JEE Main - Quadratic equations in real and complex systems Online Test (MCQ Quiz)

1. What is the standard form of a quadratic equation?

A) ax + b = 0 B) ax^2 + bx + c = 0 C) ax^3 + bx^2 + cx + d = 0 D) y = mx + c

2. For a quadratic equation ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0, what does the discriminant (Δ) represent?

A) The sum of the roots B) The product of the roots C) The nature of the roots D) The coefficient of the x term

3. If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is positive (Δ > 0), what can be said about its roots?

A) The roots are real and equal B) The roots are real and distinct C) The roots are complex conjugates D) One root is real, and the other is complex

4. If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is zero (Δ = 0), what can be said about its roots?

A) The roots are real and distinct B) The roots are real and equal C) The roots are complex conjugates D) The roots are purely imaginary

5. If the discriminant (Δ = b^2 - 4ac) of a quadratic equation with real coefficients is negative (Δ < 0), what can be said about its roots?

A) The roots are real and distinct B) The roots are real and equal C) The roots are complex conjugates D) The roots are zero

6. For the quadratic equation ax^2 + bx + c = 0, what is the sum of the roots (α + β)?

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

7. For the quadratic equation ax^2 + bx + c = 0, what is the product of the roots (αβ)?

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

8. Which formula gives the roots of a quadratic equation ax^2 + bx + c = 0?

A) x = [-b ± √(b^2 - 4ac)] / 2a B) x = [-b ± √(b^2 + 4ac)] / 2a C) x = [b ± √(b^2 - 4ac)] / 2a D) x = [-b ± √(4ac - b^2)] / 2a

9. If one root of the quadratic equation x^2 - 5x + k = 0 is 2, what is the value of k?

A) 3 B) 6 C) 8 D) 10

10. Find the quadratic equation whose roots are 3 and 4.

A) x^2 - 7x + 12 = 0 B) x^2 + 7x + 12 = 0 C) x^2 - 12x + 7 = 0 D) x^2 + 12x + 7 = 0

11. What are the roots of the quadratic equation x^2 + 4 = 0?

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

12. What are the roots of the quadratic equation x^2 - 6x + 13 = 0?

A) 3 + 2i, 3 - 2i B) 3 + i, 3 - i C) 6 + 2i, 6 - 2i D) 2 + 3i, 2 - 3i

13. If the roots of the quadratic equation px^2 + qx + r = 0 are equal, what is the condition on its coefficients?

A) q^2 - 4pr > 0 B) q^2 - 4pr = 0 C) q^2 - 4pr < 0 D) q^2 = 4pr

14. Consider the quadratic equation ax^2 + bx + c = 0. If the coefficients a, b, and c are complex numbers, what can be said about the nature of the roots?

A) The roots must be complex conjugates. B) The roots are not necessarily complex conjugates. C) The roots must be real. D) The roots must be equal.

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

A) Real and distinct B) Real and equal C) Complex and not conjugates D) Complex conjugates

16. If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are α + 1 and β + 1 is:

A) a(x-1)^2 + b(x-1) + c = 0 B) a(x+1)^2 + b(x+1) + c = 0 C) a(x-1) + b(x-1) + c = 0 D) ax^2 + b(x-1) + c = 0

