)(2x + 1)^4 + C
D.
(1/2)(2x + 1)^4 + C
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Solution
Using substitution, the integral is (1/4)(2x + 1)^4 + C.
Correct Answer: A — (1/4)(2x + 1)^4 + C
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Q. Find the integral of (2x + 3)dx. (2022)
A.
x^2 + 3x + C
B.
x^2 + 3x + 1
C.
x^2 + 3 + C
D.
2x^2 + 3x + C
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Solution
Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.
Correct Answer: A — x^2 + 3x + C
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Q. Find the integral of cos(2x)dx. (2023)
A.
(1/2)sin(2x) + C
B.
sin(2x) + C
C.
(1/2)cos(2x) + C
D.
2sin(2x) + C
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Solution
The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. Find the integral of cos(x). (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
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Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. Find the integral of cos(x)dx. (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
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Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. Find the integral of e^x dx. (2022)
A.
e^x + C
B.
e^x
C.
x e^x + C
D.
ln(e^x) + C
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Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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