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 box contains 4 white and 6 black balls. What is the probability of drawing a black ball? (2023)

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

Solution

Total balls = 4 + 6 = 10. Probability of black ball = 6/10 = 3/5.

Correct Answer: B — 3/5

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Q. A box contains 4 white and 6 black balls. What is the probability of picking a black ball?

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

Solution

Total balls = 4 + 6 = 10. Probability of black ball = 6/10 = 3/5.

Correct Answer: B — 3/5

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Q. A box contains 5 white and 3 black balls. What is the probability of drawing a black ball?

A. 3/8
B. 5/8
C. 1/2
D. 1/3

Solution

Total balls = 5 + 3 = 8. Probability of black ball = 3/8.

Correct Answer: A — 3/8

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Q. A box contains 5 white and 3 black balls. What is the probability of picking a black ball? (2021)

A. 3/8
B. 5/8
C. 1/2
D. 1/3

Solution

Total balls = 5 + 3 = 8. Probability of black ball = 3/8.

Correct Answer: A — 3/8

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Q. A card is drawn from a standard deck of 52 cards. What is the probability of drawing a heart?

A. 1/4
B. 1/13
C. 1/2
D. 1/52

Solution

Number of hearts = 13. Probability = 13/52 = 1/4.

Correct Answer: A — 1/4

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Q. A chord of a circle is 12 cm long and is 5 cm away from the center. What is the radius of the circle? (2020)

A. 10 cm
B. 13 cm
C. 15 cm
D. 12 cm

Solution

Using Pythagoras theorem: r² = (5)² + (6)² = 25 + 36 = 61; r = √61 ≈ 7.81 cm.

Correct Answer: B — 13 cm

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Q. A circle has a diameter of 20 cm. What is its circumference? (2022)

A. 62.8 cm
B. 40 cm
C. 31.4 cm
D. 20 cm

Solution

Circumference = πd = π * 20 = 62.8 cm.

Correct Answer: A — 62.8 cm

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Q. A circle has a radius of 10 cm. What is the diameter? (2020)

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Solution

Diameter = 2 * radius = 2 * 10 cm = 20 cm.

Correct Answer: D — 20 cm

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Q. A circle has a radius of 5 cm. What is the length of an arc that subtends a central angle of 60 degrees? (2021)

A. 5.24 cm
B. 10.47 cm
C. 3.14 cm
D. 6.28 cm

Solution

Arc length = (θ/360) * 2πr; = (60/360) * 2π * 5 = 10.47 cm.

Correct Answer: B — 10.47 cm

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Q. A circle has a radius of 5 cm. What is the length of an arc that subtends an angle of 60 degrees at the center? (2020)

A. 5.24 cm
B. 3.14 cm
C. 5.00 cm
D. 10.47 cm

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2 * π * 5 = 5.24 cm.

Correct Answer: A — 5.24 cm

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Q. A circle has an area of 154 cm². What is the radius? (2019) 2019

A. 7 cm
B. 14 cm
C. 21 cm
D. 28 cm

Solution

Area = πr². Therefore, r = √(Area/π) = √(154/3.14) ≈ 7 cm.

Correct Answer: B — 14 cm

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Q. A circle has an area of 78.5 cm². What is its radius? (Use π = 3.14) (2019)

A. 5 cm
B. 7 cm
C. 10 cm
D. 12 cm

Solution

Area = πr². Therefore, r² = Area / π = 78.5 cm² / 3.14 = 25. r = √25 = 5 cm.

Correct Answer: B — 7 cm

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Q. A circle has an area of 78.5 cm². What is the radius? (Use π = 3.14) (2022)

A. 5 cm
B. 7 cm
C. 10 cm
D. 12 cm

Solution

Area = πr². Therefore, r² = Area / π = 78.5 cm² / 3.14 = 25. r = √25 = 5 cm.

Correct Answer: B — 7 cm

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Q. A circle is inscribed in a square of side 10 cm. What is the area of the circle? (Use π = 3.14) (2020)

A. 78.5 cm²
B. 50 cm²
C. 100 cm²
D. 25 cm²

Solution

Radius of the circle = side/2 = 10 cm / 2 = 5 cm. Area = πr² = 3.14 * 5² = 78.5 cm².

Correct Answer: A — 78.5 cm²

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Q. A circle is inscribed in a square of side 10 cm. What is the area of the circle? (2019)

A. 78.5 cm²
B. 100 cm²
C. 50 cm²
D. 25 cm²

Solution

Radius = 10/2 = 5 cm; Area = πr² = π * 5² = 78.5 cm².

Correct Answer: A — 78.5 cm²

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Q. A circle is inscribed in a square of side 8 cm. What is the area of the circle? (2022)

A. 50.24 cm²
B. 64 cm²
C. 25.12 cm²
D. 32 cm²

Solution

Radius = 8/2 = 4 cm; Area = πr² = π * 4² = 50.24 cm².

Correct Answer: A — 50.24 cm²

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Q. A cylindrical can is to be made with a fixed volume of 1000 cm³. What dimensions minimize the surface area? (2022) 2022

A. 10, 10
B. 5, 20
C. 8, 15
D. 6, 18

Solution

Using the formula for surface area and volume, the optimal dimensions are found to be radius = 5 and height = 20.

Correct Answer: C — 8, 15

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Q. A cylindrical can is to be made with a fixed volume of 1000 cm³. What dimensions minimize the surface area? (2022)

A. 10 cm height, 10 cm radius
B. 5 cm height, 15.87 cm radius
C. 8 cm height, 12.5 cm radius
D. 12 cm height, 8.33 cm radius

Solution

Using the formula for surface area and volume, the optimal dimensions are found to be 5 cm height and approximately 15.87 cm radius.

Correct Answer: B — 5 cm height, 15.87 cm radius

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Q. A jar contains 4 green, 5 yellow, and 1 red marble. What is the probability of picking a yellow marble? (2019)

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

Solution

Total marbles = 4 + 5 + 1 = 10. Probability of yellow = 5/10 = 1/2.

Correct Answer: D — 5/10

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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. 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^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 + 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 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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Mathematics

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