17. If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are 1/α and 1/β is:

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

18. If α and β are the roots of ax^2 + bx + c = 0, then the quadratic equation whose roots are α^2 and β^2 is:

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

19. The equation x^2 + |x| + 1 = 0 has:

A) Two real roots B) No real roots C) One real root D) Two complex roots

20. The equation x^2 - |x| - 2 = 0 has:

A) Two real roots B) Four real roots C) No real roots D) One real root

21. If one root of the equation (k+1)x^2 - 3(k+4)x + 4(k+3) = 0 is 1, then find the value of k.

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

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

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

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

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

24. The number of real solutions of the equation x^2 + 2|x| + 1 = 0 is:

A) 0 B) 1 C) 2 D) Infinite

25. If the equation x^2 - bx + 1 = 0 has real roots, then:

A) b < -2 or b > 2 B) -2 < b < 2 C) b = 2 or b = -2 D) b = 0

26. What is the value of k for which the equation x^2 - kx + 4 = 0 has equal roots?

A) 4 B) -4 C) ±4 D) ±2

27. If one root of the equation ax^2 + bx + c = 0 is 3 times the other, then:

A) 3b^2 = 16ac B) b^2 = 3ac C) 3b^2 = ac D) b^2 = 9ac

28. The roots of the quadratic equation (a+b)x^2 + 2(a+b)x + c = 0 are:

A) Real and distinct B) Real and equal C) Complex D) One real and one complex

29. If the roots of the equation x^2 + ax + b = 0 are α and β, and the roots of x^2 + cx + d = 0 are α + k and β + k, then:

A) a-c = k B) a+c = k C) c-a = k D) a=c

30. If α and β are the roots of x^2 - 2x + 4 = 0, then the value of α^3 is:

A) 8 B) -8 C) 8i D) -8i

31. If α and β are the roots of x^2 + x + 1 = 0, then the value of α^3 + β^3 is:

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

32. The equation ax^2 + bx + c = 0 has real roots if:

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

33. If the sum of the roots of the equation ax^2 + bx + c = 0 is equal to the product of the roots, then:

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

34. The quadratic equation that has roots 2+i and 2-i is:

A) x^2 - 4x + 5 = 0 B) x^2 + 4x + 5 = 0 C) x^2 - 4x - 5 = 0 D) x^2 + 4x - 5 = 0

35. If one root of the quadratic equation x^2 - px + q = 0 is sin(30°) and the other is cos(60°), find p and q.

A) p=1, q=1/4 B) p=1/2, q=1/4 C) p=1, q=1/2 D) p=1/4, q=1

36. The values of x satisfying the equation x^2 - |x| - 6 = 0 are:

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

37. If the roots of the equation (p^2 + q^2)x^2 - 2(p+q)x + 2 = 0 are real, then:

A) p=q B) p=-q C) p=2q D) q=2p

38. If α, β are roots of ax^2 + bx + c = 0, then the equation whose roots are α+k and β+k is given by:

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) ax^2 + b(x-k) + c = 0

39. If α and β are the roots of x^2 - 3x + 1 = 0, then the value of (α^2 + 1)(β^2 + 1) is:

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

40. The number of real roots of the equation x^2 + 1/x^2 - (x + 1/x) - 2 = 0 is:

A) 0 B) 1 C) 2 D) 4

41. If the roots of the equation x^2 + ax + b = 0 are α and β, and the roots of x^2 + cx + d = 0 are αβ and 1/(αβ), then:

A) a = c B) b = d C) b = 1/d D) a = c^2

42. If α and β are the roots of ax^2 + bx + c = 0, then the value of (α + 1/β)(β + 1/α) is:

A) (b^2 - ac)/a^2 B) (ac - b^2)/a^2 C) (b^2 + ac)/a^2 D) (b^2 - ac)/c^2

43. The minimum value of the expression x^2 - 4x + 7 is:

A) 3 B) 4 C) 7 D) 1

44. If one root of the equation ax^2 + bx + c = 0 is 0, then:

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

45. The roots of the equation (x^2 - 5x + 5)^(x^2 - 11x + 30) = 1 are:

A) 1, 2, 3, 5, 6 B) 1, 5, 6 C) 2, 3, 5, 6 D) 1, 2, 3, 5

46. If the roots of the equation x^2 - 6x + k = 0 are α and β, and α^2 + β^2 = 20, then k is:

A) 2 B) 4 C) 6 D) 8

47. The number of real solutions for the equation |x^2 - 1| = |x - 1| is:

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

48. If α and β are the roots of x^2 - (k-2)x + (k+1) = 0, find k such that α+β = αβ.

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

49. If the roots of the equation ax^2 + bx + c = 0 are in the ratio m:n, then:

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

50. If α and β are roots of x^2 + 2x + 4 = 0, then α^3 + β^3 is:

A) 16 B) -16 C) 32 D) -32

Quadratic equations in real and complex systems - Online Test

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