Algebra Study Resources for Competitive Exam Success

Algebra

Binomial theorem Complex numbers Determinants Inverse Trigonometric Functions Linear Inequalities Logarithms Matrices Permutations & combinations Quadratic equations Sequence & series Sets & Relations

Q. For which value of k does the equation x^2 - 4x + k = 0 have roots that differ by 2?

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

Solution

Let the roots be r and r+2. Then, r + (r+2) = 4 and r(r+2) = k leads to k = 4.

Correct Answer: C — 4

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Q. For which value of k does the equation x^2 - kx + 9 = 0 have roots that are both positive?

A. k < 6
B. k > 6
C. k = 6
D. k = 0

Solution

For both roots to be positive, k must be greater than 6, as the sum of the roots must be positive.

Correct Answer: B — k > 6

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Q. For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?

A. k < 4
B. k = 4
C. k > 4
D. k ≤ 4

Solution

The discriminant must be negative: k^2 - 16 < 0, hence k > 4.

Correct Answer: C — k > 4

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Q. From a group of 10 people, how many ways can a committee of 4 be formed?

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

Solution

The number of ways to choose 4 people from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. From a set of 8 different books, how many ways can you choose 3 books?

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

Solution

The number of ways to choose 3 books from 8 is given by 8C3 = 56.

Correct Answer: B — 84

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Q. How many different ways can 3 red, 2 blue, and 1 green balls be arranged in a line?

A. 60
B. 120
C. 30
D. 10

Solution

The total arrangements = 6! / (3! * 2! * 1!) = 60.

Correct Answer: B — 120

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Q. How many different ways can 4 people be seated at a round table?

A. 6
B. 12
C. 24
D. 30

Solution

The number of arrangements of n people at a round table is (n-1)!. For 4 people, it is 3! = 6.

Correct Answer: B — 12

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Q. How many different ways can 4 students be selected from a group of 10?

A. 210
B. 120
C. 100
D. 90

Solution

The number of ways to choose 4 from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. How many different ways can 6 people be seated at a round table?

A. 720
B. 120
C. 600
D. 480

Solution

The number of arrangements of 6 people at a round table is (6-1)! = 5! = 120.

Correct Answer: A — 720

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Q. How many subsets can be formed from the set S = {a, b, c, d}?

A. 4
B. 8
C. 16
D. 2

Solution

The number of subsets of a set with n elements is 2^n. Here, n = 4, so the number of subsets is 2^4 = 16.

Correct Answer: B — 8

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Q. How many subsets does the set A = {a, b, c} have?

A. 2
B. 3
C. 4
D. 8

Solution

The number of subsets of a set with n elements is 2^n. Here, n = 3, so the number of subsets is 2^3 = 8.

Correct Answer: D — 8

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Q. How many subsets does the set {a, b, c} have?

A. 2
B. 3
C. 4
D. 8

Solution

The number of subsets of a set with n elements is 2^n. Here, n = 3, so the number of subsets is 2^3 = 8.

Correct Answer: D — 8

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Q. How many ways can 2 boys and 2 girls be selected from 5 boys and 4 girls?

A. 60
B. 40
C. 50
D. 30

Solution

The number of ways = 5C2 * 4C2 = 10 * 6 = 60.

Correct Answer: A — 60

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Q. How many ways can 2 boys and 2 girls be selected from 6 boys and 4 girls?

A. 60
B. 30
C. 40
D. 50

Solution

The number of ways is C(6,2) * C(4,2) = 15 * 6 = 90.

Correct Answer: A — 60

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Q. How many ways can 3 different books be chosen from a set of 7 books?

A. 35
B. 21
C. 42
D. 28

Solution

The number of ways to choose 3 books from 7 is 7C3 = 7! / (3! * 4!) = 35.

Correct Answer: A — 35

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Q. How many ways can 3 different fruits be chosen from 8 fruits?

A. 56
B. 84
C. 28
D. 36

Solution

The number of ways is C(8,3) = 56.

Correct Answer: B — 84

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Q. How many ways can 3 different fruits be selected from 5 available fruits?

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

Solution

The number of ways to choose 3 from 5 is given by 5C3 = 10.

Correct Answer: B — 15

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Q. How many ways can 3 different letters be chosen from the word 'COMBINATION'?

A. 120
B. 220
C. 60
D. 80

Solution

The number of ways is C(11, 3) = 165.

Correct Answer: C — 60

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Q. How many ways can 3 letters be chosen from the word 'COMBINATION'?

A. 120
B. 60
C. 30
D. 10

Solution

The number of ways to choose 3 letters from 11 distinct letters is 11C3 = 165.

Correct Answer: B — 60

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Q. How many ways can 3 men and 2 women be arranged in a line if the men must be together?

A. 60
B. 120
C. 30
D. 24

Solution

Treat the 3 men as one unit. So, we have 3 units (MMM, W, W). Arrangements = 4! * 3! = 24 * 6 = 144.

Correct Answer: B — 120

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Q. How many ways can 3 red balls and 2 blue balls be arranged in a row?

A. 10
B. 30
C. 60
D. 20

Solution

The arrangements = 5! / (3! * 2!) = 10.

Correct Answer: A — 10

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Q. How many ways can 3 red, 2 blue, and 1 green balls be arranged in a line?

A. 60
B. 120
C. 30
D. 90

Solution

The arrangements = 6! / (3! * 2! * 1!) = 60.

Correct Answer: A — 60

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Q. How many ways can 3 red, 2 blue, and 1 green balls be arranged in a row?

A. 60
B. 120
C. 30
D. 90

Solution

The total arrangements = 6! / (3! * 2! * 1!) = 60.

Correct Answer: A — 60

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Q. How many ways can 4 different books be chosen from a shelf of 10 books?

A. 210
B. 120
C. 240
D. 300

Solution

The number of ways to choose 4 books from 10 is C(10, 4) = 210.

Correct Answer: A — 210

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Q. How many ways can 4 different colored balls be arranged in a line?

A. 16
B. 24
C. 32
D. 48

Solution

The number of arrangements is 4! = 24.

Correct Answer: B — 24

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Q. How many ways can 4 different colored balls be placed in 3 different boxes?

A. 81
B. 64
C. 27
D. 12

Solution

Each ball can go into any of the 3 boxes, so the total ways = 3^4 = 81.

Correct Answer: A — 81

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Q. How many ways can 4 different fruits be selected from a basket of 10 fruits?

A. 210
B. 120
C. 300
D. 150

Solution

The number of ways to choose 4 fruits from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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

A. 210
B. 126
C. 70
D. 84

Solution

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

Correct Answer: A — 210

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

A. 210
B. 120
C. 60
D. 30

Solution

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

Correct Answer: A — 210

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Q. How many ways can 4 different prizes be awarded to 3 students?

A. 12
B. 24
C. 36
D. 48

Solution

The number of ways to award 4 different prizes to 3 students is 3^4 = 81.

Correct Answer: C — 36

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Algebra

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