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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 + 2x + k = 0 has roots that are equal, what is the value of k?

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

Solution

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

Correct Answer: D — -2

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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 + 4x + c = 0 has one root equal to -2, what is the value of c?

A. 0
B. 2
C. 4
D. 6

Solution

If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.

Correct Answer: A — 0

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Q. If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?

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

Solution

Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

Correct Answer: B — 4

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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 + 9 = 0 is solved, what is the nature of the roots?

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 + 6x + k = 0 has roots -2 and -4, what is the value of k?

A. 8
B. 12
C. 16
D. 20

Solution

Using Vieta's formulas, k = (-2)(-4) = 8.

Correct Answer: B — 12

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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?

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

Solution

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.

Correct Answer: A — k > 9

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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 + bx + 9 = 0 has roots 3 and -3, what is the value of b?

A. 0
B. 6
C. -6
D. 9

Solution

The sum of the roots is 3 + (-3) = 0, so b = -0.

Correct Answer: C — -6

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Q. If the quadratic equation x^2 + kx + 16 = 0 has equal roots, what is the value of k?

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

Solution

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

Correct Answer: A — -8

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Q. If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the condition on k?

A. k^2 >= 64
B. k^2 < 64
C. k > 16
D. k < 16

Solution

For real roots, the discriminant must be non-negative: k^2 - 4*1*16 >= 0, leading to k^2 >= 64.

Correct Answer: A — k^2 >= 64

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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 + kx + 9 = 0 has no real roots, what is the condition on k?

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

Solution

The discriminant must be less than zero: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6.

Correct Answer: B — k > 6

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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of m?

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

Solution

Using Vieta's formulas, m = -(1 + (-3)) = 2.

Correct Answer: A — 2

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

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

Solution

Using Vieta's formulas, the product of the roots is n = 1 * (-3) = -3.

Correct Answer: A — -3

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Q. If the quadratic equation x^2 + mx + n = 0 has roots 2 and -3, what is the value of m + n?

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

Solution

Using Vieta's formulas, m = -(-1) = 1 and n = 2*(-3) = -6, thus m + n = 1 - 6 = -5.

Correct Answer: B — 5

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

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

Solution

Using Vieta's formulas, p = -(2 + 3) = -5 and q = 2*3 = 6. Thus, p + q = -5 + 6 = 1.

Correct Answer: C — 7

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

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

Solution

The sum of the roots is -p = 2 + 3 = 5, so p = -5.

Correct Answer: A — -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 + 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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Major Competitive Exams

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