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Q. If the roots of the equation x^2 - kx + 8 = 0 are equal, what is the value of k?

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

Solution

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

Correct Answer: A — 4

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Q. If the roots of the equation x² + 2x + k = 0 are 1 and -3, what is the value of k? (2020)

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

Solution

Using the sum and product of roots: 1 + (-3) = -2 and 1 * (-3) = -3, thus k = 3.

Correct Answer: C — 3

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Q. If the roots of the equation x² + 2x + k = 0 are real and distinct, what is the condition for k? (2020)

A. k > 1
B. k < 1
C. k > 4
D. k < 4

Solution

The discriminant must be greater than zero: 2² - 4*1*k > 0, which simplifies to k < 1.

Correct Answer: C — k > 4

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Q. If the roots of the equation x² + 5x + 6 = 0 are a and b, what is the value of a + b? (2019)

A. 5
B. 6
C. 10
D. 4

Solution

The sum of the roots is given by -b/a = -5/1 = -5.

Correct Answer: A — 5

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Q. If the roots of the equation x² + 5x + k = 0 are -2 and -3, find k. (2020)

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

Solution

Using the product of roots: k = (-2)(-3) = 6.

Correct Answer: A — 6

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Q. If the roots of the equation x² + 5x + k = 0 are 1 and 4, find k. (2020)

A. 4
B. 5
C. 6
D. 7

Solution

Using the sum and product of roots: k = 1*4 = 4, and sum = 1 + 4 = 5, thus k = 7.

Correct Answer: D — 7

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Q. If the roots of the equation x² + 5x + k = 0 are real and distinct, what is the condition for k? (2020)

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

Solution

The discriminant must be greater than zero: 5² - 4*1*k > 0, thus k < 25.

Correct Answer: A — k > 25

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Q. If the roots of the equation x² + 5x + q = 0 are 1 and 4, find q. (2019)

A. 5
B. 4
C. 6
D. 7

Solution

Using the product of roots: q = 1 * 4 = 4.

Correct Answer: A — 5

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Q. If the roots of the equation x² + 7x + k = 0 are -3 and -4, find k. (2022)

A. 12
B. 7
C. 15
D. 20

Solution

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find k = 12.

Correct Answer: A — 12

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Q. If the roots of the equation x² + 7x + k = 0 are 1 and 6, what is the value of k? (2020)

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

Solution

Using the product of roots, k = 1*6 = 6.

Correct Answer: C — 8

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Q. If the roots of the equation x² + 7x + p = 0 are -3 and -4, find p. (2019)

A. 12
B. 7
C. 15
D. 10

Solution

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find p = 12.

Correct Answer: A — 12

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Q. If the roots of the equation x² + 7x + p = 0 are -3 and -4, find the value of p. (2019)

A. 12
B. 7
C. 10
D. 15

Solution

Using the product of roots: p = (-3)(-4) = 12.

Correct Answer: A — 12

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Q. If the roots of the equation x² + 7x + p = 0 are -3 and -4, what is the value of p? (2019)

A. 12
B. 7
C. 15
D. 10

Solution

Using the product of roots: p = (-3)(-4) = 12.

Correct Answer: A — 12

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Q. If the roots of the equation x² + mx + n = 0 are 1 and -1, find m and n. (2020)

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

Solution

The sum of the roots is 0 (m = 0) and the product is -1 (n = 1).

Correct Answer: A — 0, 1

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Q. If the roots of the equation x² + px + 12 = 0 are 3 and 4, find p. (2020)

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

Solution

Using the sum of the roots: p = -(3 + 4) = -7.

Correct Answer: A — -7

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Q. If the roots of the equation x² + px + 12 = 0 are 3 and 4, what is the value of p? (2020)

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

Solution

The sum of the roots is -p = 3 + 4 = 7, hence p = -7.

Correct Answer: A — -7

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Q. If the roots of the equation x² + px + q = 0 are 3 and -2, what is the value of p? (2019)

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

Solution

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

Correct Answer: C — 5

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Q. If the roots of the equation x² - 6x + p = 0 are 2 and 4, what is the value of p? (2023)

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

Solution

The product of the roots gives p = 2 * 4 = 8.

Correct Answer: B — 12

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Q. If the roots of the polynomial x^3 - 3x^2 + 3x - 1 = 0 are a, b, and c, what is the value of a + b + c?

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

Solution

By Vieta's formulas, the sum of the roots a + b + c = -(-3) = 3.

Correct Answer: B — 3

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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are 3 and -2, what is the value of c if a = 1 and b = -1?

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

Solution

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

Correct Answer: A — -6

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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are 4 and -1, what is the value of b if a = 1 and c = -4? (2023)

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

Solution

Using the sum of roots, b = - (4 + (-1)) = -3.

Correct Answer: A — -3

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

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 quadratic equation ax^2 + bx + c = 0 are equal, which of the following must be true? (2019)

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

Solution

For the roots to be equal, the discriminant must be zero, which means b^2 = 4ac.

Correct Answer: C — b^2 = 4ac

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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are p and q, which of the following is correct?

A. p + q = -b/a and pq = c/a
B. p + q = c/a and pq = -b/a
C. p - q = -b/a and pq = c/a
D. p * q = -b/a and p + q = c/a

Solution

According to Vieta's formulas, the sum of the roots p + q = -b/a and the product pq = c/a.

Correct Answer: A — p + q = -b/a and pq = c/a

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Q. If the roots of the quadratic equation x^2 + 2x + k = 0 are equal, what is the value of k? (2022)

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

Solution

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

Correct Answer: D — -1

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Q. If the roots of the quadratic 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 quadratic equation x^2 + mx + n = 0 are 3 and 4, what is the value of m?

A. 7
B. 12
C. 10
D. 8

Solution

The sum of the roots is 3 + 4 = 7, hence m = -7.

Correct Answer: A — 7

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Q. If the roots of the quadratic equation x^2 + px + q = 0 are -2 and -3, what is the value of p? (2019)

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. If the roots of the quadratic equation x^2 + px + q = 0 are -2 and -3, what is the value of p + q? (2019)

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

Solution

Using Vieta's formulas, p = -(-2 - 3) = 5 and q = (-2)(-3) = 6. Therefore, p + q = 5 + 6 = 11.

Correct Answer: B — -6

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Q. If the roots of the quadratic equation x^2 + px + q = 0 are equal, what is the relationship between p and q?

A. p^2 = 4q
B. p^2 > 4q
C. p^2 < 4q
D. p + q = 0

Solution

For equal roots, the discriminant must be zero: p^2 - 4q = 0, hence p^2 = 4q.

Correct Answer: A — p^2 = 4q

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Major Competitive Exams

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