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. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the nature of the roots? (2022)

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. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the value of x? (2023)

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

Solution

The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.

Correct Answer: A — -1

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Q. If the quadratic equation x^2 + 2x + k = 0 has one root equal to -1, what is the value of k? (2022)

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

Solution

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus k = 1.

Correct Answer: B — 1

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots 1 and -3, what is the value of k? (2022)

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

Solution

The product of the roots is 1 * (-3) = -3, hence k = -3.

Correct Answer: A — -3

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition on k? (2023)

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

Solution

For both roots to be negative, k must be greater than 0.

Correct Answer: A — k > 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition for k? (2023)

A. k > 1
B. k < 1
C. k > 0
D. k < 0

Solution

For both roots to be negative, k must be greater than 0.

Correct Answer: C — k > 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, what is the condition on k? (2019)

A. k < 0
B. k > 0
C. k < 4
D. k > 4

Solution

For both roots to be positive, k must be less than 4 (from the condition of the sum and product of roots).

Correct Answer: C — k < 4

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Q. If the quadratic equation x^2 + 4x + 4 = 0 is solved, what is the nature of its roots? (2019)

A. Two distinct real roots
B. One real root
C. Two complex roots
D. No roots

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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Q. If the quadratic equation x^2 + 5x + 6 = 0 is solved, what is the product of the roots? (2022)

A. 6
B. 5
C. 4
D. 3

Solution

The product of the roots is given by c/a = 6/1 = 6.

Correct Answer: A — 6

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Q. If the quadratic equation x^2 + 5x + k = 0 has one root as 2, what is the value of k? (2019)

A. 6
B. 8
C. 10
D. 12

Solution

Substituting x = 2 into the equation gives 2^2 + 5(2) + k = 0, leading to 4 + 10 + k = 0, thus k = -14.

Correct Answer: A — 6

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Q. If the quadratic equation x^2 + 6x + 9 = 0 is solved, what is the nature of its roots? (2019)

A. Two distinct real roots
B. One real root
C. Two complex roots
D. No roots

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

A. 0
B. 1
C. 4
D. 9

Solution

For both roots to be positive, k must be greater than 9 (since the sum of roots is -b/a and must be positive).

Correct Answer: C — 4

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Q. If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

A. 6
B. 7
C. 8
D. 9

Solution

For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.

Correct Answer: C — 8

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Q. If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)

A. k > 8
B. k < -8
C. k > -8
D. k < 8

Solution

The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.

Correct Answer: B — k < -8

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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and -2, what is the value of p + q? (2023)

A. 1
B. 5
C. 7
D. 3

Solution

Using Vieta's formulas, p = -(3 + (-2)) = -1 and q = 3 * (-2) = -6. Therefore, p + q = -1 - 6 = -7.

Correct Answer: B — 5

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

A. -1
B. 1
C. 5
D. -5

Solution

Using the sum of roots formula, p = -(3 + (-2)) = -1. Therefore, p = 1.

Correct Answer: C — 5

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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and 4, what is the value of p + q? (2023)

A. 7
B. 12
C. 10
D. 11

Solution

Using Vieta's formulas, p = -(3 + 4) = -7 and q = 3 * 4 = 12. Therefore, p + q = -7 + 12 = 5.

Correct Answer: B — 12

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Q. If the quadratic equation x^2 - 10x + 25 = 0 is solved, what is the value of x? (2022)

A. 5
B. 10
C. 0
D. 25

Solution

The equation can be factored as (x - 5)^2 = 0, giving the root x = 5.

Correct Answer: A — 5

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Q. If the quadratic equation x^2 - 8x + 15 = 0 is solved, what are the roots? (2022)

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

Solution

Factoring gives (x - 3)(x - 5) = 0, hence the roots are 3 and 5.

Correct Answer: A — 3 and 5

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Q. If the roots of the equation ax^2 + bx + c = 0 are equal, what is the condition on a, b, and c? (2020)

A. b^2 - 4ac > 0
B. b^2 - 4ac = 0
C. b^2 - 4ac < 0
D. a + b + c = 0

Solution

The condition for equal roots is given by the discriminant: b^2 - 4ac = 0.

Correct Answer: B — b^2 - 4ac = 0

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Q. If the roots of the equation x^2 + 2x + 1 = 0 are equal, what is the value of the discriminant?

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

Solution

The discriminant is given by b^2 - 4ac. Here, b = 2, a = 1, c = 1, so the discriminant is 2^2 - 4*1*1 = 0.

Correct Answer: A — 0

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Q. If the roots of the equation x^2 + 2x + k = 0 are -1 and -3, what is the value of k? (2022)

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

Solution

The sum of the roots is -1 + -3 = -4, and the product is (-1)(-3) = 3. Thus, k = 3.

Correct Answer: C — 4

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Q. If the roots of the equation x^2 + 3x + k = 0 are -1 and -2, what is the value of k? (2023)

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

Solution

Using Vieta's formulas, k = (-1)(-2) = 2.

Correct Answer: A — 2

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Q. If the roots of the equation x^2 + 3x + k = 0 are real and distinct, what is the condition on k? (2022)

A. k < 0
B. k > 0
C. k < 9
D. k > 9

Solution

The discriminant must be positive: 3^2 - 4*1*k > 0 leads to 9 - 4k > 0, thus k < 9.

Correct Answer: C — k < 9

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Q. If the roots of the equation x^2 + 4x + k = 0 are -2 and -2, what is the value of k? (2023)

A. 0
B. 4
C. 8
D. 16

Solution

Using the formula for the sum of roots, -2 + -2 = -4, and product of roots, (-2)(-2) = 4, we find k = 4.

Correct Answer: B — 4

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Q. If the roots of the equation x^2 + 4x + k = 0 are equal, what is the value of k?

A. 4
B. 8
C. 16
D. 0

Solution

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 leads to k = 4.

Correct Answer: B — 8

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Q. If the roots of the equation x^2 + 5x + 6 = 0 are a and b, what is the value of ab? (2023)

A. 6
B. 5
C. 11
D. 1

Solution

The product of the roots ab is given by c/a. Here, c = 6 and a = 1, so ab = 6.

Correct Answer: A — 6

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Q. If the roots of the equation x^2 + 5x + c = 0 are 2 and 3, what is the value of c? (2022)

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

Solution

Using the product of the roots, c = 2 * 3 = 6.

Correct Answer: A — 6

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Q. If the roots of the equation x^2 + 5x + k = 0 are -2 and -3, what is the value of k?

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

Solution

The product of the roots is (-2)(-3) = 6, so k = 6.

Correct Answer: A — 6

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Q. If the roots of the equation x^2 + 5x + k = 0 are real and distinct, what is the condition on k? (2023)

A. k < 25
B. k > 25
C. k = 25
D. k ≤ 25

Solution

The discriminant must be greater than zero: 5^2 - 4(1)(k) > 0, leading to k < 25.

Correct Answer: A — k < 25

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Algebra

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