1. 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
2. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?
(a) -11 (b) 11 (c) 12 (d) 14
3. Both roots of the quadratic equation \(ax^2+bx+c = 0\) will be zero when?
4. The roots of the equation \(ax^2+bx+c=0\) will be equal in magnitude but opposite in sign if-
(a) \(c=0, a≠0\) (b) \(b=0, a≠0\) (c) \(c=0, a=0\) (d) \(b=0, a=0\)
5. 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
6. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.
(a) -8 (b) -1 (c) 3 (d) 8
7. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when –
(a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2≠ 4ac \) (d) \(b^2<4ac\)
8. A root of the equation \(ax^2 + bx + c = 0\) being zero requires the condition—
(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these
9. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is zero.
(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of the above
10. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).
11. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).
12. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.
13. 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
14. For what value of \(k\) will \(\cfrac{2}{3}\) be a root of the quadratic equation \(7x^2+kx-3=0\)?
15. 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.
16. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).
17. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).
18. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).
19. 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
20. The roots of the equation \(ax^2 + bx - c = 0\) (where \(a \ne 0\)) will be equal if...
(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c=\cfrac{b^2}{4a}\) (d) \(c=\cfrac{-b^2}{4a}\)
21. For what value of \(k\), will \(\frac{2}{3}\) be a root of the quadratic equation \(7x^2 + kx - 3 = 0\)?
22. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]
23. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .
24. 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?
25. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
26. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]
27. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?
28. If the equation \(ax^2 + bx + c = 0\) has equal roots, then what is the value of \(c\)?
(a) \(\cfrac{-b}{2a}\) (b) \(\cfrac{b}{2a}\) (c) \(\cfrac{-b^2}{4a}\) (d) \(\cfrac{b}{4a}\)
29. If the equation \(ax^2 + 2bx + c = 0\) has equal roots, then what is the value of \(c\)?
(a) \(\cfrac{b^2}{a}\) (b) \(\cfrac{b^2}{4a}\) (c) \(\cfrac{a^2}{b}\) (d) \(\cfrac{a^2}{4b}\)
30. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?
