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

2. If \(x : a = y : b = z : c\), then show that \(\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\).

3. If \(x : a = y : b = z : c\), then show that \[ \frac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \frac{xyz}{abc} \]

4. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that \(x^2 : y^2 = (1 - a^2) : (1 - b^2)\).

5. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3}{a^2} + \cfrac{y^3}{b^2} + \cfrac{z^3}{c^2} = \cfrac{(x + y + z)^3}{(a + b + c)^2}\)

6. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \cfrac{xyz}{abc}\)

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

8. If \(x : a = y : b = z : c\), then prove that \[\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\]

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

10. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

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

12. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

13. Given: \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] Show that: \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

14. If \(A = B = 45^\circ\), then let us verify that: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\)

15. If \(A = B = 45^\circ\), then let us verify that: \(\cos (A + B) = \cos A \cos B - \sin A \sin B\)

16. If \(x : y = (a + 2) : (a - 2)\), then show that \((x^2 + y^2) : (x^2 - y^2) = 4a : (a^2 + 4)\).

17. A person bought \(y\) pencils for ₹\(x\). If each pencil had cost ₹1 less, then with the same amount ₹\(x\), he would have received one additional pencil. Prove that: \[ 2y = \sqrt{1 + 4x} - 1 \]

18. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

19. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

20. If a cone has volume \(V\) cubic units, total surface area \(S\) square units, height \(h\) units, and base radius \(r\) units, then show that: \[ S = 2V\left(\frac{1}{h} + \frac{1}{r}\right) \]

21. A right circular cone and a cylinder have equal curved surface areas. If the height and radius of the cone are \(h\) and \(r\), and the height and radius of the cylinder are \(H\) and \(r\), then show that: \[ h^2 = (2H + r)(2H - r) \]

22. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that: \[ AB + CD = BC + DA \]

23. If \(a : b = b : c\), then prove that \[ (a + b)^2 : (b + c)^2 = a : c \]

24. Given: \[ \frac{x}{b + c} = \frac{y}{c + a} = \frac{z}{a + b} \] Prove that: \[ \frac{a}{y + z - x} = \frac{b}{z + x - y} = \frac{c}{x + y - z} \]

25. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

26. If \(x + y = z\) and \(2x - z = y\), then what is the value of \(x : y : z\)?

(a) 3:2:1 (b) 1:2:3 (c) 4:5:6 (d) 2:1:3

27. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

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

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

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