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. What is the 5th term in the expansion of (2 + 3x)^6? (2023)

A. 486
B. 540
C. 729
D. 810

Solution

The 5th term is given by 6C4 * (2)^(6-4) * (3x)^4 = 15 * 4 * 81x^4 = 4860x^4.

Correct Answer: B — 540

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Q. What is the 5th term in the expansion of (3x - 2)^7? (2023)

A. -1680x^3
B. 1680x^3
C. -2520x^3
D. 2520x^3

Solution

The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.

Correct Answer: A — -1680x^3

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Q. What is the 5th term of the sequence defined by a_n = 2n^2 + 3n - 1?

A. 45
B. 47
C. 49
D. 51

Solution

To find the 5th term, substitute n = 5 into the formula: a_5 = 2(5^2) + 3(5) - 1 = 2(25) + 15 - 1 = 50 + 15 - 1 = 64.

Correct Answer: B — 47

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Q. What is the 8th term of the sequence defined by a_n = 3n - 2?

A. 22
B. 24
C. 26
D. 28

Solution

Substituting n = 8 into the formula: a_8 = 3(8) - 2 = 24 - 2 = 22.

Correct Answer: A — 22

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Q. What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.

Correct Answer: B — 60 degrees

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Q. What is the angle subtended at the center of a circle by an arc of length 5 cm if the radius is 10 cm? (2022)

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 45 degrees

Solution

Arc length = (θ/360) * 2πr; 5 = (θ/360) * 2π * 10; θ = (5 * 360) / (20π) = 30 degrees.

Correct Answer: B — 60 degrees

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Q. What is the area of a circle with a circumference of 31.4 cm? (2020)

A. 25 cm²
B. 50 cm²
C. 75 cm²
D. 100 cm²

Solution

Circumference = 2πr; 31.4 = 2πr; r = 5 cm; Area = πr² = 25 cm².

Correct Answer: A — 25 cm²

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Q. What is the area of a circle with a diameter of 10 cm? (2023)

A. 25π cm²
B. 50π cm²
C. 75π cm²
D. 100π cm²

Solution

Radius = diameter/2 = 10/2 = 5 cm. Area = πr² = π(5)² = 25π cm².

Correct Answer: A — 25π cm²

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Q. What is the area of a circle with a diameter of 16 cm? (2023) 2023

A. 64π cm²
B. 32π cm²
C. 16π cm²
D. 8π cm²

Solution

Radius = diameter/2 = 16/2 = 8 cm. Area = πr² = π(8)² = 64π cm².

Correct Answer: A — 64π cm²

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Q. What is the area of a circle with a diameter of 16 cm? (Use π = 3.14) (2022)

A. 201.06 cm²
B. 100.48 cm²
C. 50.24 cm²
D. 25.12 cm²

Solution

Radius = diameter/2 = 16 cm / 2 = 8 cm. Area = πr² = 3.14 * 8² = 201.06 cm².

Correct Answer: A — 201.06 cm²

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Q. What is the area of a circle with a radius of 10 m? (2023)

A. 314 m²
B. 100 m²
C. 200 m²
D. 150 m²

Solution

Area = πr² = π * 10² = 314 m².

Correct Answer: A — 314 m²

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Q. What is the area of a circle with a radius of 3 cm? (2021)

A. 28.26 cm²
B. 9.42 cm²
C. 12.56 cm²
D. 18.84 cm²

Solution

Area = πr² = π * 3² = 28.26 cm².

Correct Answer: C — 12.56 cm²

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Q. What is the area of a circle with a radius of 3 m? (2020)

A. 28.26 m²
B. 9.42 m²
C. 18.84 m²
D. 12.56 m²

Solution

Area = πr²; = π * 3² = 28.26 m².

Correct Answer: A — 28.26 m²

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Q. What is the area of a circle with radius 7? (2022)

A. 154
B. 144
C. 160
D. 150

Solution

Area = πr^2 = π(7^2) = 49π ≈ 154.

