1. The roots of the equation \(x^2 = 6x\) are ______ and ______.
2. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.
(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0
3. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be –
(a) 2 (b) -2 (c) 6 (d) -6
4. The product of the roots of the equation \(x^2−7x+3=0\) is—?
(a) 7 (b) -7 (c) 3 (d) -3
5. 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
6. 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) \(\cfrac{1}{2},-\cfrac{1}{2}\)
7. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).
8. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).
9. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).
10. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?
11. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.
12. Without solving, determine all values of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) has real and equal roots.
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. If the product of the roots of the equation \(x^2 – 5x + k = 12\) is \(–3\), find the value of \(k\).
19. The roots of the equation \(x^2 - x + 2 = 0\) are not real.
20. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + mx + 1 = 0\).
21. The roots of the equation \(x^2 - 4x + 4 = 0\) are -
(a) 4,1 (b) 2,2 (c) -4,-1 (d) -2,-2
22. The roots of the equation \(x^2 - 2x + 1 = 0\) will be -
(a) 2,1 (b) 1,1 (c) 1,3 (d) 2,2
23. The roots of the equation \(x^2 - 4x + 3 = 0\) are -
(a) 3,1 (b) -3,-1 (c) 4,1 (d) -4,-1
24. 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) \(\pm 2\)
25. If the equation \(x^2 - x = k(2x - 1)\) has a sum of roots equal to \(0\), then what is the value of \(k\)?
(a) -2 (b) 2 (c) \(\cfrac{1}{2}\) (d) \(-\cfrac{1}{2}\)
26. 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) \(-\cfrac{3}{2}, \cfrac{1}{3}\)
27. If the sum of the roots of the equation \(x^2 - (k + 6)x + 2(2k - 1) = 0\) is half of their product, then what is the value of \(k\)?
(a) 6 (b) 7 (c) 1 (d) 5
28. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), determine the value of \(k\).
(a) -2 (b) -8 (c) 8 (d) 12
29. If one of the roots of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), determine the value of \(q\).
30. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]
