1. If \(\cfrac{a + b − c}{a + b} = \cfrac{b + c − a}{b + c} = \cfrac{c + a − b}{c + a}\) and \(a + b + c ≠ 0\), then prove that \(a = b = c\).

2. Assume, \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) (where \(k \ne 0\)) So, \(a = bk,\; b = ck,\; c = dk\) \(\therefore b = dk \cdot k = dk^2\) and \(a = dk^2 \cdot k = dk^3\)

Left-hand side \[ = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = \{(dk^3)^2 + (dk^2)^2 + (dk)^2\} \cdot \{(dk^2)^2 + (dk)^2 + d^2\} \] \[ = \{d^2k^6 + d^2k^4 + d^2k^2\} \cdot \{d^2k^4 + d^2k^2 + d^2\} = d^2k^2(k^4 + k^2 + 1) \cdot d^2(k^4 + k^2 + 1) \] \[ = d^4k^2(k^4 + k^2 + 1)^2 \] Right-hand side \[ = (ab + bc + cd)^2 = (dk^3 \cdot dk^2 + dk^2 \cdot dk + dk \cdot d)^2 = (d^2k^5 + d^2k^3 + d^2k)^2 = \{d^2k(k^4 + k^2 + 1)\}^2 = d^4k^2(k^4 + k^2 + 1)^2 \] ∴ Left-hand side = Right-hand side (Hence Proved)

3. Given that \[ \frac{a + b - c}{a + b} = \frac{b + c - a}{b + c} = \frac{c + a - b}{c + a} \] and \[ a + b + c \ne 0 \] prove that \(a = b = c\).

4. If \(a + b + c = 0\), then prove that \[ \frac{bc}{2a^2 + bc} + \frac{ca}{2b^2 + ca} + \frac{ab}{2c^2 + ab} = 1 \]

5. If \(a + b + c = 0\), then prove that: \[ \cfrac{1}{2a^2 + bc} + \cfrac{1}{2b^2 + ca} + \cfrac{1}{2c^2 + ab} = 0 \]

6. If \(a + c = 2b\) and \(q^2 = pr\), then prove that \(p^{b-c} \cdot q^{c-a} \cdot r^{a-b} = 1\).

7. If \(\cfrac{b}{a + b} = \cfrac{a + c - b}{b + c - a} = \cfrac{a + b + c}{2a + b + 2c}\), where \(a + b + c \ne 0\), then show that \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4}\).

8. If \(x = \cfrac{4ab}{a + b}\), show that \[ \cfrac{x + 2a}{x - 2a} + \cfrac{x + 2b}{x - 2b} = 2 \] [Given: \(a \ne 0\), \(b \ne 0\), and \(a \ne b\)]

9. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then prove that \(c + \frac{1}{a} = 1\).

10. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

11. If \(a + b + c = 0\), then the value of \(\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}\) is?

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

12. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then show that \(c + \frac{1}{a} = 1\).

13. If a rectangular parallelepiped has volume \(v\), and its length, width, and height are \(a, b, c\) respectively, with a total surface area \(s\), prove that: \(\cfrac{1}{v}=\cfrac{2}{3}\left(\cfrac{1}{a}+\cfrac{1}{b}+\cfrac{1}{c}\right)\).

14. 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.

15. If \(x=\cfrac{4ab}{a+b}\), prove that \(\cfrac{x+2a}{x-2a}+\cfrac{x+2b}{x-2b}=2\)
[Given \(a\ne 0, b\ne 0, a\ne b\)]

16. If \[ \cfrac{a + b}{b + c} = \cfrac{c + d}{d + a} \] then prove that either \(c = a\) or \(a + b + c + d = 0\)

17. If \(\cfrac{b}{a+b} = \cfrac{a+c-b}{b+c-a} = \cfrac{a+b+c}{2a+b+2c}\), (where \(a + b + c \ne 0\)), then show that \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4}\).

18. 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 \(2b^2 = 9ac\).

19. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

20. If \(b + c = a^2\), \(c + a = b^2\), and \(a + b = c^2\), then find the value of \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c}\).

(a) 2 (b) \(\infty\) (c) 0 (d) 1

21. If \(a, b, c,\) and \(d\) are in continued geometric proportion, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

22. If \(a, b, c\) are in continued geometric progression, prove that \(\cfrac{1}{b} = \cfrac{1}{b - a} + \cfrac{1}{b - c}\).

23. If \(a + \cfrac{1}{b} = 1\) and \(b + \cfrac{1}{c} = 1\), then find the values of \((c + \cfrac{1}{a})\) and \((abc + 1)\).

(a) 1 and 0 (b) 0 and 1 (c) 0 and 0 (d) 1 and 1

24. If the roots of the quadratic equation \((a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

25. If \(a = \cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(ab = 1\), then prove that \(\cfrac{a^2+ab+b^2}{a^2-ab+b^2}=\cfrac{12}{11}\).

26. Prove that the quadratic equation \(x^2(a^2+b^2) + 2(ac+bd)x + (c^2+d^2) = 0\) has no real roots when \(ad \ne bc\).

27. If the equation \(ax^2+bx+c=0\) has \(a=0\) and \(b\ne 0\), then the equation is a _____ equation.

28. In the quadratic equation \(ax^2 + bx + c = 0\), if one root is three times the other, prove that \(3b^2 = 16ac\).

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

30. If \(a, b, c,\) and \(d\) are in continuous proportion, then prove that \((b-c)^2+(c-a)^2+(b-d)^2=(a-d)^2\).