Q. Find the distance between the points (3, 7) and (3, 1).
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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. Find the distance between the points (5, 5) and (5, 1).
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Solution
Using the distance formula: d = √[(5 - 5)² + (1 - 5)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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Q. Find the eigenvalues of the matrix G = [[2, 1], [1, 2]]. (2020)
A.
1, 3
B.
2, 2
C.
3, 1
D.
0, 4
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Solution
The characteristic polynomial is det(G - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0. The eigenvalues are λ = 1 and λ = 3.
Correct Answer:
A
— 1, 3
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Q. Find the eigenvalues of the matrix G = [[5, 4], [2, 3]]. (2020)
A.
1, 7
B.
2, 6
C.
3, 5
D.
4, 4
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Solution
The eigenvalues are found by solving the characteristic equation det(G - λI) = 0. This gives λ^2 - 8λ + 7 = 0, which factors to (λ - 1)(λ - 7) = 0, hence λ = 1, 7.
Correct Answer:
A
— 1, 7
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Q. Find the equation of the line passing through the points (2, 3) and (4, 7). (2020)
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 3x - 3
D.
y = 2x + 3
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 the differential equation dy/dx = 3x^2.
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x^2 + C
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Solution
Integrating both sides gives y = (3/3)x^3 + C = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Find the general solution of the differential equation dy/dx = 4y.
A.
y = Ce^(4x)
B.
y = 4Ce^x
C.
y = Ce^(x/4)
D.
y = 4Ce^(x)
Show solution
Solution
This is a separable differential equation. Integrating gives y = Ce^(4x), where C is the constant.
Correct Answer:
A
— y = Ce^(4x)
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Q. Find the general solution of the equation dy/dx = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = Ce^(x^3/3)
D.
y = Ce^(x^2)
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Solution
This is a separable equation. Separating and integrating gives y = Ce^(x^3).
Correct Answer:
A
— y = Ce^(x^3)
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Q. Find the integral of (1/x) dx.
A.
ln
B.
x
C.
+ C
D.
x + C
.
1/x + C
.
e^x + C
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Solution
The integral of (1/x) is ln|x| + C, where C is the constant of integration.
Correct Answer:
A
— ln
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Q. Find the integral of e^(2x) dx.
A.
(1/2)e^(2x) + C
B.
2e^(2x) + C
C.
e^(2x) + C
D.
(1/2)e^(x) + C
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Solution
The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. Find the integral of x^2 with respect to x.
A.
(1/3)x^3 + C
B.
(1/2)x^3 + C
C.
(1/4)x^4 + C
D.
x^3 + C
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Solution
The integral of x^2 is (1/3)x^3 + C, where C is the constant of integration.
Correct Answer:
A
— (1/3)x^3 + C
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Q. Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)
A.
[[3/2, -7/4], [-1/2, 2/4]]
B.
[[3/2, -7/4], [-1/4, 2/4]]
C.
[[6, -7], [-2, 4]]
D.
[[6, 7], [2, 4]]
Show solution
Solution
The inverse of D is (1/det(D)) * adj(D) = (1/(4*6 - 7*2)) * [[6, -7], [-2, 4]] = [[3/2, -7/4], [-1/2, 2/4]].
Correct Answer:
A
— [[3/2, -7/4], [-1/2, 2/4]]
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Q. Find the inverse of the matrix F = [[4, 7], [2, 6]]. (2021)
A.
[[3, -7], [-1, 4]]
B.
[[6, -7], [-2, 4]]
C.
[[3, 7], [-1, 2]]
D.
[[2, -7], [-1, 4]]
Show solution
Solution
The inverse of F is given by (1/det(F)) * adj(F). Here, det(F) = 4*6 - 7*2 = 10, and adj(F) = [[6, -7], [-2, 4]]. Thus, F^(-1) = (1/10) * [[6, -7], [-2, 4]] = [[3/5, -7/10], [-1/5, 2/5]].
Correct Answer:
A
— [[3, -7], [-1, 4]]
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Q. Find the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)
A.
0
B.
1
C.
Infinity
D.
Undefined
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Solution
Using the fact that sin(x) approaches x as x approaches 0, we have lim (x -> 0) (x^3)/(sin(x)) = 0.
Correct Answer:
A
— 0
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Q. Find the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
A.
3/5
B.
0
C.
1
D.
Infinity
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Solution
As x approaches infinity, the leading terms dominate. Thus, lim (x -> ∞) (3x^2)/(5x^2) = 3/5.
