Major Competitive Exams

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Q. Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

A. 1
B. 0
C. π/2
D. 2

Solution

The area under the curve y = cos(x) from x = 0 to x = π/2 is given by ∫(from 0 to π/2) cos(x) dx = [sin(x)] from 0 to π/2 = 1 - 0 = 1.

Correct Answer: A — 1

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Q. Calculate the area under the curve y = x^2 + 2x from x = 0 to x = 2.

A. 4
B. 6
C. 8
D. 10

Solution

The area under the curve is given by ∫(from 0 to 2) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 2 = (8/3 + 4) = 20/3.

Correct Answer: B — 6

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Q. Calculate the area under the curve y = x^3 from x = 0 to x = 2.

A. 4
B. 8
C. 6
D. 2

Solution

The area under the curve is given by ∫(from 0 to 2) x^3 dx = [x^4/4] from 0 to 2 = (16/4) - (0) = 4.

Correct Answer: B — 8

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Q. Calculate the area under the curve y = x^4 from x = 0 to x = 2.

A. 4
B. 8
C. 16
D. 32

Solution

The area under the curve y = x^4 from x = 0 to x = 2 is given by ∫(from 0 to 2) x^4 dx = [x^5/5] from 0 to 2 = (32/5) - 0 = 32/5.

Correct Answer: B — 8

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Q. Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

A. 36
B. 48
C. 54
D. 64

Solution

The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.

Correct Answer: B — 48

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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/2)^6.

A. 15/4
B. 45/8
C. 15/8
D. 5/4

Solution

The coefficient of x^2 is C(6,2) * (1/2)^2 = 15 * 1/4 = 15/4.

Correct Answer: B — 45/8

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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/2)^8. (2021)

A. 28
B. 56
C. 70
D. 84

Solution

The coefficient of x^2 is C(8,2) * (1/2)^6 = 28 * 1/64 = 28/64 = 7/16.

Correct Answer: C — 70

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Q. Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)

A. 15
B. 30
C. 20
D. 10

Solution

The coefficient of x^2 is given by 6C2 * (1)^4 = 15.

Correct Answer: A — 15

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Q. Calculate the coefficient of x^2 in the expansion of (x + 4)^6.

A. 96
B. 144
C. 192
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. Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.

A. 20
B. 30
C. 40
D. 50

Solution

The coefficient of x^3 is given by C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.

Correct Answer: A — 20

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Q. Calculate the coefficient of x^3 in the expansion of (x - 1)^5.

A. -5
B. 10
C. -10
D. 5

Solution

The coefficient of x^3 is C(5,3) * (-1)^2 = 10.

Correct Answer: B — 10

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Q. Calculate the coefficient of x^4 in the expansion of (3x - 2)^6.

A. 540
B. 720
C. 810
D. 960

Solution

The coefficient of x^4 is C(6,4) * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: A — 540

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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)

A. 15/8
B. 45/8
C. 5/8
D. 1/8

Solution

The coefficient of x^4 is C(6,4)(1/2)^2 = 15 * 1/4 = 15/4.

Correct Answer: B — 45/8

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Q. Calculate the coefficient of x^4 in the expansion of (x + 2)^6.

A. 15
B. 60
C. 90
D. 120

Solution

The coefficient of x^4 is C(6,4) * 2^2 = 15 * 4 = 60.

Correct Answer: C — 90

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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)

A. 54
B. 81
C. 108
D. 729

Solution

The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.

Correct Answer: C — 108

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Q. Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

A. 150
B. 600
C. 750
D. 1000

Solution

The coefficient of x^4 is C(6,4) * 5^2 = 15 * 25 = 375.

Correct Answer: C — 750

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Q. Calculate the coefficient of x^5 in the expansion of (x + 2)^7.

A. 21
B. 42
C. 63
D. 84

Solution

The coefficient of x^5 is C(7,5) * (2)^2 = 21 * 4 = 84.

Correct Answer: C — 63

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Q. Calculate the coefficient of x^5 in the expansion of (x - 3)^7. (2021)

A. -189
B. -243
C. -126
D. -21

Solution

The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -189.

Correct Answer: A — -189

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Q. Calculate the derivative of f(x) = 5x^5. (2016)

A. 25x^4
B. 5x^4
C. 20x^4
D. 10x^4

Solution

The derivative f'(x) = d/dx(5x^5) = 25x^4.

Correct Answer: A — 25x^4

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Q. Calculate the derivative of f(x) = e^(2x).

A. 2e^(2x)
B. e^(2x)
C. 2xe^(2x)
D. e^(x)

Solution

Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).

Correct Answer: A — 2e^(2x)

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Q. Calculate the derivative of f(x) = ln(x^2 + 1).

A. 2x/(x^2 + 1)
B. 1/(x^2 + 1)
C. 2/(x^2 + 1)
D. x/(x^2 + 1)

Solution

Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).

Correct Answer: A — 2x/(x^2 + 1)

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Q. Calculate the derivative of f(x) = x^2 * e^x.

A. (2x + x^2)e^x
B. 2xe^x
C. x^2e^x
D. (x^2 + 2x)e^x

Solution

Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.

Correct Answer: D — (x^2 + 2x)e^x

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Q. Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023

A. e^x(2x + 1)
B. e^x(2x - 1)
C. 2xe^x
D. x^2e^x

Solution

Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).

Correct Answer: A — e^x(2x + 1)

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Q. Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)

A. -10
B. 10
C. 0
D. 5

Solution

Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.

Correct Answer: A — -10

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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)

A. 1
B. 2
C. 3
D. 4

Solution

Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.

Correct Answer: A — 1

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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

A. 1
B. 2
C. 3
D. 4

Solution

Det(D) = 3(0*3 - 2*1) - 2(1*3 - 2*2) + 1(1*1 - 0*2) = 3(0 - 2) - 2(3 - 4) + 1(1) = -6 + 2 + 1 = -3.

Correct Answer: A — 1

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Q. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)

A. 10
B. 8
C. 6
D. 12

Solution

Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Calculate the determinant of D = [[4, 2], [3, 1]]. (2020)

A. -2
B. 2
C. 10
D. 12

Solution

Determinant of D = (4*1) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]]. (2020)

A. 0
B. 1
C. 2
D. 3

Solution

Determinant of D = 4(8*3 - 9*2) - 5(7*3 - 9*1) + 6(7*2 - 8*1) = 0.

Correct Answer: A — 0

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Q. Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

A. -14
B. 14
C. 0
D. 10

Solution

Using the determinant formula, det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = -24 + 40 - 15 = 1.

Correct Answer: A — -14

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Major Competitive Exams

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