1. 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.

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

3. 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

4. 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

5. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

6. Determine for which value of \(a\) the equation \((a - 2)x^2 + 3x + 5 = 0\) will not be a quadratic equation.

(a) \(a=0\) (b) \(a=2\) (c) \(a=4\) (d) \(a=-2\)

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

8. The equation \((a - 2)x^2 + 3x + 5 = 0\) will not be a quadratic equation for the value of \(a =\) ______.

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

10. 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

11. 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

12. 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

13. 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

14. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and curved surface area will be equal.

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 \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

(a) \(\cfrac{\pi^c}{2}\) (b) \(\cfrac{\pi^c}{3}\) (c) \(\pi^c\) (d) \(\cfrac{\pi^c}{4}\)

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

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

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

20. Determine the values of \(\theta\) for which \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true, given that \(0^\circ < \theta < 90^\circ\).

21. Let us determine for which value of \(a\) the equation \((a - 2)x^2 + 3x + 5 = 0\) will **not** be a quadratic equation.

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

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

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

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

26. 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.

27. 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?

28. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and its curved surface area will be equal.

29. Determine the value(s) of \(\theta\) \((0^\circ \le \theta \le 90^\circ)\) for which the equation \(\sin^2\theta - 3\sin\theta + 2 = 0\) holds true.

30. Determine the value(s) of \(\theta\) within the range \(0^\circ \leq \theta \leq 90^\circ\) for which the equation \(2\cos^2\theta - 3\cos\theta + 1 = 0\) holds true.