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

2. Calculate the value(s) of \(k\) for which the quadratic equation \(2x^2 + 3x + k = 0\) will have real and equal roots.

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

4. For which value(s) of \(k\) will the quadratic equation \((3k + 1)x^2 + 2(k + 1)x + k = 0\) have real and equal roots?

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

6. Calculate the value(s) of \(k\) for which the quadratic equation \(x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0\) will have real and equal roots.

7. For which value or values of \(k\) will the following equation have real and equal roots: \[ (3k + 1)x^2 + 2(k + 1)x + k = 0 \]

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

9. 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

10. 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?

11. For what value of \(k\) will the system of equations \[ x + 5y = 8 \quad \text{and} \quad 2x - ky = 13 \] have no solution?

(a) 1 (b) 5 (c) 10 (d) -10

12. For what value of \(k\) will the system of equations \[ x - ky = k \quad \text{and} \quad x + (k - 2)y = 2 \] have no solution?

(a) -1 (b) 1 (c) 2 (d) None of the above

13. For what value of \(k\) will the system of equations \[ 3x + y = \frac{1}{k - 4}, \quad 2x + 3y = 5 \] have no solution?

(a) 3 (b) 2 (c) 4 (d) None of the above

14. For what value of \(k\), the system of equations \(2x + ky + 1 = 0\) and \(3x + 2y = 2\) has no solution?

(a) \(\cfrac{1}{2}\) (b) \(-\cfrac{1}{2}\) (c) \(\cfrac{4}{3}\) (d) \(-\cfrac{4}{3}\)

15. For what value of \(k\) will the system of equations \(x + (k - 2)y = 5\) and \((k - 3)x + 2y = 7\) have no common solution?

(a) 2 (b) 3 (c) 4 (d) 5

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

17. 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

18. 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.

19. Write the value of \(k\) for which the quadratic equation \(9x^2 + 3kx + 4 = 0\) has real and equal roots.

20. Determine the value of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) will have equal and real roots.

21. 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

22. 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

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

24. The value of \(k\) for which the system of equations \(2x + 5y = 8\) and \(2x - ky = 3\) will have no solution, is —

(a) 5 (b) -5 (c) 6 (d) -6

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

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

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

28. 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 \(k\).

29. 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

30. 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 \(9\).