Q. Find the determinant of \( G = \begin{pmatrix} 4 & 2 \\ 3 & 1 \end{pmatrix} \). (2020)
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Solution
The determinant is \( 4*1 - 2*3 = 4 - 6 = -2 \).
Correct Answer:
A
— -2
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Q. Find the dimensions of a box with a square base that maximizes volume given a surface area of 600 sq. units. (2020)
A.
10, 10
B.
15, 15
C.
12, 12
D.
20, 20
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Solution
Let x be the side of the base and h the height. The surface area constraint gives 2x^2 + 4xh = 600. Max volume occurs at x = 12.
Correct Answer:
C
— 12, 12
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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 dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2020)
A.
5, 10
B.
7, 7
C.
10, 5
D.
8, 6.25
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Solution
For a fixed area, the perimeter is minimized when the rectangle is a square. Thus, side = √50.
Correct Answer:
B
— 7, 7
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Q. Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)
A.
5/√29
B.
10/√29
C.
15/√29
D.
20/√29
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Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.
Correct Answer:
B
— 10/√29
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Q. Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)
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Solution
Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.
Correct Answer:
A
— 5
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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
Show solution
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 (2, 3) and is perpendicular to the line y = 4x - 1.
A.
y = -1/4x + 4
B.
y = 4x - 5
C.
y = -4x + 11
D.
y = 1/4x + 2
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Solution
The slope of the given line is 4, so the perpendicular slope is -1/4. Using point-slope form, we get y - 3 = -1/4(x - 2) which simplifies to y = -1/4x + 11/4.
Correct Answer:
C
— y = -4x + 11
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Q. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 1/3x + 2.
A.
y = -3x + 9
B.
y = 3x - 3
C.
y = -1/3x + 4
D.
y = 1/3x + 1
Show solution
Solution
The slope of the given line is 1/3, so the perpendicular slope is -3. Using point-slope form, y - 3 = -3(x - 2) gives y = -3x + 9.
Correct Answer:
A
— y = -3x + 9
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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 (2, -3) and has a slope of 4.
A.
y = 4x - 11
B.
y = 4x + 5
C.
y = 4x - 3
D.
y = 4x + 3
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Solution
Using point-slope form, y + 3 = 4(x - 2) simplifies to y = 4x - 11.
Correct Answer:
A
— y = 4x - 11
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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
Show solution
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
Show solution
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
Show solution
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 general solution of the equation y' = 5y + 3.
A.
y = Ce^(5x) - 3/5
B.
y = Ce^(5x) + 3/5
C.
y = 3/5 + Ce^(-5x)
D.
y = 5x + C
Show solution
Solution
The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Correct Answer:
A
— y = Ce^(5x) - 3/5
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Q. Find the general solution of the equation y'' - 3y' + 2y = 0.
A.
y = C1 e^(2x) + C2 e^(x)
B.
y = C1 e^(x) + C2 e^(2x)
C.
y = C1 e^(3x) + C2 e^(0)
D.
y = C1 e^(0) + C2 e^(3x)
Show solution
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r-1)(r-2)=0. Thus, the general solution is y = C1 e^(x) + C2 e^(2x).
Correct Answer:
B
— y = C1 e^(x) + C2 e^(2x)
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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) with respect to 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). (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
Show solution
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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