1. 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

2. 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

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

5. 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

6. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

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 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

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

10. 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

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

12. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

13. 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) \(-\cfrac{3}{2}, \cfrac{1}{3}\)

14. 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) \(-\cfrac{2}{3}\)

15. 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

16. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is 2, find the value of \(k\).

17. If the sum and the product of the roots of the equation \(kx^2 + 2x + 3k = 0\) \((k \ne 0)\) are equal, find the value of \(k\).

18. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is zero, find the value of \(k\).

19. If one of the roots of each of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), find the value of \(q\).

20. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

21. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

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

22. 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

23. 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

24. 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

25. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

26. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?

27. 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?

28. 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?

29. Find the value of \(k\) if the roots of the equation \(x^2 - 2kx + 4 = 0\) are equal.

(a) \(\pm 1\) (b) \(\pm 3\) (c) \(\pm 4\) (d) \(\pm 2\)

30. 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) \(-\cfrac{1}{2}\)