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. How many ways can a committee of 4 be formed from 10 people? (2019)

A. 210
B. 120
C. 100
D. 30

Solution

The number of ways to form a committee of 4 from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. If 2x = 10, what is the value of x? (2015)

A. 5
B. 10
C. 2
D. 20

Solution

Dividing both sides by 2 gives x = 10/2 = 5.

Correct Answer: A — 5

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Q. If 3x = 12, what is the value of x? (2019)

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

Solution

x = 12 / 3 = 4.

Correct Answer: B — 4

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Q. If a = 2 and b = 3, what is the value of a^3 + b^3? (2021)

A. 35
B. 27
C. 10
D. 8

Solution

a^3 = 2^3 = 8 and b^3 = 3^3 = 27, so 8 + 27 = 35.

Correct Answer: A — 35

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Q. If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.7071, θ = 45 degrees.

Correct Answer: B — 45 degrees

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Q. If A = 3i + 4j and B = 2i - j, what is A + B? (2021)

A. 5i + 3j
B. 5i + 5j
C. 1i + 5j
D. 1i + 3j

Solution

A + B = (3i + 4j) + (2i - j) = (3 + 2)i + (4 - 1)j = 5i + 3j.

Correct Answer: A — 5i + 3j

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Q. If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (2023)

A. 0
B. 8
C. 12
D. 16

Solution

B × A = |i  j  k| |-1  3  0| |4  2  0| = 0, hence A · (B × A) = 0.

Correct Answer: A — 0

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Q. If A = 5i + 12j, what is the unit vector in the direction of A? (2021)

A. (5/13)i + (12/13)j
B. (12/13)i + (5/13)j
C. (5/12)i + (12/5)j
D. (12/5)i + (5/12)j

Solution

Unit vector = A / |A| = (5i + 12j) / √(5^2 + 12^2) = (5/13)i + (12/13)j.

Correct Answer: A — (5/13)i + (12/13)j

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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is A - B? (2023)

A. -3i - 3j - 3k
B. -3i - 3j + 3k
C. 3i + 3j + 3k
D. 3i + 3j - 3k

Solution

A - B = (1 - 4)i + (2 - 5)j + (3 - 6)k = -3i - 3j - 3k.

Correct Answer: A — -3i - 3j - 3k

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Q. If a circle has a circumference of 31.4 cm, what is its radius? (2022)

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

Solution

Circumference = 2πr; 31.4 = 2 * π * r; r = 31.4 / (2 * π) = 5 cm.

Correct Answer: A — 5 cm

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Q. If a circle has a radius of 3 cm, what is its circumference? (Use π = 3.14) (2023)

A. 6.28 cm
B. 9.42 cm
C. 12.56 cm
D. 15.70 cm

Solution

Circumference = 2πr = 2 * 3.14 * 3 = 18.84 cm.

Correct Answer: C — 12.56 cm

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Q. If a circle's radius is tripled, how does its area change? (2021)

A. Increases by 3 times
B. Increases by 6 times
C. Increases by 9 times
D. Remains the same

Solution

Area = πr². If radius is tripled, new area = π(3r)² = 9πr², which is 9 times the original area.

Correct Answer: C — Increases by 9 times

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Q. If a circle's radius is tripled, how does its area change? (2023) 2023

A. Increases by 3 times
B. Increases by 6 times
C. Increases by 9 times
D. Remains the same

Solution

Area = πr². If radius is tripled, new area = π(3r)² = 9πr², which is 9 times the original area.

Correct Answer: C — Increases by 9 times

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Q. If a coin is tossed twice, what is the probability of getting at least one head?

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

Solution

Total outcomes = 4 (HH, HT, TH, TT). Outcomes with at least one head = 3. Probability = 3/4.

Correct Answer: C — 3/4

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Q. If a die is rolled, what is the probability of getting an even number? (2022)

A. 1/2
B. 1/3
C. 1/6
D. 2/3

Solution

Even numbers on a die = 2, 4, 6. Probability = 3/6 = 1/2.

Correct Answer: A — 1/2

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Q. If a rectangle has a length of 12 cm and a width of 4 cm, what is its perimeter? (2019)

A. 32 cm
B. 28 cm
C. 24 cm
D. 30 cm

Solution

Perimeter = 2(length + width) = 2(12 + 4) = 32 cm.

Correct Answer: A — 32 cm

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Q. If a student guesses on a multiple-choice test with 4 options, what is the probability of guessing correctly? (2022)

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

Solution

Probability of guessing correctly = 1/4.

Correct Answer: A — 1/4

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Q. If a student is selected at random from a group of 20 students, 12 of whom are girls, what is the probability that the student is a girl?

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

Solution

Probability of selecting a girl = 12/20 = 3/5.

Correct Answer: A — 3/5

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Q. If C = [[0, 1], [1, 0]], what is det(C)? (2022)

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

Solution

Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.

Correct Answer: C — -1

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Q. If cos(θ) = 0.5, what is the value of θ in degrees? (2017)

A. 30°
B. 45°
C. 60°
D. 90°

Solution

cos(60°) = 0.5, so θ = 60°

Correct Answer: C — 60°

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Q. If cos(θ) = 0.5, what is the value of θ? (2020)

A. 0°
B. 30°
C. 60°
D. 90°

Solution

cos(60°) = 0.5, so θ = 60°.

Correct Answer: C — 60°

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Q. If cos(θ) = 0.6, what is sin(θ) using Pythagorean identity? (2017)

A. 0.4
B. 0.5
C. 0.6
D. 0.8

Solution

sin²(θ) + cos²(θ) = 1; sin²(θ) = 1 - 0.6² = 0.64; sin(θ) = √0.64 = 0.8.

Correct Answer: A — 0.4

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Q. If cos(θ) = 1/2, what is θ in degrees?

A. 30°
B. 45°
C. 60°
D. 90°

Solution

cos(60°) = 1/2, so θ = 60°

Correct Answer: C — 60°

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Q. If E = [[2, 1, 3], [1, 0, 2], [4, 1, 1]], what is det(E)? (2020)

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

Solution

The determinant of E can be calculated using the rule of Sarrus or cofactor expansion, resulting in 0.

Correct Answer: A — -1

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Q. If F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is det(F)? (2021)

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

Solution

Det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Correct Answer: A — -14

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Q. If F = [[2, 0], [0, 3]], what is det(F)? (2020)

A. 0
B. 6
C. 5
D. 2

Solution

The determinant of F is calculated as (2*3) - (0*0) = 6.

Correct Answer: B — 6

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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)

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

Solution

Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.

Correct Answer: A — -10

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Q. If F = [[2, 1], [1, 3]], what is the value of det(F)? (2022)

A. 5
B. 6
C. 7
D. 8

Solution

The determinant of F is (2*3) - (1*1) = 6 - 1 = 5.

Correct Answer: A — 5

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Q. If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)

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

Solution

f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.

Correct Answer: A — 1, 2

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Q. If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)

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

Solution

The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: B — 1

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Mathematics

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