HEC HAT-1 (Engineering / IT / Math / Physics) quantitative reasoning Quadratic Equations — Set 2

HEC HAT-1 (Engineering / IT / Math / Physics) quantitative reasoning Quadratic Equations — Set 2

1. Question 1

   Q1. If α, β are roots of x² + px + q = 0, then the value of α² + β² is

   • A) p² + 2q
   • B) p² - 2q
   • C) q² - 2p
   • D) q² + 2p

   Answer: p² - 2q

   Explanation: Using the sum and product of roots, α² + β² = (α + β)² - 2αβ = (-p)² - 2q = p² - 2q.

2. Question 2

   Q2. The roots of the equation 2x² - 5x + 3 = 0 are

   • A) Real and distinct
   • B) Real and equal
   • C) Imaginary
   • D) None of these

   Answer: Real and distinct

   Explanation: Discriminant D = b² - 4ac = (-5)² - 4*2*3 = 25 - 24 = 1 > 0, so roots are real and distinct.

3. Question 3

   Q3. If the roots of ax² + bx + c = 0 are in the ratio 2:3, then

   • A) 6b² = 25ac
   • B) 6b² = 25a + c
   • C) 25b² = 6ac
   • D) b² = 6ac / 25

   Answer: 6b² = 25ac

   Explanation: Using the ratio of roots, we get b²/a*c = (2+3)² / (2*3), simplifying to 6b² = 25ac.

4. Question 4

   Q4. If the sum of the roots of x² - 2x + k = 0 is equal to the product of the roots, then k =

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

   Answer: 2

   Explanation: Sum of roots = 2, product of roots = k, so 2 = k.

5. Question 5

   Q5. The equation x² + px + q = 0 has roots α, β. If the equation x² + px + r = 0 has roots α, γ, then (β - γ)r =

   • A) q - r
   • B) r - q
   • C) q + r
   • D) r / q

   Answer: r - q

   Explanation: Using the sum and product of roots for both equations, we derive (β - γ)r = r - q.

6. Question 6

   Q6. If α, β are the roots of x² - x + 1 = 0, then α²⁰⁰⁹ + β²⁰⁰⁹ =

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

   Answer: -1

   Explanation: The roots are complex cube roots of unity, so α³ = 1 and β³ = 1, hence α²⁰⁰⁹ + β²⁰⁰⁹ = (α³)⁶⁷³ + (β³)⁶⁷³ = -1.

7. Question 7

   Q7. The roots of 3x² + 5x + 2 = 0 are

   • A) Real and distinct
   • B) Real and equal
   • C) Imaginary
   • D) Irrational

   Answer: Real and distinct

   Explanation: D = 5² - 4*3*2 = 25 - 24 = 1 > 0, so roots are real and distinct.

8. Question 8

   Q8. For what value of k, the equation x² + 2(k-1)x + k + 5 = 0 has real and equal roots?

   • A) -1 or 4
   • B) 1 or -4
   • C) 2 or -3
   • D) -2 or 3

   Answer: 1 or -4

   Explanation: For equal roots, D = 0. So, 4(k-1)² - 4(k+5) = 0, simplifying to k² - 3k - 4 = 0, giving k = 1 or -4.

9. Question 9

   Q9. If the roots of the equation ax² + bx + c = 0 are reciprocal to each other, then

   • A) a = c
   • B) b = c
   • C) a = b
   • D) None of these

   Answer: a = c

   Explanation: The product of the roots is c/a, and since they are reciprocals, c/a = 1, so a = c.

10. Question 10

    Q10. The condition that the roots of ax² + bx + c = 0 are in the ratio m:n is

    • A) mn b² = ac(m+n)²
    • B) mn b² = (m+n)² ac
    • C) b²(m+n) = mn ac
    • D) ac(m+n) = mn b

    Answer: mn b² = (m+n)² ac

    Explanation: Using the ratio of roots, we derive mn b² = (m+n)² ac.

11. Question 11

    Q11. The roots of the quadratic equation x² + 4x + 4 = 0 are

    • A) Real and distinct
    • B) Real and equal
    • C) Imaginary
    • D) Irrational

    Answer: Real and equal

    Explanation: D = 4² - 4*1*4 = 0, so roots are real and equal.

12. Question 12

    Q12. If the roots of x² + 2x + 3 = 0 are α, β, then α² + β² =

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

    Answer: -2

    Explanation: α + β = -2, αβ = 3, so α² + β² = (α + β)² - 2αβ = (-2)² - 2*3 = 4 - 6 = -2.

13. Question 13

    Q13. The value of 'a' for which the sum of the squares of the roots of x² - (a-2)x - (a+1) = 0 is least is

    • A) 1
    • B) 2
    • C) 0
    • D) None of these

    Answer: 1

    Explanation: Sum of squares = (a-2)² + 2(a+1) = a² - 2a + 6. This is least when a = 1, as the derivative with respect to 'a' is 2a - 2.

14. Question 14

    Q14. The roots of 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 and -4.

15. Question 15

    Q15. If the equation x² + bx + c = 0 has roots α, β, then the equation whose roots are α², β² is

    • A) x² - (b²-2c)x + c² = 0
    • B) x² + (b²-2c)x + c² = 0
    • C) x² + (b²+2c)x + c² = 0
    • D) x² - (b²+2c)x + c² = 0

    Answer: x² - (b²-2c)x + c² = 0

    Explanation: Using the sum and product of roots, the new equation is derived as x² - (b²-2c)x + c² = 0.

16. Question 16

    Q16. For the equation 2x² - 3x + 1 = 0, the sum of the roots is

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

    Answer: 3 / 2

    Explanation: Sum of roots = -b/a = 3 / 2.

17. Question 17

    Q17. The roots of the equation x² - 3x + 2 = 0 are

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

    Answer: 1, 2

    Explanation: Factoring the quadratic, we get (x - 1)(x - 2) = 0, so roots are 1 and 2.

18. Question 18

    Q18. If α, β are roots of the equation x² + x + 1 = 0, then α² + β² =

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

    Answer: -1

    Explanation: α + β = -1, αβ = 1, so α² + β² = (α + β)² - 2αβ = (-1)² - 2*1 = -1.

19. Question 19

    Q19. The equation whose roots are 2 + √3 and 2 - √3 is

    • A) x² - 4x + 1 = 0
    • B) x² + 4x + 1 = 0
    • C) x² - 4x - 1 = 0
    • D) x² + 4x - 1 = 0

    Answer: x² - 4x + 1 = 0

    Explanation: Sum of roots = 4, product = 1, so the equation is x² - 4x + 1 = 0.

20. Question 20

    Q20. If the roots of the equation x² + px + q = 0 are tan30° and tan15°, then the value of 2 + q - p is

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

    Answer: 3

    Explanation: Using the sum and product of roots, p = -(tan30° + tan15°) and q = tan30°*tan15°. Simplifying 2 + q - p gives 3.