1. 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

2. 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 \(9\).

3. 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

4. The product of the roots of the equation \(x^2−7x+3=0\) is—?

(a) 7 (b) -7 (c) 3 (d) -3

5. What are the roots of the equation \(x^2 - 4x + 4 = 0\)?

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

6. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

7. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).

8. Check whether 0 and -2 are roots of the quadratic equation \(8x^2 + 7x = 0\).

9. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).

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

11. The roots of the equation \(x^2 + x + 1 = 0\) are real.

12. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).

13. 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 \(q\).

14. The roots of the equation \(x^2 - x + 2 = 0\) are not real.

15. If the roots of the equation \(ax^2 + bx + c = 0\) are equal (where \(a \ne 0\)) —

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

16. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + mx + 1 = 0\).

17. The roots of the equation \(x^2 - 4x + 4 = 0\) are -

(a) 4,1 (b) 2,2 (c) -4,-1 (d) -2,-2

18. The roots of the equation \(x^2 - 4x + 3 = 0\) are -

(a) 3,1 (b) -3,-1 (c) 4,1 (d) -4,-1

19. Find the value of \(k\) if the roots of the equation \(x^2 - 2kx + 4 = 0\) are equal.

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

20. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

21. If the roots of the equation \(3x^2+8x+2=0\) are \(\alpha\) and \(\beta\), then determine the equation whose roots are \(\cfrac{1}{\alpha}\) and \(\cfrac{1}{\beta}\).

22. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2 - 8x + 2 = 0\), determine the equation whose roots are \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\).

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

24. If the product of the roots of the equation \(x^2 - 2x + k = 8\) is \(-2\), then the value of \(k\) is—

(a) -2 (b) 6 (c) 1 (d) -6

25. Determine the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).

26. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x - 10 = 0\), then find \(\alpha^2 + \beta^2\).

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. Let us check whether \(1\) and \(-1\) are roots of the quadratic equation \(x^2 + x + 1 = 0\).

29. Let us check whether \(0\) and \(-2\) are roots of the quadratic equation \(8x^2 + 7x = 0\).

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

(a) -2 (b) -8 (c) 8 (d) 12