1. 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

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

3. 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

4. What is the value of \(k\) such that the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\)

5. 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

6. If the equation \(kx^2 + 6x + 4k = 0\) has equal values for the sum and product of its roots, then what is the value of \(k\)?

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

7. If the sum of the roots of the equation \(x^2 - (k + 6)x + 2(2k - 1) = 0\) is half of their product, then what is the value of \(k\)?

(a) 6 (b) 7 (c) 1 (d) 5

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

9. What is the value of \(k\) if the sum and product of the roots of the equation \(kx^2 + 2x + 3k = 0\) \((k \ne 0)\) are equal?

10. If the equation \(x^2 + k(4x + k - 1) + 2 = 0\) has equal roots, then what is the value of \(k\)?

(a) \(-\cfrac{2}{3}, 1\) (b) \(\cfrac{2}{3}, 1\) (c) \(\cfrac{2}{3}, \cfrac{1}{3}\) (d) \(-\cfrac{3}{2}, \cfrac{1}{3}\)

11. If the roots of the quadratic equation \(5x^2+13x+k=0\) are reciprocals of each other, then the value of \(k\) is:

(a) 3 (b) 4 (c) 5 (d) -5

12. 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.

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

(a) -2 (b) -8 (c) 8 (d) 12

14. If the roots of the quadratic equation \(2x^2 + 5x + k - 3 = 0\) are reciprocals of each other, then what is the value of \(k\)?

(a) 2 (b) 1 (c) 5 (d) 3

15. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –

(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

16. For what value of \(k\) will the equation \(2x^2 + 3x + k = 0\) have real and equal roots?

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

18. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?

19. If the equation \(x^2 - x = k(2x - 1)\) has a sum of roots equal to \(0\), then what is the value of \(k\)?

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

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

21. 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 \(\cfrac{32}{25}\).

22. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), determine the value of \(k\).

(a) -2 (b) -8 (c) 8 (d) 12

23. For what value of \(k\) will \(\cfrac{2}{3}\) be a root of the quadratic equation \(7x^2+kx-3=0\)?

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

25. If the product of the roots of the equation \(x^2 - 2x + k = 8\) is \(-2\), then the value of \(k\) is—

(a) -2 (b) 6 (c) 1 (d) -6

26. 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

27. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

28. Calculate the value(s) of \(k\) for which the quadratic equation \(49x^2 + kx + 1 = 0\) will have real and equal roots.

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

30. For which value(s) of \(k\) will the quadratic equation \(9x^2 - 24x + k = 0\) have real and equal roots?