JEE Main - Nature of roots and formation of quadratic equations Online Test (MCQ Quiz)

1. If the roots of the quadratic equation ax^2 + bx + c = 0 are real and distinct, what is the condition on the discriminant (D = b^2 - 4ac)?

A) D > 0 B) D < 0 C) D = 0 D) D >= 0

2. For a quadratic equation ax^2 + bx + c = 0, if the roots are real and equal, the discriminant is:

A) D > 0 B) D < 0 C) D = 0 D) D = 1

3. If the roots of the quadratic equation ax^2 + bx + c = 0 are imaginary, then the discriminant (D) satisfies:

A) D > 0 B) D < 0 C) D = 0 D) D = 1

4. Consider the quadratic equation x^2 - 5x + 6 = 0. What is the nature of its roots?

A) Real and distinct B) Real and equal C) Imaginary D) One real, one imaginary

5. What are the roots of the quadratic equation x^2 - 6x + 9 = 0?

A) 3 and 3 B) 3 and -3 C) 6 and 9 D) 1 and 9

6. Determine the nature of the roots for the equation 2x^2 + 3x + 4 = 0.

A) Real and distinct B) Real and equal C) Imaginary D) Zero

7. If one root of the quadratic equation x^2 - kx + 4 = 0 is 2, what is the value of k?

A) 3 B) 4 C) 5 D) 6

8. If the roots of the quadratic equation ax^2 + bx + c = 0 are alpha and beta, then the sum of the roots is given by:

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

9. For a quadratic equation ax^2 + bx + c = 0 with roots alpha and beta, the product of the roots is:

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

10. Form a quadratic equation whose roots are 2 and 3.

A) x^2 - 5x + 6 = 0 B) x^2 + 5x + 6 = 0 C) x^2 - 6x + 5 = 0 D) x^2 + 6x + 5 = 0

11. If alpha and beta are the roots of x^2 - 7x + 10 = 0, find alpha + beta.

A) -7 B) 7 C) 10 D) -10

12. Given the quadratic equation x^2 + 4x + k = 0 has equal roots, find the value of k.

A) 2 B) 4 C) -4 D) 16

13. If one root of the equation x^2 - 3x + k = 0 is 2, find the other root.

A) 1 B) 2 C) 3 D) 5

14. What is the sum of the roots of the quadratic equation 3x^2 - 6x + 5 = 0?

A) 2 B) -2 C) 5/3 D) -5/3

15. What is the product of the roots of the quadratic equation 2x^2 + 5x - 7 = 0?

A) 5/2 B) -5/2 C) 7/2 D) -7/2

16. If the roots of ax^2 + bx + c = 0 are reciprocal of each other, then:

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

17. If the roots of ax^2 + bx + c = 0 are opposite to each other, then:

A) a = b B) b = 0 C) a = c D) a = -c

18. Form a quadratic equation with roots 1/2 and 1/3.

A) 6x^2 - 5x + 1 = 0 B) 6x^2 + 5x + 1 = 0 C) x^2 - 5x + 6 = 0 D) x^2 + 5x + 6 = 0

19. If one root of x^2 + px + q = 0 is 2, and the other root is 3, what are the values of p and q?

A) p = -5, q = 6 B) p = 5, q = 6 C) p = -5, q = -6 D) p = 5, q = -6

20. The quadratic equation x^2 - (alpha + beta)x + alpha*beta = 0 has roots:

A) alpha and beta B) -alpha and -beta C) 1/alpha and 1/beta D) alpha and 1/beta

21. If the discriminant of ax^2 + bx + c = 0 is positive, the roots are:

A) Real and equal B) Real and distinct C) Imaginary D) Zero

22. For the equation 4x^2 - 12x + 9 = 0, the nature of the roots is:

A) Real and distinct B) Real and equal C) Imaginary D) One positive, one negative

23. If the roots of x^2 + kx + 1 = 0 are real and equal, find the value of k.

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

24. If alpha and beta are the roots of x^2 - 2x + 1 = 0, find alpha*beta.

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

25. Form a quadratic equation whose roots are -2 and 5.

A) x^2 - 3x - 10 = 0 B) x^2 + 3x - 10 = 0 C) x^2 - 3x + 10 = 0 D) x^2 + 3x + 10 = 0

