1. The roots of the equation \(x^2 - 4x + 4 = 0\) are -
(a) 4,1 (b) 2,2 (c) -4,-1 (d) -2,-2
2. The roots of the equation \(x^2 - 4x + 3 = 0\) are -
(a) 3,1 (b) -3,-1 (c) 4,1 (d) -4,-1
3. What should be the value of \(m\) so that the roots of the quadratic equation \(4x^2 + 4(3m - 1)x + (m + 7) = 0\) are reciprocals of each other?
4. 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) \(\pm 2\)
5. If the equation \(x^2 + k(4x + k - 1) + 2 = 0\) has equal roots, then what is the value of \(k\)?
(a) \(-\cfrac{2}{3}, 1\) (b) \(\cfrac{2}{3}, 1\) (c) \(\cfrac{2}{3}, \cfrac{1}{3}\) (d) \(-\cfrac{3}{2}, \cfrac{1}{3}\)
6. Find the value of \(m\) if the roots of the quadratic equation \(4x^2+4(3m+1)x+(m-7)-20=0\) are distinct.
7. What should be the value of \(m\) so that the roots of the quadratic equation \(4x^2 + 4(3m - 1)x + (m + 7) = 0\) are reciprocals of each other?
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. 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
10. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).
11. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).
12. Check whether \(\cfrac{5}{6}\) and \(\cfrac{4}{3}\) are roots of the equation \(x + \cfrac{1}{x} = \cfrac{13}{6}\).
13. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).
14. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).
15. The roots of the equation \(x^2 + x + 1 = 0\) are real.
16. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).
17. 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 \(q\).
18. The roots of the equation \(x^2 - x + 2 = 0\) are not real.
19. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?
20. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + mx + 1 = 0\).
21. 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
22. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?
23. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]
24. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(7x^2 + 5x - 4 = 0\), determine the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).
25. Determine the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).
26. In the equation \(x^2+4x+3=(x+3)\), the coefficient of \(x°\) is ____.
27. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x - 10 = 0\), then find \(\alpha^2 + \beta^2\).
28. Let us check whether \(1\) and \(-1\) are roots of the quadratic equation \(x^2 + x + 1 = 0\).
29. For which value(s) of \(k\) will the quadratic equation \(9x^2 - 24x + k = 0\) have real and equal roots?
30. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —
(a) >0 (b) =0 (c) <0 (d) none of the above
