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

2. 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) \(a^2+ac\)

3. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), then the value of \(\left(\cfrac{1}{α} + \cfrac{1}{β}\right)\) is —

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

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

5. If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then find the value of \[ \frac{a\alpha^2}{b\alpha + c} - \frac{a\beta^2}{b\beta + c} \]

6. 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

7. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\).

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

8. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

9. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

10. If the equation \(ax^2 + bx + c = 0\) has equal roots, then what is the value of \(c\)?

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

11. If the equation \(ax^2 + 2bx + c = 0\) has equal roots, then what is the value of \(c\)?

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

12. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

13. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).

14. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

15. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

16. 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

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

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

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

20. 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?

21. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

22. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

23. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

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

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

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

27. If \(\alpha, \beta\) are the roots of the equation \(3x^2+8x+2=0\), then the value of \( \cfrac{1}{\alpha}+\cfrac{1}{\beta}\) is –

(a) \(-\cfrac{3}{8}\) (b) 4 (c) \(\cfrac{2}{3}\) (d) -4

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

29. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \[ 5x^2 + 2x - 3 = 0, \] find the value of \(\alpha^2 + \beta^2\).

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