JEE Main - Relations between roots and coefficients Online Test (MCQ Quiz)

1. For a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, if $\alpha$ and $\beta$ are the roots, what is the sum of the roots?

A) $\frac{c}{a}$ B) $\frac{-b}{a}$ C) $\frac{b}{c}$ D) $\frac{-c}{a}$

2. For a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, if $\alpha$ and $\beta$ are the roots, what is the product of the roots?

A) $\frac{c}{a}$ B) $\frac{-b}{a}$ C) $\frac{b}{c}$ D) $\frac{-c}{a}$

3. If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, find $\alpha + \beta$.

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

4. If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, find $\alpha \beta$.

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

5. Consider the quadratic equation $2x^2 + 4x + 3 = 0$. If its roots are $\alpha$ and $\beta$, what is $\alpha + \beta$?

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

6. Consider the quadratic equation $2x^2 + 4x + 3 = 0$. If its roots are $\alpha$ and $\beta$, what is $\alpha \beta$?

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

7. If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the quadratic equation whose roots are $\alpha + 1$ and $\beta + 1$ is:

A) $(x-1)^2 + p(x-1) + q = 0$ B) $x^2 + p(x-1) + q = 0$ C) $x^2 + (p+1)x + (q+1) = 0$ D) $x^2 + (p-1)x + (q-1) = 0$

8. If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the quadratic equation whose roots are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:

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

9. If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

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

10. If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\alpha^2 + \beta^2$.

A) 5 B) 7 C) 9 D) 13

11. If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find the value of $\alpha^2 + \beta^2$.

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

12. If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 3 = 0$, find the value of $\alpha^3 + \beta^3$.

A) 6 B) -6 C) 0 D) 12

13. If the sum of the roots of the quadratic equation $kx^2 - 6x - 2 = 0$ is 3, find the value of $k$.

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

14. If the product of the roots of the quadratic equation $3x^2 + kx - 5 = 0$ is -5/3, find the value of $k$.

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

15. If one root of the quadratic equation $x^2 - kx + 8 = 0$ is 2, find the value of $k$.

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

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

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

17. If the roots of the quadratic equation $x^2 - px + q = 0$ are $\alpha$ and $\beta$, and the roots of $x^2 - rx + s = 0$ are $\alpha + \delta$ and $\beta + \delta$, then what is $r-p$?

A) $\delta$ B) 2$\delta$ C) 3$\delta$ D) $\delta^2$

18. If $\alpha$ and $\beta$ are the roots of $x^2 + bx + c = 0$, then the equation whose roots are $\alpha - k$ and $\beta - k$ is:

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

19. If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $(\alpha + 1)(\beta + 1)$ is:

A) $\frac{c+b+a}{a}$ B) $\frac{c-b+a}{a}$ C) $\frac{c+b-a}{a}$ D) $\frac{c-b-a}{a}$

20. If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 5 = 0$, find the value of $(\alpha - 1)(\beta - 1)$.

A) 7 B) -7 C) 5 D) 3

21. If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $\alpha^2 \beta + \alpha \beta^2$ is:

A) $\frac{-bc}{a^2}$ B) $\frac{bc}{a^2}$ C) $\frac{abc}{a^2}$ D) $\frac{-abc}{a^2}$

22. If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 4 = 0$, find the value of $\alpha^2 \beta + \alpha \beta^2$.

A) -4 B) 4 C) 8 D) -8

23. If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the equation whose roots are $\alpha^2$ and $\beta^2$ is:

A) $x^2 - (p^2 - 2q)x + q^2 = 0$ B) $x^2 + (p^2 - 2q)x + q^2 = 0$ C) $x^2 - (p^2 + 2q)x + q^2 = 0$ D) $x^2 + (p^2 + 2q)x + q^2 = 0$

24. If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$.

A) $x^2 - 13x + 36 = 0$ B) $x^2 + 13x + 36 = 0$ C) $x^2 - 5x + 36 = 0$ D) $x^2 + 5x + 36 = 0$

25. If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$.

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

26. If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $(\alpha - \beta)^2$ is:

A) $\frac{b^2 - 4ac}{a^2}$ B) $\frac{4ac - b^2}{a^2}$ C) $\frac{b^2 + 4ac}{a^2}$ D) $\frac{4ac + b^2}{a^2}$

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

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

28. If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 3 = 0$, find the value of $(\alpha - \beta)^2$.

A) 4 B) -4 C) 8 D) -8

29. If $\alpha$ and $\beta$ are the roots of $x^2 - 7x + 12 = 0$, then the value of $\alpha^3 + \beta^3$ is:

A) 189 B) -189 C) 27 D) -27

30. If $\alpha$ and $\beta$ are the roots of $x^2 - 6x + 8 = 0$, then the value of $\alpha^3 + \beta^3$ is:

A) 144 B) -144 C) 216 D) -216

31. If $\alpha$ and $\beta$ are the roots of $x^2 + 3x + 5 = 0$, then the value of $(\alpha + 2)(\beta + 2)$ is:

A) 7 B) -7 C) 5 D) 3

32. If $\alpha$ and $\beta$ are the roots of $x^2 - 4x + 3 = 0$, find the equation whose roots are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$.

