1. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).
2. The roots of the equation \(x^2 - 18x + 8 = 0\) are ā
(a) Real , Rational , Unequal (b) equal,Rational (c) Real , Rational , equal (d) None of the above
3. 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) \(\cfrac{1}{2},-\cfrac{1}{2}\)
4. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).
5. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?
6. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).
7. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).
8. The roots of the equation \(x^2 + x + 1 = 0\) are real.
9. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).
10. 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 \(q\).
11. The roots of the equation \(x^2 - x + 2 = 0\) are not real.
12. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + mx + 1 = 0\).
13. The roots of the equation \(x^2 - 4x + 4 = 0\) are -
(a) 4,1 (b) 2,2 (c) -4,-1 (d) -2,-2
14. The roots of the equation \(x^2 - 4x + 3 = 0\) are -
(a) 3,1 (b) -3,-1 (c) 4,1 (d) -4,-1
15. 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) \(\pm 2\)
16. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).
17. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]
18. 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.
19. Find the value of \(k\) if the roots of the equation \(x^2 - x = k(2x - 1)\) are equal and opposite in sign.
20. Determine the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).
21. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x - 10 = 0\), then find \(\alpha^2 + \beta^2\).
22. If the quadratic equation \(x^2 - x = k(2x - 1)\) has roots that are equal in magnitude but opposite in sign, determine the value of \(k\).
23. Let us check whether \(1\) and \(-1\) are roots of the quadratic equation \(x^2 + x + 1 = 0\).
24. The roots of the equation \(x^2 = 6x\) are ______ and ______.
25. The roots of the equation \(x^2 + x + 1 = 0\) are real.
26. The roots of the equation \(x^2 - x + 2 = 0\) are not real.
27. If \(Ī±\) and \(Ī²\) are the roots of the equation \(x^2 - 22x + 105 = 0\), find the value of \((Ī± - Ī²)\).
28. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).
29. 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? Iāve got both options ready.
30. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then proveāwithout solvingāthat the value of \(p\) will be either \(1\) or \(9\).
