HEC USAT-E (Pre-Engineering) Mathematics: Quadratic Equations MCQs

Practice Quadratic Equations MCQs for HEC USAT-E (Pre-Engineering) Mathematics — topic-wise sets with solved answers.

HEC USAT-E (Pre-Engineering) Mathematics: Quadratic Equations MCQs — sample questions

  1. Question 1

    Q1. If α, β are roots of x² + 2x + 3 = 0, then α² + β² = ?

    • A) -2
    • B) 2
    • C) -4
    • D) 4

    Answer: -2

    Explanation: Using the sum and product of roots, α + β = -2 and αβ = 3. Then, α² + β² = (α + β)² - 2αβ = (-2)² - 2*3 = -2.

  2. Question 2

    Q2. The roots of the equation 2x² - 5x + 2 = 0 are?

    • A) 1/2, 2
    • B) 2, 2
    • C) 1, 2
    • D) 1, 1/2

    Answer: 1/2, 2

    Explanation: Using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, with a = 2, b = -5, and c = 2.

  3. Question 3

    Q3. For what value of k, the equation x² + kx + 4 = 0 has equal roots?

    • A) 4
    • B) -4
    • C) ±4
    • D) 2

    Answer: ±4

    Explanation: For equal roots, the discriminant b² - 4ac = 0. So, k² - 4*1*4 = 0, hence k = ±4.

  4. Question 4

    Q4. If the roots of x² - 3x + k = 0 are real, then?

    • A) k ≤ 9/4
    • B) k ≥ 9/4
    • C) k = 9/4
    • D) k < 9/4

    Answer: k ≤ 9/4

    Explanation: For real roots, the discriminant b² - 4ac ≥ 0. So, (-3)² - 4*1*k ≥ 0, hence k ≤ 9/4.

  5. Question 5

    Q5. The sum of the roots of the equation x² - 6x + 8 = 0 is?

    • A) 6
    • B) -6
    • C) 8
    • D) -8

    Answer: 6

    Explanation: Using the sum of roots formula, -b/a = -(-6)/1 = 6.

  6. Question 6

    Q6. If α, β are the roots of the equation x² + x + 1 = 0, then 1/α + 1/β = ?

    • A) 1
    • B) -1
    • C) 0
    • D) -2

    Answer: -1

    Explanation: Using the sum and product of roots, α + β = -1 and αβ = 1. Then, 1/α + 1/β = (α + β) / αβ = -1 / 1 = -1.

  7. Question 7

    Q7. The product of the roots of 3x² - 2x - 1 = 0 is?

    • A) -1/3
    • B) 1/3
    • C) -1
    • D) 1

    Answer: -1/3

    Explanation: Using the product of roots formula, c/a = -1/3.

  8. Question 8

    Q8. For the equation x² + 2x - 3 = 0, the roots are?

    • A) 1, -3
    • B) -1, 3
    • C) 1, 3
    • D) -1, -3

    Answer: 1, -3

    Explanation: Factoring the quadratic equation x² + 2x - 3 = 0 into (x + 3)(x - 1) = 0.

  9. Question 9

    Q9. The roots of the equation x² - 4x + 4 = 0 are?

    • A) 2, 2
    • B) -2, -2
    • C) 2, -2
    • D) -2, 2

    Answer: 2, 2

    Explanation: The equation is a perfect square trinomial (x - 2)² = 0, so the roots are 2, 2.

  10. Question 10

    Q10. If the roots of the equation ax² + bx + c = 0 are reciprocal, then?

    • A) a = c
    • B) a = -c
    • C) b = c
    • D) b = -c

    Answer: a = c

    Explanation: If the roots are reciprocal, then their product is 1. So, c/a = 1, hence a = c.

  11. Question 11

    Q11. The value of k for which the equation x² + kx + 1 = 0 has real roots is?

    • A) k ≥ 2 or k ≤ -2
    • B) k > 2 or k < -2
    • C) k = 2 or k = -2
    • D) -2 < k < 2

    Answer: k ≥ 2 or k ≤ -2

    Explanation: For real roots, the discriminant b² - 4ac ≥ 0. So, k² - 4*1*1 ≥ 0, hence k ≥ 2 or k ≤ -2.

  12. Question 12

    Q12. If α and β are the roots of x² - x - 1 = 0, then α/β + β/α = ?

    • A) 3
    • B) -3
    • C) 1
    • D) -1

    Answer: -3

    Explanation: Using sum and product of roots: α+β=1 and αβ=-1. Then α/β+β/α = (α²+β²)/αβ = ((α+β)²-2αβ)/αβ = (1+2)/(-1) = -3.

  13. Question 13

    Q13. For the quadratic equation 2x² + 3x - 1 = 0, the sum of roots is?

    • A) -3/2
    • B) 3/2
    • C) 1/2
    • D) -1/2

    Answer: -3/2

    Explanation: Using the sum of roots formula, -b/a = -3/2.

  14. Question 14

    Q14. The equation x² + x + 1 = 0 has?

    • A) Real and distinct roots
    • B) Real and equal roots
    • C) No real roots
    • D) One real root

    Answer: No real roots

    Explanation: The discriminant b² - 4ac = 1² - 4*1*1 = -3 < 0, so the equation has no real roots.

  15. Question 15

    Q15. If the roots of the equation x² - px + q = 0 are 2 and 3, then p + q = ?

    • A) 7
    • B) 5
    • C) 8
    • D) 11

    Answer: 11

    Explanation: Sum of roots gives p=5 and product gives q=6. Therefore p+q = 5+6 = 11.

  16. Question 16

    Q16. The roots of the quadratic equation x² - 7x + 12 = 0 are?

    • A) 3, 4
    • B) -3, -4
    • C) 3, -4
    • D) -3, 4

    Answer: 3, 4

    Explanation: Factoring the quadratic equation into (x - 3)(x - 4) = 0.

  17. Question 17

    Q17. For the equation 3x² + 2x - 5 = 0, the product of the roots is?

    • A) -5/3
    • B) 5/3
    • C) -3/5
    • D) 3/5

    Answer: -5/3

    Explanation: Using the product of roots formula, c/a = -5/3.

  18. Question 18

    Q18. The sum of the roots of the quadratic equation 2x² - 4x + 1 = 0 is?

    • A) 2
    • B) -2
    • C) 1/2
    • D) -1/2

    Answer: 2

    Explanation: Using the sum of roots formula, -b/a = -(-4)/2 = 2.

  19. Question 19

    Q19. If α, β are roots of the equation x² - 2x + 4 = 0, then α³ + β³ = ?

    • A) -8
    • B) 8
    • C) -16
    • D) 16

    Answer: -16

    Explanation: With α+β=2 and αβ=4, use α³+β³ = (α+β)³-3αβ(α+β) = 8-24 = -16.

  20. Question 20

    Q20. The value of p for which the roots of x² + px + 2 = 0 are equal is?

    • A) ±2√2
    • B) ±√2
    • C) ±4
    • D) ±2

    Answer: ±2√2

    Explanation: For equal roots, the discriminant b² - 4ac = 0. So, p² - 4*1*2 = 0, hence p = ±2√2.

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