Mathematics for Competitive Exam Success

Mathematics

3D Geometry Applications of Derivatives Binomial Theorem Circles Complex Numbers Conic Sections Coordinate Geometry Differential Equations Differentiation Integration Limits & Continuity Matrices & Determinants Permutations & Combinations Probability Quadratic Equations Sequences & Series Sets & Relations Straight Lines Trigonometry Vector Algebra

Q. A jar contains 6 green and 4 yellow marbles. What is the probability of picking a yellow marble?

A. 2/5
B. 1/2
C. 4/10
D. 3/5

Solution

Total marbles = 6 + 4 = 10. Probability of yellow marble = 4/10 = 2/5.

Correct Answer: A — 2/5

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Q. A jar contains 7 red, 5 blue, and 8 green marbles. What is the probability of selecting a blue marble?

A. 5/20
B. 1/4
C. 1/5
D. 1/3

Solution

Total marbles = 7 + 5 + 8 = 20. Probability of blue marble = 5/20 = 1/4.

Correct Answer: A — 5/20

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Q. A jar contains 8 red and 4 blue marbles. What is the probability of selecting a blue marble?

A. 1/3
B. 1/2
C. 1/4
D. 1/5

Solution

Total marbles = 8 + 4 = 12. Probability of blue marble = 4/12 = 1/3.

Correct Answer: A — 1/3

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Q. A jar contains 8 red, 6 blue, and 4 green marbles. What is the probability of selecting a blue marble?

A. 1/3
B. 1/4
C. 2/5
D. 3/10

Solution

Total marbles = 8 + 6 + 4 = 18. Probability of blue marble = 6/18 = 1/3.

Correct Answer: D — 3/10

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Q. A rectangle has a perimeter of 40 cm. What dimensions maximize the area? (2022)

A. 10 cm by 10 cm
B. 8 cm by 12 cm
C. 5 cm by 15 cm
D. 6 cm by 14 cm

Solution

For maximum area, the rectangle should be a square. Thus, each side = 40/4 = 10 cm.

Correct Answer: A — 10 cm by 10 cm

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Q. A rectangle has a perimeter of 40 cm. What dimensions will maximize the area? (2022)

A. 10 cm by 10 cm
B. 15 cm by 5 cm
C. 20 cm by 0 cm
D. 12 cm by 8 cm

Solution

For maximum area, the rectangle should be a square. Thus, each side = 40/4 = 10 cm.

Correct Answer: A — 10 cm by 10 cm

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Q. A rectangle has a perimeter of 40 units. What dimensions maximize the area? (2022)

A. 10, 10
B. 8, 12
C. 6, 14
D. 5, 15

Solution

For a fixed perimeter, the area is maximized when the rectangle is a square. Thus, side = 10 units.

Correct Answer: A — 10, 10

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Q. A rectangle has a perimeter of 40 units. What dimensions maximize the area? (2022) 2022

A. 10, 10
B. 5, 15
C. 8, 12
D. 6, 14

Solution

For maximum area, the rectangle should be a square. Thus, each side = 40/4 = 10.

Correct Answer: A — 10, 10

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Q. A student has a 70% chance of passing an exam. What is the probability of failing the exam?

A. 0.3
B. 0.7
C. 0.5
D. 0.2

Solution

Probability of failing = 1 - Probability of passing = 1 - 0.7 = 0.3.

Correct Answer: A — 0.3

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Q. At what point does the function f(x) = x^3 - 3x^2 + 4 have a local minimum? (2020)

A. (1, 2)
B. (2, 1)
C. (0, 4)
D. (3, 0)

Solution

To find local minima, we find f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x = 0 and x = 2. Checking the second derivative, f''(2) = 6 > 0, so (2, 1) is a local minimum.

Correct Answer: A — (1, 2)

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Q. At which point does the function f(x) = -x^3 + 3x^2 + 4 have a local maximum? (2023)

A. (0, 4)
B. (1, 6)
C. (2, 5)
D. (3, 4)

Solution

Setting f'(x) = -3x^2 + 6x = 0 gives x = 0 and x = 2. f''(2) < 0 indicates a local maximum at (2, 5).

Correct Answer: B — (1, 6)

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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) = 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. (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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Mathematics

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