A) $3x^2 - 4x + 1 = 0$ B) $3x^2 + 4x + 1 = 0$ C) $x^2 - 4x + 3 = 0$ D) $x^2 + 4x + 3 = 0$

33. If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $(\alpha + \beta)^2$ is:

A) $p^2$ B) $q^2$ C) $p^2 - 4q$ D) $p^2 + 4q$

34. If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $(\alpha \beta)^2$ is:

A) $p^2$ B) $q^2$ C) $p^2 - 4q$ D) $p^2 + 4q$

35. If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ is:

A) $\frac{p^2 - 2q}{q}$ B) $\frac{p^2 + 2q}{q}$ C) $\frac{2q - p^2}{q}$ D) $\frac{2q + p^2}{q}$

36. If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

A) 25/6 B) -25/6 C) 13/6 D) -13/6

37. If the roots of $x^2 + ax + b = 0$ are $\alpha$ and $\beta$, then the equation whose roots are $\alpha^2$ and $\beta^2$ is:

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

38. If the roots of $x^2 - 6x + 5 = 0$ are $\alpha$ and $\beta$, find the equation whose roots are $\alpha^2$ and $\beta^2$.

A) $x^2 - 26x + 25 = 0$ B) $x^2 + 26x + 25 = 0$ C) $x^2 - 11x + 25 = 0$ D) $x^2 + 11x + 25 = 0$

39. If the roots of $x^2 + px + q = 0$ are $\alpha$ and $\beta$, then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is:

A) $q x^2 - (p^2 - 2q)x + q = 0$ B) $q x^2 + (p^2 - 2q)x + q = 0$ C) $q x^2 - (p^2 + 2q)x + q = 0$ D) $q x^2 + (p^2 + 2q)x + q = 0$

40. If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 4 = 0$, find the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

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

41. If $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$, then the value of $\alpha^3+\beta^3$ is:

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

42. If $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$, then the value of $\alpha^4+\beta^4$ is:

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

43. If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the equation whose roots are $\alpha+2$ and $\beta+2$.

A) $x^2 - 7x + 10 = 0$ B) $x^2 + 7x + 10 = 0$ C) $x^2 - 3x + 4 = 0$ D) $x^2 + 3x + 4 = 0$

44. If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the equation whose roots are $\alpha-1$ and $\beta-1$.

A) $x^2 - 3x + 2 = 0$ B) $x^2 + 3x + 2 = 0$ C) $x^2 - 5x + 4 = 0$ D) $x^2 + 5x + 4 = 0$

45. If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the equation whose roots are $\alpha^3$ and $\beta^3$ is:

A) $x^2 + (p^3 - 3pq)x + q^3 = 0$ B) $x^2 - (p^3 - 3pq)x + q^3 = 0$ C) $x^2 + (p^3 + 3pq)x + q^3 = 0$ D) $x^2 - (p^3 + 3pq)x + q^3 = 0$

46. If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 3 = 0$, find the equation whose roots are $\alpha^3$ and $\beta^3$.

A) $x^2 + 6x + 27 = 0$ B) $x^2 - 6x + 27 = 0$ C) $x^2 + 6x - 27 = 0$ D) $x^2 - 6x - 27 = 0$

47. If the roots of $x^2 - px + q = 0$ are $\alpha$ and $\beta$, then the value of $\alpha^2 + \beta^2 + \alpha \beta$ is:

A) $p^2 - q$ B) $p^2 + q$ C) $p^2 - 3q$ D) $p^2 + 3q$

48. If the roots of $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, then the value of $\alpha^2 + \beta^2 + \alpha \beta$ is:

A) 13 B) 7 C) 19 D) 25

49. If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$.

A) 5/4 B) -5/4 C) 13/4 D) -13/4

50. If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 3 = 0$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$.

A) -2/9 B) 2/9 C) 4/9 D) -4/9

Relations between roots and coefficients - Online Test

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