Algebra Study Resources for Competitive Exam Success

Algebra

AP, GP, HP Binomial Theorem Complex Numbers Logarithms Permutations & Combinations Quadratic Equations Real Numbers Theory of Equations

Q. For the quadratic equation 2x^2 + 4x + k = 0 to have equal roots, what should be the value of k? (2020)

A. -4
B. 0
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.

Correct Answer: A — -4

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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real and equal roots, what is the condition on k? (2020)

A. k < 0
B. k = 0
C. k = 8
D. k > 8

Solution

For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.

Correct Answer: C — k = 8

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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)

A. k > 4
B. k < 4
C. k >= 4
D. k <= 4

Solution

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

Correct Answer: D — k <= 4

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Q. For the quadratic equation 2x^2 + 4x - 6 = 0, what is the value of the discriminant? (2020)

A. 16
B. 4
C. 0
D. 36

Solution

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

Correct Answer: A — 16

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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have equal roots, what must be the value of k? (2019)

A. 0
B. 2
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

Correct Answer: C — 4

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Q. For the quadratic equation 5x^2 + 3x - 2 = 0, what is the value of the roots using the quadratic formula? (2023)

A. -1, 2/5
B. 1, -2/5
C. 2, -1/5
D. 0, -2

Solution

Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.

Correct Answer: A — -1, 2/5

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Q. For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)

A. p > 2
B. p < 2
C. p = 2
D. p >= 2

Solution

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

Correct Answer: D — p >= 2

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Q. For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the condition on k? (2023)

A. k < 1
B. k > 1
C. k >= 1
D. k <= 1

Solution

The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.

Correct Answer: D — k <= 1

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Q. For the quadratic equation x^2 + 6x + 9 = 0, what type of roots does it have? (2019)

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

Correct Answer: B — Real and equal

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Q. For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)

A. k < 9
B. k = 9
C. k > 9
D. k ≤ 9

Solution

The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.

Correct Answer: A — k < 9

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Q. For the quadratic equation x^2 + 6x + k = 0 to have real roots, what must be the condition on k? (2020)

A. k < 9
B. k = 9
C. k > 9
D. k ≤ 9

Solution

The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.

Correct Answer: D — k ≤ 9

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Q. For the quadratic equation x^2 + px + q = 0, if the roots are -2 and -3, what is the value of p? (2020)

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

Solution

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

Correct Answer: A — 5

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Q. For the quadratic equation x^2 - 4x + 4 = 0, what type of roots does it have? (2019)

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0, indicating that the roots are real and equal.

Correct Answer: B — Real and equal

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Q. For the quadratic equation x^2 - 6x + k = 0 to have one root equal to 3, what is the value of k? (2023)

A. 3
B. 6
C. 9
D. 12

Solution

If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.

Correct Answer: C — 9

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Q. For the quadratic equation x^2 - 8x + 15 = 0, what are the roots? (2023)

A. 3 and 5
B. 2 and 6
C. 1 and 7
D. 4 and 4

Solution

The roots can be found by factorization: (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

Correct Answer: A — 3 and 5

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Q. For which value of k does the equation x^2 + kx + 16 = 0 have equal roots? (2019)

A. -8
B. -4
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0. Solving gives k = -8.

Correct Answer: B — -4

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Q. From a deck of 52 cards, how many ways can you choose 5 cards?

A. 2598960
B. 1001
C. 3125
D. 1024

Solution

The number of ways to choose 5 cards from 52 is 52C5 = 2598960.

Correct Answer: A — 2598960

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Q. From a group of 8 people, how many ways can a team of 3 be selected? (2022)

A. 56
B. 24
C. 36
D. 48

Solution

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

Correct Answer: A — 56

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

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

Solution

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

Correct Answer: A — 210

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Q. How many ways can 10 different items be selected from a group of 15? (2023)

A. 3003
B. 5005
C. 1001
D. 2002

Solution

The number of ways to choose 10 items from 15 is given by 15C10 = 15C5 = 3003.

Correct Answer: A — 3003

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Q. How many ways can 2 boys and 2 girls be selected from a group of 5 boys and 5 girls? (2023)

A. 100
B. 120
C. 80
D. 60

Solution

The number of ways is 5C2 * 5C2 = 10 * 10 = 100.

Correct Answer: B — 120

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Q. How many ways can 2 boys and 3 girls be selected from 6 boys and 4 girls? (2023)

A. 60
B. 80
C. 100
D. 120

Solution

The number of ways is 6C2 * 4C3 = 15 * 4 = 60.

Correct Answer: A — 60

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Q. How many ways can 2 boys and 3 girls be selected from a group of 6 boys and 4 girls? (2023)

A. 60
B. 80
C. 100
D. 120

Solution

The number of ways is 6C2 * 4C3 = 15 * 4 = 60.

Correct Answer: A — 60

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Q. How many ways can 2 boys and 3 girls be selected from a group of 6 boys and 8 girls? (2020)

A. 280
B. 300
C. 240
D. 360

Solution

The number of ways is 6C2 * 8C3 = 15 * 56 = 840.

Correct Answer: A — 280

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Q. How many ways can 2 men and 3 women be selected from a group of 5 men and 6 women? (2020)

A. 100
B. 60
C. 120
D. 80

Solution

The number of ways to select 2 men from 5 is 5C2 and 3 women from 6 is 6C3. Total = 5C2 * 6C3 = 10 * 20 = 200.

Correct Answer: A — 100

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

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

Solution

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

Correct Answer: B — 60

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Q. How many ways can 4 men and 3 women be arranged in a line if the men must be together? (2019)

A. 5040
B. 720
C. 840
D. 1200

Solution

Treat the 4 men as one unit. Thus, we have 4 units (1 man unit + 3 women) to arrange: 4! * 4! = 24 * 24 = 576.

Correct Answer: A — 5040

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Q. How many ways can 4 students be selected from a class of 10? (2020)

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

Solution

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

Correct Answer: A — 210

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Q. How many ways can 5 different flags be arranged on a pole? (2019)

A. 60
B. 120
C. 100
D. 24

Solution

The number of arrangements of 5 different flags is 5! = 120.

Correct Answer: B — 120

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Q. How many ways can 6 people be seated around a circular table?

A. 720
B. 120
C. 60
D. 30

Solution

The number of arrangements around a circular table is (n-1)! = 5! = 120.

Correct Answer: A — 720

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Showing 31 to 60 of 334 (12 Pages)

Algebra

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