26. If one root of the equation x^2 - 5x + c = 0 is 3, find the value of c.

A) 3 B) 4 C) 5 D) 6

27. If alpha and beta are the roots of ax^2 + bx + c = 0, then the equation whose roots are alpha+k and beta+k is:

A) a(x-k)^2 + b(x-k) + c = 0 B) a(x+k)^2 + b(x+k) + c = 0 C) a(x-k) + b(x-k) + c = 0 D) a(x+k) + b(x+k) + c = 0

28. If alpha and beta are the roots of ax^2 + bx + c = 0, then the equation whose roots are 1/alpha and 1/beta is:

A) cx^2 + bx + a = 0 B) ax^2 + bx + c = 0 C) cx^2 - bx + a = 0 D) ax^2 - bx + c = 0

29. If the roots of x^2 + ax + b = 0 are real and distinct, then:

A) a^2 - 4b > 0 B) a^2 - 4b < 0 C) a^2 - 4b = 0 D) a^2 - 4b >= 0

30. Consider the equation x^2 + 2x + 3 = 0. The sum of squares of its roots (alpha^2 + beta^2) is:

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

31. If one root of the equation x^2 - 7x + k = 0 is 4, find the value of k.

A) 12 B) 10 C) 14 D) 16

32. For the equation x^2 + 5x + 6 = 0, the roots are:

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

33. If the roots of x^2 + px + 12 = 0 are equal, find the value of p.

A) sqrt(48) B) -sqrt(48) C) 6 D) -6

34. If alpha and beta are roots of x^2 - 5x + 6 = 0, find the value of 1/alpha + 1/beta.

A) 5/6 B) 6/5 C) -5/6 D) -6/5

35. If the roots of the equation x^2 + bx + c = 0 are alpha and beta, then the sum of roots alpha+beta is related to coefficients as:

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

36. If the roots of the equation x^2 + bx + c = 0 are alpha and beta, then the product of roots alpha*beta is related to coefficients as:

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

37. What is the condition for the quadratic equation ax^2 + bx + c = 0 to have real roots?

A) b^2 - 4ac > 0 B) b^2 - 4ac < 0 C) b^2 - 4ac = 0 D) b^2 - 4ac >= 0

38. If the roots of x^2 - 6x + k = 0 are real and distinct, what is the range of k?

A) k < 9 B) k > 9 C) k = 9 D) k <= 9

39. If the roots of x^2 - kx + 4 = 0 are real and equal, find the value of k.

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

40. Form a quadratic equation with roots -1 and -1.

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

41. If one root of the equation kx^2 - 3x + 2 = 0 is 1, find the value of k.

A) 1 B) 2 C) 3 D) 4

42. If alpha and beta are the roots of x^2 - 5x + 6 = 0, find alpha^2 + beta^2.

A) 13 B) 25 C) 36 D) 1

43. If the roots of ax^2 + bx + c = 0 are real and distinct, then b^2 - 4ac is:

A) Equal to 0 B) Greater than 0 C) Less than 0 D) Greater than or equal to 0

44. What is the nature of the roots of the equation x^2 + x + 1 = 0?

A) Real and distinct B) Real and equal C) Imaginary D) Zero

45. If the roots of x^2 + 4x + k = 0 are imaginary, find the range of k.

A) k > 4 B) k < 4 C) k = 4 D) k >= 4

46. If one root of x^2 - 3x + k = 0 is 1, find the other root.

A) 1 B) 2 C) 3 D) 4

47. Form a quadratic equation whose roots are 2+sqrt(3) and 2-sqrt(3).

A) x^2 - 4x + 1 = 0 B) x^2 + 4x + 1 = 0 C) x^2 - x + 4 = 0 D) x^2 + x - 4 = 0

48. If alpha and beta are the roots of x^2 - 5x + 6 = 0, find the value of (alpha+1)(beta+1).

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

49. If the roots of x^2 + bx + c = 0 are alpha and beta, then the equation whose roots are alpha^2 and beta^2 is:

A) x^2 - (b^2-2ac)x + c^2 = 0 B) x^2 + (b^2-2ac)x + c^2 = 0 C) x^2 - (b^2-2ac)x + a^2 = 0 D) x^2 + (b^2-2ac)x + a^2 = 0

50. If the roots of the equation x^2 - px + q = 0 are alpha and beta, then alpha^2 + beta^2 = ?

A) p^2 - 2q B) p^2 + 2q C) q^2 - 2p D) q^2 + 2p

Nature of roots and formation of quadratic equations - Online Test

Test Results