PMA Long Course Mathematics Quadratic Equations — Set 3

Quadratic Equations MCQs set 3 for PMA Long Course Mathematics — 20 solved questions.

PMA Long Course Mathematics Quadratic Equations — Set 3

  1. Question 1

    Q1. If the sum of roots is 3 and product is 2, the quadratic equation is?

    • A) x² + 3x + 2 = 0
    • B) x² - 3x + 2 = 0
    • C) x² + 3x - 2 = 0
    • D) x² - 3x - 2 = 0

    Answer: x² - 3x + 2 = 0

    Explanation: Using the formula x² - (sum of roots)x + (product of roots) = 0, we get x² - 3x + 2 = 0.

  2. Question 2

    Q2. If α, β are roots of ax² + bx + c = 0, then 1/α + 1/β = ?

    • A) -b/c
    • B) b/c
    • C) -c/b
    • D) c/b

    Answer: -b/c

    Explanation: Using α + β = -b/a and αβ = c/a, 1/α + 1/β = (α + β) / αβ = (-b/a) / (c/a) = -b/c.

  3. Question 3

    Q3. For the equation x² - 2x + k = 0, if roots are real, then?

    • A) k ≤ 1
    • B) k ≥ 1
    • C) k = 1
    • D) k = 0

    Answer: k ≤ 1

    Explanation: For real roots, discriminant b² - 4ac ≥ 0, so (-2)² - 4*1*k ≥ 0, giving k ≤ 1.

  4. Question 4

    Q4. The value of k for which x² + 3x + k = 0 has one root as zero?

    • A) 0
    • B) 1
    • C) 2
    • D) 3

    Answer: 0

    Explanation: If one root is zero, then c = 0, so k = 0.

  5. Question 5

    Q5. For what value of m, the equation x² - (m + 3)x + m = 0 has equal roots?

    • A) 9 or 1
    • B) -9 or 1
    • C) 9 or -1
    • D) -9 or -1

    Answer: 9 or 1

    Explanation: Using discriminant b² - 4ac = 0, where b = -(m + 3) and c = m, we get (m + 3)² - 4m = 0.

  6. Question 6

    Q6. The roots of 3x² + 2x - 1 = 0 are?

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

    Answer: -1, 1/3

    Explanation: Using quadratic formula with a = 3, b = 2, c = -1.

  7. Question 7

    Q7. The quadratic equation with one root as 1 + √2 is?

    • A) x² - 2x - 1 = 0
    • B) x² + 2x + 1 = 0
    • C) x² - 2x + 1 = 0
    • D) x² + 2x - 1 = 0

    Answer: x² - 2x - 1 = 0

    Explanation: If one root is 1 + √2, the other is 1 - √2; using sum and product of roots.

  8. Question 8

    Q8. For the quadratic equation x² + 2x + c = 0, if roots differ by 2, then c = ?

    • A) 0
    • B) 1/4
    • C) 3/4
    • D) 5/4

    Answer: 3/4

    Explanation: Using the difference of roots = √(b² - 4ac) / a, and given difference is 2, we derive c.

  9. Question 9

    Q9. The roots of the equation 5x² - 6x + 1 = 0 are?

    • A) 1, 1/5
    • B) 1/5, 1
    • C) -1, 1/5
    • D) -1/5, -1

    Answer: 1/5, 1

    Explanation: Using quadratic formula with a = 5, b = -6, c = 1.

  10. Question 10

    Q10. If α, β are roots of the equation x² + 3x - 2 = 0, then α/β + β/α = ?

    • A) 13/2
    • B) -13/2
    • C) 13/-2
    • D) -2/13

    Answer: -13/2

    Explanation: Using α + β = -3 and αβ = -2, we find α/β + β/α = (α² + β²) / αβ.

  11. Question 11

    Q11. For the equation 2x² + kx + 3 = 0, if roots are real and equal, then k = ?

    • A) ±2√3
    • B) ±2√6
    • C) ±√6
    • D) ±4√3

    Answer: ±2√6

    Explanation: Using discriminant b² - 4ac = 0 for equal roots, where b = k, a = 2, c = 3.

  12. Question 12

    Q12. If the sum of the roots of the equation x² + px + q = 0 is 3, then p = ?

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

    Answer: -3

    Explanation: Using the sum of roots = -b/a, we get -p/1 = 3, so p = -3.

  13. Question 13

    Q13. The product of the roots of the equation ax² + bx + c = 0 is?

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

    Answer: c/a

    Explanation: Using the product of roots = c/a, we directly get c/a.

  14. Question 14

    Q14. If α, β are the roots of x² - x - 1 = 0, then 1/α + 1/β = ?

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

    Answer: -1

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

  15. Question 15

    Q15. The roots of the 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, we get (x + 3)(x + 4) = 0, so roots are -3, -4.

  16. Question 16

    Q16. If the roots of the equation x² + 2x + k = 0 are real, then?

    • A) k ≤ 1
    • B) k ≥ 1
    • C) k = 1
    • D) k < 1

    Answer: k ≤ 1

    Explanation: For real roots, discriminant (b² - 4ac) ≥ 0. So, 4 - 4k ≥ 0, giving k ≤ 1.

  17. Question 17

    Q17. The sum of the roots of the equation 3x² - 2x + 1 = 0 is?

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

    Answer: 2/3

    Explanation: Using the sum of roots = -b/a, we get -(-2)/3 = 2/3.

  18. Question 18

    Q18. If the product of the roots of the equation x² + 2x + c = 0 is 2, then c = ?

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

    Answer: 2

    Explanation: Using the product of roots = c/a, we get c/1 = 2, so c = 2.

  19. Question 19

    Q19. For what value of p, the equation x² + px + 9 = 0 has equal roots?

    • A) ±6
    • B) 6
    • C) -6
    • D) ±3

    Answer: ±6

    Explanation: For equal roots, discriminant (b² - 4ac) = 0. So, p² - 4(1)(9) = 0, giving p = ±6.

  20. Question 20

    Q20. If α, β are roots of x² - 3x + 2 = 0, then α + β = ?

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

    Answer: 3

    Explanation: Using the sum of roots = -b/a, we get -(-3)/1 = 3.