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 4 different flags be arranged on a pole? (2021)

A. 24
B. 12
C. 16
D. 20

Solution

The number of arrangements of 4 distinct flags is 4! = 24.

Correct Answer: A — 24

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Q. How many ways can 4 different fruits be selected from 8 available fruits? (2021)

A. 70
B. 80
C. 90
D. 100

Solution

The number of ways to choose 4 fruits from 8 is given by 8C4 = 70.

Correct Answer: A — 70

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Q. How many ways can 4 different fruits be selected from 8?

A. 70
B. 56
C. 80
D. 90

Solution

The number of ways to select 4 fruits from 8 is given by 8C4 = 70.

Correct Answer: B — 56

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Q. How many ways can 4 different letters be chosen from the word 'COMPUTE'? (2022)

A. 35
B. 70
C. 56
D. 84

Solution

The number of ways is 8C4 = 70.

Correct Answer: C — 56

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Q. How many ways can 4 letters be chosen from the word 'COMBINATION'?

A. 210
B. 120
C. 150
D. 180

Solution

The number of ways to choose 4 letters from 11 is 11C4 = 330.

Correct Answer: A — 210

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Q. How many ways can 4 people be seated in a row of 6 chairs? (2021)

A. 360
B. 720
C. 240
D. 480

Solution

The number of ways is 6P4 = 6! / (6-4)! = 360.

Correct Answer: A — 360

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Q. How many ways can 4 prizes be distributed among 10 students if each student can receive at most one prize? (2019)

A. 5040
B. 720
C. 1000
D. 1200

Solution

The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.

Correct Answer: A — 5040

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Q. How many ways can 5 different letters be arranged if 2 letters are always together?

A. 48
B. 120
C. 240
D. 720

Solution

Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Correct Answer: C — 240

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Q. How many ways can 5 different letters be arranged if 2 letters must always be together?

A. 60
B. 120
C. 240
D. 720

Solution

Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Correct Answer: C — 240

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Q. How many ways can 6 different letters be arranged if 2 letters are always together?

A. 120
B. 240
C. 720
D. 1440

Solution

Treat the 2 letters as one unit. So, we have 5 units to arrange: 5! = 120. The 2 letters can be arranged in 2! = 2 ways. Total = 120 * 2 = 240.

Correct Answer: D — 1440

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Q. How many ways can 6 people be seated in a row? (2017)

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 people in a row is 6! = 720.

Correct Answer: A — 720

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Q. How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

A. 720
B. 120
C. 5040
D. 840

Solution

Treat the 3 specific books as one unit. So, we have 5 units to arrange: 5! * 3! = 360.

Correct Answer: C — 5040

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Q. How many ways can 8 different books be arranged on a shelf if 3 specific books must be together?

A. 720
B. 5040
C. 40320
D. 2880

Solution

Treat the 3 specific books as one unit. So, we have 6 units to arrange: 6! = 720. The 3 books can be arranged in 3! = 6 ways. Total = 720 * 6 = 4320.

Correct Answer: B — 5040

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Q. How many ways can 8 different colored balls be arranged in a line?

A. 40320
B. 720
C. 1000
D. 100

Solution

The number of arrangements is 8! = 40320.

Correct Answer: A — 40320

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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 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 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 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, 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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Mathematics

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