Correct Answer: A — 154

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Q. What is the area of a rectangle with vertices at (1, 1), (1, 5), (4, 1), and (4, 5)? (2021)

A. 12
B. 15
C. 10
D. 8

Solution

Area = length * width = (4-1) * (5-1) = 3 * 4 = 12.

Correct Answer: A — 12

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Q. What is the area of a sector of a circle with a radius of 4 cm and a central angle of 90 degrees? (2014)

A. 6.28 cm²
B. 12.56 cm²
C. 3.14 cm²
D. 9.42 cm²

Solution

Area of sector = (θ/360) * πr²; = (90/360) * π * 4² = 12.56 cm².

Correct Answer: B — 12.56 cm²

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Q. What is the area of a sector of a circle with a radius of 4 cm and an angle of 90 degrees? (2023)

A. 4π cm²
B. 2π cm²
C. 8π cm²
D. 6π cm²

Solution

Area of sector = (θ/360) × πr²; = (90/360) × π(4)² = 2π cm².

Correct Answer: B — 2π cm²

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Q. What is the area of a sector of a circle with a radius of 7 cm and an angle of 90 degrees? (2022)

A. 38.5 cm²
B. 12.25 cm²
C. 15.4 cm²
D. 25.5 cm²

Solution

Area of sector = (θ/360) * πr² = (90/360) * π * 7² = 38.5 cm².

Correct Answer: A — 38.5 cm²

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Q. What is the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0)? (2022)

A. 0.5
B. 1
C. 2
D. 0

Solution

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.

Correct Answer: A — 0.5

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Q. What is the area of the base of a cone with radius 3? (2023)

A. 9π
B. 6π
C. 3π
D. 12π

Solution

Area of base = πr² = π(3²) = 9π square units.

Correct Answer: A — 9π

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Q. What is the area of the base of a cylinder with radius 4 units? (2020)

A. 16π
B. 8π
C. 12π
D. 20π

Solution

Area of base = πr² = π(4)² = 16π square units.

Correct Answer: A — 16π

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Q. What is the argument of the complex number -1 + 0i? (2021)

A. π
B. 0
C. π/2
D. 3π/2

Solution

The argument of -1 + 0i is π (180 degrees) since it lies on the negative x-axis.

Correct Answer: A — π

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Q. What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

A. -243
B. 243
C. -81
D. 81

Solution

The constant term is C(5,5)(-3)^5 = -243.

Correct Answer: A — -243

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Q. What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

A. 32
B. 48
C. 64
D. 80

Solution

The coefficient of x^0 is C(5,5) * (3x)^0 * (2)^5 = 1 * 32 = 32.

Correct Answer: C — 64

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Q. What is the coefficient of x^0 in the expansion of (x + 1/x)^8? (2021)

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

Solution

The coefficient of x^0 occurs when k=4, which gives us 8C4 = 70.

Correct Answer: C — 70

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Q. What is the coefficient of x^1 in the expansion of (x + 1)^10? (2021)

A. 10
B. 90
C. 100
D. 1

Solution

The coefficient of x^1 is C(10,1)(1)^9 = 10.

Correct Answer: A — 10

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Q. What is the coefficient of x^1 in the expansion of (x - 1)^5?

A. -5
B. 5
C. 10
D. 15

Solution

The coefficient of x^1 is given by 5C1 * (-1)^4 = 5 * 1 = 5.

Correct Answer: A — -5

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Q. What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

A. 10
B. 20
C. 30
D. 40

Solution

The coefficient of x^2 is C(5,2) * (2)^2 * (-1)^3 = 10 * 4 * (-1) = -40.

Correct Answer: B — 20

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Q. What is the coefficient of x^2 in the expansion of (x + 3)^5?

A. 45
B. 60
C. 75
D. 90

Solution

The coefficient of x^2 is given by 5C2 * (3)^3 = 10 * 27 = 270.

Correct Answer: A — 45

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Q. What is the coefficient of x^2 in the expansion of (x + 4)^5?

A. 80
B. 100
C. 120
D. 160

Solution

The coefficient of x^2 is C(5,2) * 4^3 = 10 * 64 = 640.

Correct Answer: A — 80

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