1. If \[ \frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b} \] then prove that each of these ratios is either \(\frac{1}{2}\) or \(-1\).

2. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

3. If \[ \frac{x - \sqrt{x^2 - 1}}{x + \sqrt{x^2 - 1}} + \frac{x - \sqrt{x^2 - 1}}{x + \sqrt{x^2 - 1}} = 14 \] then determine the possible values of \(x\).

4. If \[ \cfrac{ax + by}{a} = \cfrac{bx - ay}{b} \] then prove that each ratio is equal to \(x\).

5. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).

6. If \[ \frac{x}{y + z} = \frac{y}{z + x} = \frac{z}{x + y} \] then prove that the value of each ratio is either \(\frac{1}{2}\) or \(-1\).

7. A point is located \(h\) meters above a lake. From this point, the angle of elevation to a cloud is \(α\), and the angle of depression to its reflection in the lake is \(β\). Prove that the distance from the point to the cloud is \[ \cfrac{2h \sec α}{\tan β − \tan α} \]

8. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

9. Let \(a\) be a positive number, and given: \[ a : \frac{27}{64} = \frac{3}{4} : a \] Find the value of \(a\).

10. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).

11. Given: \[ x \cos \theta = 3 \quad \text{and} \quad 4 \tan \theta = y \] Find the relation between \(x\) and \(y\) eliminating \(\theta\).

12. If \(x \propto y\), then – **x is directly proportional to y.** This means: \[ x = ky \quad \text{for some constant } k \] Or, the ratio \(\frac{x}{y}\) remains constant.

(a) \(x^2\propto y^3\) (b) \(x^3\propto y^2\) (c) \(x \propto y^3\) (d) \(x^2\propto y^2\)

13. In a circle centered at \(O\), two chords \(AB\) and \(CD\) intersect each other at point \(P\). Prove that \[ \angle AOD + \angle BOC = 2\angle BPC \]

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

15. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).

16. If \[ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = 2\frac{51}{79} \] then find the value of \(\sin\theta\).

17. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).

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

19. If \((5x - 3y) : (2x + 4y) = 11:12\), determine the ratio \(x:y\).

(a) 19 : 40 (b) 40 : 19 (c) 19 : 21 (d) 40 : 21

20. The capital ratio of three friends is \(\frac{1}{6}:\frac{1}{5}:\frac{1}{4}\). If the total profit at the end of the year is ₹37,000, determine the profit share of the third friend.

(a) ₹ 15,000 (b) ₹ 12,000 (c) ₹ 10,000 (d) ₹ 20,000

21. The capital ratio of A, B, and C is \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\). If B's profit at the end of the year is 240 rupees, determine the total business profit.

(a) ₹ 720 (b) ₹ 780 (c) ₹ 760 (d) ₹ 750

22. The inverse ratio of the mixed ratio of \( ab:c^2, bc:a^2 \) and \( ca:b^2 \) is \( 1:1 \).

23. The capital ratio of A, B, and C is \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\). If B's profit at the end of the year is ₹240, determine the total business profit.

(a) ₹ 720 (b) ₹ 760 (c) ₹ 780 (d) ₹ 750

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

25. Let the interest rate be \( r\% \). \(\therefore\) According to the question, \[ 5000\left(1+\cfrac{r}{100}\right)^2 = 5408 \] \[ \left(1+\cfrac{r}{100}\right)^2 = \cfrac{5408}{5000} = \cfrac{676}{625} \] \[ \left(1+\cfrac{r}{100}\right)^2 = \left(\cfrac{26}{25}\right)^2 \] \[ 1+\cfrac{r}{100} = \cfrac{26}{25} \] \[ \cfrac{r}{100} = \cfrac{26}{25} - 1 = \cfrac{1}{25} \] \[ r = \cfrac{100}{25} = 4 \] \(\therefore\) The interest rate is \( 4\% \).

26. In a partnership business, the capital ratio of A, B, and C is \(\frac{1}{6}:\frac{1}{5}:\frac{1}{4}\). If the total profit at the end of the year is 3700 currency units, determine C's profit.

27. In a joint business, the capital ratio of A, B, and C is \(\frac{1}{2}:\frac{1}{3}:\frac{1}{4}\). After 4 months, A withdraws half of his capital, and after another 4 months, the total profit amounts to 61,050 rupees. Determine each person's share of the profit.

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

29. In \( \triangle ABC \), the vertex \( A \) has a perpendicular \( AD \) drawn to the side \( BC \). If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that \( \triangle ABC \) is a right-angled triangle.

30. In triangle \(ABC\), a straight line parallel to side \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. If \(AQ = 2AP\), determine the ratio \(PB:QC\).