Q. Find the angle between the vectors A = i + j and B = j - i. (2022)
-
A.
90 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).
-
A.
60°
-
B.
45°
-
C.
30°
-
D.
90°
Solution
A · B = 1*2 + 2*3 = 8. |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. cos(θ) = 8/(√5*√13).
Correct Answer:
B
— 45°
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Q. Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).
-
A.
60°
-
B.
45°
-
C.
30°
-
D.
90°
Solution
A · B = 4*1 + 3*2 = 10; |A| = √(4^2 + 3^2) = 5; |B| = √(1^2 + 2^2) = √5; cos(θ) = 10/(5√5) = 2/√5; θ = 45°.
Correct Answer:
B
— 45°
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Q. Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.
-
A.
0.5
-
B.
1
-
C.
0.25
-
D.
0.75
Solution
The area between the curves is given by ∫(from 0 to 1) (x - x^2) dx = [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.
Correct Answer:
A
— 0.5
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Q. Find the area under the curve y = 3x^2 from x = 1 to x = 2.
Solution
The area under the curve is given by ∫(from 1 to 2) 3x^2 dx = [x^3] from 1 to 2 = (8 - 1) = 7.
Correct Answer:
B
— 6
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Q. Find the coefficient of x^2 in the expansion of (2x - 3)^4.
Solution
Using the binomial theorem, the coefficient of x^2 in (2x - 3)^4 is given by 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.
Correct Answer:
C
— 54
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Q. Find the coefficient of x^2 in the expansion of (3x - 2)^5.
-
A.
-60
-
B.
-90
-
C.
90
-
D.
60
Solution
The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.
Correct Answer:
B
— -90
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^6.
-
A.
96
-
B.
144
-
C.
216
-
D.
256
Solution
The coefficient of x^2 is given by C(6, 2)(4)^4 = 15 * 256 = 3840.
Correct Answer:
A
— 96
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Q. Find the coefficient of x^3 in the expansion of (3x - 4)^5.
-
A.
-540
-
B.
-720
-
C.
720
-
D.
540
Solution
The coefficient of x^3 in (3x - 4)^5 is given by 5C3 * (3)^3 * (-4)^2 = 10 * 27 * 16 = -720.
Correct Answer:
B
— -720
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Q. Find the coefficient of x^3 in the expansion of (x + 1)^8.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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)
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)
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).
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).
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.
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)
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).
Solution
Using the distance formula: d = √[(3 - 3)² + (7 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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