1. If the product of the roots of the quadratic equation \(3x^2 – 4x + k = 0\) is 5, then what will be the value of \(k\)?

(a) 5 (b) -12 (c) 15 (d) -20

2. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is 4, then what is the value of \(b\)?

3. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.

4. If \(\alpha, \beta\) are the roots of the equation \(3x^2+8x+2=0\), then the value of \( \cfrac{1}{\alpha}+\cfrac{1}{\beta}\) is –

(a) \(-\cfrac{3}{8}\) (b) 4 (c) \(\cfrac{2}{3}\) (d) -4

5. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0

6. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

7. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2 + 2x - 3 = 0\), then the value of \(\alpha^2 + \beta^2\) will be \(\cfrac{32}{25}\).

8. If the roots of \(5x^2 - 6x + c = 0\) are distinct, then the value of \(c\) will be -.

9. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation whose roots are \(\alpha\) and \(\beta\), find the value of \(\alpha^3 + \beta^3\).

10. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).

11. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{α^2}{β} + \frac{β^2}{α}\).

12. If the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\), then the value of \(k\) will be \(12\).

13. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:

(a) >0 (b) <0 (c) 0 (d) None of these

14. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when –

(a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2≠ 4ac \) (d) \(b^2<4ac\)

15. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –

(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

16. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be –

(a) 2 (b) -2 (c) 6 (d) -6

17. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants, and \(a \ne 0\)) will be the equation of the x-axis when:

(a) b = a (b) b = -a (c) b = 2 (d) b = 0

18. If the product of the roots of the equation \(x^2 – 5x + k = 12\) is \(–3\), find the value of \(k\).

19. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

20. The ratio of the product and the sum of the roots of the equation \(ax^2 - bx + c = 0\) is – ____

21. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

22. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

23. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2-3x+6=0\), determine the value of \(\left(\cfrac{1}{\alpha}+\cfrac{1}{\beta}\right)\).

24. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?

25. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).

26. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(7x^2 + 5x - 4 = 0\), determine the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

27. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2+8x+2=0\), find the value of \(\cfrac{1}{\alpha^2}+\cfrac{1}{\beta^2}\).

28. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

29. For which value(s) of \(k\) will the quadratic equation \(3x^2 - 5x + 2k = 0\) have real and equal roots?

30. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative method—that is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.