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

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

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

4. Prove that:The squares of the roots of the equation \(x^2 + x + 1 = 0\) are also roots of the same equation \(x^2 + x + 1 = 0\). Would you like me to walk through the proof again in English, or are you asking for a translated version of your original Bengali explanation? I’ve got both options ready.

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

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

7. Form the equation whose roots are the squares of the roots of the equation \[ x^2 + x + 1 = 0 \]

8. The roots of the equation \(x^2 - 18x + 8 = 0\) are —

(a) Real , Rational , Unequal (b) equal,Rational (c) Real , Rational , equal (d) None of the above

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

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

11. 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 \(q\).

12. 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 \(\cfrac{1}{\beta}\).

13. Determine the equation whose roots are the cubes of the roots of the equation \( 2x^2 - 3x + 1 = 0 \).

14. Determine the equation whose roots are the cubes of the roots of the equation \( 2x^2 - 3x + 1 = 0 \).

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

16. Form the equation whose roots are the **reciprocals** of the roots of the equation \[ x^2 + px + 1 = 0 \]

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

18. If \(α\) and \(β\) are the roots of the equation \(x^2 - 22x + 105 = 0\), find the value of \((α - β)\).

19. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

20. 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 \(9\).

21. If the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\), then the value of \(k\) will be \(12\).

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

23. 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

24. 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

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

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

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

28. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

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

30. Check whether \(\cfrac{5}{6}\) and \(\cfrac{4}{3}\) are roots of the equation \(x + \cfrac{1}{x} = \cfrac{13}{6}\).