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. 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 = {2, 4, 6} and B = {1, 2, 3}, what is A ∩ B?

A. {2}
B. {4, 6}
C. {1, 2, 3}
D. {}

Solution

The intersection A ∩ B includes elements that are in both A and B. Here, A ∩ B = {2}.

Correct Answer: A — {2}

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Q. If A = {x | x is a prime number less than 10}, what is A?

A. {2, 3, 5, 7}
B. {1, 2, 3, 4}
C. {2, 4, 6, 8}
D. {3, 5, 7, 9}

Solution

The prime numbers less than 10 are 2, 3, 5, and 7. Therefore, A = {2, 3, 5, 7}.

Correct Answer: A — {2, 3, 5, 7}

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Q. If A = {x | x is an even number} and B = {x | x is a multiple of 3}, what is A ∪ B?

A. {0, 2, 4, 6, ...}
B. {0, 3, 6, 9, ...}
C. {0, 2, 3, 4, 6, 9, ...}
D. {0, 1, 2, 3, ...}

Solution

The union A ∪ B includes all even numbers and all multiples of 3. Thus, A ∪ B = {0, 2, 3, 4, 6, 9, ...}.

Correct Answer: C — {0, 2, 3, 4, 6, 9, ...}

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Q. If a bag contains 4 green and 6 yellow marbles, what is the probability of picking a green marble?

A. 2/5
B. 4/10
C. 4/6
D. 1/3

Solution

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

Correct Answer: A — 2/5

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Q. If a bag contains 4 green and 6 yellow marbles, what is the probability of picking a yellow marble?

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

Solution

Total marbles = 4 + 6 = 10. Probability of yellow = 6/10 = 3/5.

Correct Answer: B — 3/5

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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 circumference of 62.8 cm, what is its radius? (2018)

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

Solution

Circumference = 2πr; 62.8 = 2 * π * r; r = 62.8 / (2 * π) = 10 cm.

Correct Answer: A — 10 cm

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Q. If a circle has a radius of 10 m, what is the diameter? (2021)

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

Solution

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

Correct Answer: C — 20 m

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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 has a radius of 4 cm, what is its circumference? (Use π = 3.14) (2023)

A. 12.56 cm
B. 25.12 cm
C. 50.24 cm
D. 6.28 cm

Solution

Circumference = 2πr = 2 * 3.14 * 4 cm = 25.12 cm.

Correct Answer: A — 12.56 cm

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Q. If a circle has a radius of 4 cm, what is its diameter? (2022)

A. 4 cm
B. 8 cm
C. 12 cm
D. 16 cm

Solution

Diameter = 2 * radius = 2 * 4 cm = 8 cm.

Correct Answer: B — 8 cm

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Q. If a circle's radius is tripled, by what factor does the area increase? (2019)

A. 2
B. 3
C. 6
D. 9

Solution

Area = πr²; If r is tripled, new area = π(3r)² = 9πr²; Factor = 9.

Correct Answer: D — 9

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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 line has the equation 7x - 3y + 21 = 0, what is its slope?

A. 7/3
B. -7/3
C. 3/7
D. -3/7

Solution

Rearranging gives y = (7/3)x + 7. The slope is -7/3.

Correct Answer: B — -7/3

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Q. If a line has the equation y = -3x + 7, what is the y-coordinate when x = 2?

A. 1
B. 4
C. 7
D. 10

Solution

Substituting x = 2 into the equation gives y = -3(2) + 7 = 1.

Correct Answer: B — 4

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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 per question, what is the probability of guessing correctly?

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 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 10 boys and 15 girls, what is the probability that the student is a girl? (2023)

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

Solution

Total students = 10 + 15 = 25. Probability of girl = 15/25 = 3/5.

Correct Answer: B — 3/5

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Mathematics

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