1. 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) \(a=-2\)

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

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 equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.

(a) -8 (b) -1 (c) 3 (d) 8

5. 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

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

7. What value of \(P\) will make the equation \((P - 3)x^2 - 5x + 10 = 0\) not a quadratic equation? \(P =\) _____

8. 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

9. 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

10. 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

11. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

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

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

14. 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.

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

16. 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.

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

18. Under what condition will one root of the quadratic equation \(ax^2 + bx + c = 0\) be zero?

(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these

19. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

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

20. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

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

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

23. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .

24. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.

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

26. 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

27. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2-3x+6=0\), determine the value of \(\left(\cfrac{1}{\alpha}+\cfrac{1}{\beta}\right)\).

28. 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

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

30. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).