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

2. 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?

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

4. 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

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

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

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

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

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

11. 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 \(k\).

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

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

14. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

15. 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

16. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

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

18. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

19. 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

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. 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 \(\frac{25}{36}\)

22. 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 \(k\).

23. What is the ratio of the sum and product of the roots of the equation \[ 7x^2 - 66x + 27 = 0? \]

24. 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

25. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

26. 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

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

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

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

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