1. Under what condition will one root of the quadratic equation \(ax^2 + bx + c = 0\) be zero?
(a) \(a=0\) (b) \(b=0\) (c) \(c=0\) (d) None of these
2. 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
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. 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
6. 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) \(b^2<4ac\)
7. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]
8. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .
9. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.
10. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).
11. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
12. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, prove that \(2b^2 = 9ac\).
13. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]
14. If the roots of the equation \(ax^2 + bx + c = 0\) are equal (where \(a \ne 0\)) —
(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c=\cfrac{b^2}{4a}\) (d) \(c=\cfrac{-b^2}{4a}\)
15. The ratio of the product and the sum of the roots of the equation \(ax^2 - bx + c = 0\) is – ____
16. 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) \(\cfrac{b}{4a}\)
17. 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) \(\cfrac{a^2}{4b}\)
18. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).
19. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:p\), prove that \(\cfrac{(p+1)^2}{p} = \cfrac{b^2}{ac}\).
20. The condition under which the roots of the equation \(ax^2 + bx + c = 0\) are reciprocals of each other is:
(a) \(a=c\) (b) \(b^2=4ac\) (c) \(c=0\) (d) \(b^2-4ab\gt 0\)
21. 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.
22. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).
23. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).
24. If one root of the quadratic equation \(ax^2 + bx + c = 0\) is twice the other, show that \(2b^2 = 9ac\).
25. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), 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 \(1:2\), then prove that \(2b^2=9ac\).
27. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
28. If one root of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) is double the other, show that \(2b^2 = 9ac\).
29. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).
30. Translate to English: If one root of the quadratic equation \(ax^2 + bx + c = 0\) is double the other, show that \(2b^2 = 9ac\).
