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

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

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

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

5. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is zero, then the value of \(k\) is \(\frac{1}{2}\).

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

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

8. If the sum of the roots of the equation \(x^2 - x = k(2x - 1)\) is zero, determine the product of the roots.

9. Find the value of \(k\) if the roots of the equation \(x^2 - x = k(2x - 1)\) are equal and opposite in sign.

10. If the quadratic equation \(x^2 - x = k(2x - 1)\) has roots that are equal in magnitude but opposite in sign, determine the value of \(k\).

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

12. If the sum of the roots of the quadratic equation \[ x^2 - x = k(2x - 1) \] is zero, then what is the value of \(k\)?

13. 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 \(q\).

14. 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

15. 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

16. 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 \(k\).

17. 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 \(q\).

18. 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.

19. 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

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

21. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

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

23. 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?

24. For what value of \(k\) will the sum of the roots of the equation \[ x^2 - x = k(2x - 1) \] be zero?

25. 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) \(\pm 2\)

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

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

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

29. 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

30. If the quadratic equation \(x^2 + px + q = 0\) has roots \(\alpha\) and \(\beta\), then find the value of \(\alpha^3 + \beta^3\).