1. If \(x = \cfrac{\sqrt{a+2b} + \sqrt{a-2b}}{\sqrt{a+2b} - \sqrt{a-2b}}\), then show that \(bx^2 - ax + b = 0\).

2. If \(x = \cfrac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}}\), then what is the value of \(bx^2 - ax + b\)?

(a) 0 (b) 1 (c) 2 (d) 3

3. If \(x = \frac{\sqrt{a + 1} + \sqrt{a - 1}}{\sqrt{a + 1} - \sqrt{a - 1}}\), then show that \(x^2 - 2ax + 1 = 0\).

4. If \(x : a = y : b = z : c\), then show that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

5. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

6. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

7. If \( x = \cfrac{\sqrt{a+2b}+\sqrt{a-2b}}{\sqrt{a+2b}+\sqrt{a-2b}} \), then determine the value of \( bx^2 - ax + b \).

(a) 1 (b) \(0\) (c) 3 (d) 4

8. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1:r\), then show that \((r+1)^2ac = b^2r\).

9. If \(\cfrac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}=\cfrac{1}{2}\), then show that \(\cfrac{a^2+ab+b^2}{a^2-ab+b^2}=\cfrac{91}{73}\).

10. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

11. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

12. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).

13. If \[ \cfrac{x^2}{by + cz} = \cfrac{y^2}{cz + ax} = \cfrac{z^2}{ax + by} = 1 \] then show that \[ \cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1 \]

14. “If the ratio of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) (where \( a \ne 0 \)) is \(1:r\), then show that \(\cfrac{(r+1)^2}{r} = \cfrac{b^2}{ac}\)”

15. If \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] then show that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

16. If \(x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1\), then show that \(\cfrac{a}{a + x} + \cfrac{b}{b + y} + \cfrac{c}{c + z} = 1\).

17. If the roots of the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), are in the ratio 1:2, then show that \(2b^2 = 9ac\).

18. Show that if one root of the quadratic equation \( ax^2 + bx + c = 0 \) is double the other, then \( 2a^2 = 9ac \).