Q. Find the coefficient of x^3 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^3 in (x + 1)^8 is given by 8C3 = 56.
Correct Answer:
C
— 84
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Q. Find the coefficient of x^4 in the expansion of (x + 1)^8.
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Solution
The coefficient of x^4 is C(8, 4) = 70.
Correct Answer:
A
— 70
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Q. Find the coefficient of x^5 in the expansion of (3x + 2)^6.
A.
486
B.
729
C.
729
D.
486
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Solution
The coefficient of x^5 in (3x + 2)^6 is C(6, 5)(3)^5(2)^1 = 6 * 243 * 2 = 2916.
Correct Answer:
A
— 486
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Q. Find the conjugate of the complex number z = 2 - 5i.
A.
2 + 5i
B.
2 - 5i
C.
-2 + 5i
D.
-2 - 5i
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Solution
The conjugate of z = 2 - 5i is z* = 2 + 5i.
Correct Answer:
A
— 2 + 5i
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Q. Find the critical points of the function f(x) = x^3 - 3x^2 + 4.
A.
x = 0, 2
B.
x = 1, 2
C.
x = 1, 3
D.
x = 0, 1
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Solution
To find critical points, set f'(x) = 0. f'(x) = 3x^2 - 6x = 3x(x - 2). Critical points are x = 0 and x = 2.
Correct Answer:
B
— x = 1, 2
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Q. Find the derivative of f(x) = 4x^3 - 2x + 1. (2022)
A.
12x^2 - 2
B.
12x^2 + 2
C.
4x^2 - 2
D.
4x^2 + 2
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Solution
Using the power rule, f'(x) = 12x^2 - 2.
Correct Answer:
A
— 12x^2 - 2
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Q. Find the derivative of f(x) = 5x^2 + 3x - 1. (2020)
A.
10x + 3
B.
5x + 3
C.
10x - 1
D.
5x^2 + 3
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Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^2 + 3x - 7. (2020)
A.
10x + 3
B.
5x + 3
C.
10x - 3
D.
5x - 3
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Solution
Using the power rule, f'(x) = 10x + 3.
Correct Answer:
A
— 10x + 3
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Q. Find the derivative of f(x) = 5x^3 - 4x + 7. (2019)
A.
15x^2 - 4
B.
15x^2 + 4
C.
5x^2 - 4
D.
5x^2 + 4
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Solution
Using the power rule, f'(x) = 15x^2 - 4.
Correct Answer:
A
— 15x^2 - 4
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Q. Find the derivative of f(x) = x^3 * ln(x). (2023)
A.
3x^2 * ln(x) + x^2
B.
3x^2 * ln(x) + x^3/x
C.
3x^2 * ln(x) + x^3
D.
3x^2 * ln(x) + 1
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Solution
Using the product rule, f'(x) = (x^3)' * ln(x) + x^3 * (ln(x))' = 3x^2 * ln(x) + x^2.
Correct Answer:
A
— 3x^2 * ln(x) + x^2
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Q. Find the derivative of f(x) = x^4 + 2x^3 - x + 1. (2023)
A.
4x^3 + 6x^2 - 1
B.
4x^3 + 2x^2 - 1
C.
3x^3 + 6x^2 - 1
D.
4x^3 + 2x - 1
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Solution
Using the power rule, f'(x) = 4x^3 + 6x^2 - 1.
Correct Answer:
A
— 4x^3 + 6x^2 - 1
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 2.
A.
4x^3 - 12x^2 + 12x
B.
4x^3 - 12x + 6
C.
12x^2 - 4x + 6
D.
4x^3 - 12x^2 + 2
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Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x.
Correct Answer:
A
— 4x^3 - 12x^2 + 12x
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Q. Find the derivative of f(x) = x^4 - 4x^3 + 6x^2 - 24x + 5. (2023)
A.
4x^3 - 12x^2 + 12x - 24
B.
4x^3 - 12x^2 + 6x - 24
C.
4x^3 - 12x^2 + 12x
D.
4x^3 - 12x^2 + 6x
Show solution
Solution
Using the power rule, f'(x) = 4x^3 - 12x^2 + 12x - 24.
Correct Answer:
A
— 4x^3 - 12x^2 + 12x - 24
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Q. Find the derivative of f(x) = x^5 - 2x^3 + x. (2019)
A.
5x^4 - 6x^2 + 1
B.
5x^4 - 6x
C.
5x^4 + 2x^2 + 1
D.
5x^4 - 2x^2
Show solution
Solution
Using the power rule, f'(x) = 5x^4 - 6x^2 + 1.
Correct Answer:
A
— 5x^4 - 6x^2 + 1
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Q. Find the derivative of g(x) = sin(x) + cos(x). (2020)
A.
cos(x) - sin(x)
B.
-sin(x) - cos(x)
C.
sin(x) + cos(x)
D.
-cos(x) + sin(x)
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Solution
Using the derivatives of sine and cosine, g'(x) = cos(x) - sin(x).
Correct Answer:
A
— cos(x) - sin(x)
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Q. Find the determinant of the matrix \( D = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \). (2019)
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Solution
Det(D) = (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer:
A
— -2
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Q. Find the distance between the points (-1, -1) and (2, 2).
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Solution
Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[9 + 9] = √18 = 3√2.
Correct Answer:
C
— 5
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Q. Find the distance between the points (-2, -3) and (4, 5).
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Solution
Using the distance formula: d = √[(4 - (-2))² + (5 - (-3))²] = √[(4 + 2)² + (5 + 3)²] = √[36 + 64] = √100 = 10.
Correct Answer:
B
— 7
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Q. Find the distance between the points (0, 0) and (x, y) where x = 6 and y = 8.
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Solution
Using the distance formula: d = √[(6 - 0)² + (8 - 0)²] = √[36 + 64] = √100 = 10.
Correct Answer:
A
— 10
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Q. Find the distance between the points (1, 1) and (4, 5). (2023)
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Solution
Using the distance formula: d = √[(4 - 1)² + (5 - 1)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the distance between the points (3, 3) and (3, 7).
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Solution
Using the distance formula: d = √[(3 - 3)² + (7 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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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
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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 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)
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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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