Correct Answer:
A
— 3/5
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Q. Find the local maxima of f(x) = -x^2 + 6x - 8. (2022)
A.
(3, 1)
B.
(2, 2)
C.
(4, 0)
D.
(1, 5)
Show solution
Solution
f'(x) = -2x + 6; setting to 0 gives x = 3; f(3) = -3^2 + 6(3) - 8 = 1.
Correct Answer:
A
— (3, 1)
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Q. Find the local minima of f(x) = x^2 - 4x + 5.
A.
(2, 1)
B.
(1, 2)
C.
(0, 5)
D.
(4, 0)
Show solution
Solution
The vertex occurs at x = 2. f(2) = 2^2 - 4*2 + 5 = 1, so local minima is (2, 1).
Correct Answer:
A
— (2, 1)
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Q. Find the maximum value of f(x) = -2x^2 + 10x - 12. (2023)
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Solution
The maximum occurs at x = -b/(2a) = 10/(2*2) = 2.5. f(2.5) = -2(2.5^2) + 10(2.5) - 12 = 6.
Correct Answer:
D
— 8
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Q. Find the maximum value of f(x) = -3x^2 + 12x - 5. (2020)
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Solution
The maximum occurs at x = -b/(2a) = -12/(-6) = 2. f(2) = -3(2^2) + 12(2) - 5 = 7.
Correct Answer:
C
— 7
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Q. Find the maximum value of f(x) = -x^2 + 4x + 5. (2021)
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Solution
The vertex is at x = -4/(2*(-1)) = 2. The maximum value is f(2) = -2^2 + 4*2 + 5 = 7.
Correct Answer:
C
— 7
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Q. Find the median of the data set: 12, 15, 11, 10, 14, 13. (2020)
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Solution
First, arrange the numbers: 10, 11, 12, 13, 14, 15. Since there are 6 numbers (even), the median is the average of the 3rd and 4th numbers: (12 + 13) / 2 = 12.5.
Correct Answer:
C
— 13
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Q. Find the median of the data set: 12, 15, 11, 14, 13, 10. (2020)
Show solution
Solution
First, arrange the numbers: 10, 11, 12, 13, 14, 15. Since there are 6 numbers (even), the median is the average of the 3rd and 4th numbers: (12 + 13) / 2 = 12.5.
Correct Answer:
C
— 13
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Q. Find the median of the following data set: 12, 15, 11, 14, 13, 10. (2019)
Show solution
Solution
First, arrange the numbers: 10, 11, 12, 13, 14, 15. The median is the average of the 3rd and 4th numbers: (12 + 13) / 2 = 12.5.
Correct Answer:
B
— 13
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Q. Find the median of the following set of numbers: 10, 20, 30, 40, 50, 60, 70, 80.
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Solution
Arrange the numbers: 10, 20, 30, 40, 50, 60, 70, 80. The median is the average of the 4th and 5th values: (40 + 50) / 2 = 45.
Correct Answer:
B
— 40
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Q. Find the median of the following set of numbers: 9, 3, 6, 2, 8. (2023)
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Solution
Arrange the numbers: 2, 3, 6, 8, 9. The median is the middle number, which is 6.
Correct Answer:
A
— 6
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Q. Find the median of the numbers: 12, 15, 10, 20, 18, 25.
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Solution
Arrange the numbers: 10, 12, 15, 18, 20, 25. Since there are 6 numbers, the median is the average of the 3rd and 4th values: (15 + 18) / 2 = 16.5.
Correct Answer:
B
— 18
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Q. Find the minimum value of f(x) = 4x^2 - 16x + 15. (2023)
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Solution
The minimum occurs at x = -b/(2a) = 16/(2*4) = 2. f(2) = 4(2^2) - 16(2) + 15 = 1.
Correct Answer:
A
— 1
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Q. Find the minimum value of f(x) = 4x^2 - 8x + 3. (2022)
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Solution
The vertex is at x = 8/(2*4) = 1. The minimum value is f(1) = 4(1)^2 - 8(1) + 3 = -1.
Correct Answer:
B
— 1
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Q. Find the minimum value of f(x) = x^2 + 6x + 10. (2020)
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Solution
The minimum occurs at x = -b/(2a) = -6/(2*1) = -3. f(-3) = (-3)^2 + 6(-3) + 10 = 1.
Correct Answer:
A
— 2
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Q. Find the minimum value of the function f(x) = 3x^2 - 12x + 9. (2022)
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Solution
The minimum occurs at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer:
C
— 3
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