1. If \(\frac{\csc θ + \sin θ}{\csc θ - \sin θ} = \frac{5}{2}\), Let \(\sin θ = x\), then \(\csc θ = \frac{1}{x}\) \[ \frac{\frac{1}{x} + x}{\frac{1}{x} - x} = \frac{5}{2} \Rightarrow \frac{1 + x^2}{1 - x^2} = \frac{5}{2} \Rightarrow 2(1 + x^2) = 5(1 - x^2) \Rightarrow 2 + 2x^2 = 5 - 5x^2 \Rightarrow 7x^2 = 3 \Rightarrow x = \sqrt{\frac{3}{7}} \] ∴ \(\sin θ = \sqrt{\frac{3}{7}}\)

2. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

3. Solve: \[ \frac{1}{x - a + b} = \frac{1}{x} - \frac{1}{a} + \frac{1}{b} \]

4. Solve: \[ \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x} \] where \(x \ne 0\) and \(x \ne -(a + b)\).

5. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

6. Solve: \[ \frac{1}{x^2} - \frac{1}{x} = 0 \]

7. If \[ \frac{3 - 5x}{x} + \frac{3 - 5y}{y} + \frac{3 - 5z}{z} = 0 \] then what is the value of \[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}? \]

8. Solve: \[ \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}, \quad x \ne 0,\ -(a + b) \]

9. Solve: \[ \frac{1}{x} - \frac{1}{x + b} = \frac{1}{a} - \frac{1}{a + b}, \quad x \ne 0,\ -b \]

10. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x - \frac{1}{x} \]

11. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x + \frac{1}{x} \]

12. If \(x = \sqrt{7} + \sqrt{6}\), then find the simplified value of \[ x^2 + \frac{1}{x^2} \]

13. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x - \frac{1}{x} \]

14. If \(x = 2 + \sqrt{3}\), then find the value of \[ x + \frac{1}{x} \]

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

15. Translate: Solve: \[ x + \frac{1}{x} = p + \frac{1}{p} \]

16. If \(x = \sqrt{3} + \sqrt{2}\), then find the simplest value of \[ x^3 + \frac{1}{x^3} \]

17. If \[ (b + c - a) x = (c + a - b) y = (a + b - c) z = 2 \] then prove that \[ \left( \frac{1}{y} + \frac{1}{z} \right) \left( \frac{1}{z} + \frac{1}{x} \right) \left( \frac{1}{x} + \frac{1}{y} \right) = abc \]

18. If \(a = 3 + 2\sqrt{2}\), then what is the value of \[ \frac{a^6 + a^4 + a^2 + 1}{a^3}? \]

(a) 100 (b) 200 (c) 204 (d) 250

19. Given: \[ \theta + \phi = \frac{7\pi}{12},\quad \tan\theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3} \] \[ \phi = \frac{7\pi}{12} - \frac{\pi}{3} = \frac{7\pi}{12} - \frac{4\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4} \] \[ \tan\phi = \tan\left(\frac{\pi}{4}\right) = 1 \]

(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)

20. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

21. If \(\frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1\), then prove that \[\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} = 1\]

22. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

23. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then prove that \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{4a}{a^2 + 4} \]

24. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

25. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

26. 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 \(-1\).

27. If \[ \frac{\sinθ}{x} = \frac{\cosθ}{y} \] then prove that \[ \sinθ - \cosθ = \frac{x - y}{\sqrt{x^2 + y^2}} \]

28. The modal class of the above frequency distribution is 15–20. So, the mode is calculated as: \[ \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \] Where: - \(l = 15\) (lower boundary of modal class) - \(f_1 = 28\) (frequency of modal class) - \(f_0 = 18\) (frequency of class before modal class) - \(f_2 = 17\) (frequency of class after modal class) - \(h = 5\) (class width) Substituting the values: \[ = 15 + \left(\frac{28 - 18}{2 \times 28 - 18 - 17}\right) \times 5 = 15 + \frac{10}{21} \times 5 = 15 + \frac{50}{21} = 15 + 2.38 = 17.38 \quad \text{(approx)} \] ✅ Therefore, the mode is approximately **17.38**.

29. In triangle △ABC, ∠C = 90°, and if BC = \(m\) and AC = \(n\), then prove that: \[ m \sin A + n \sin B = \sqrt{m^2 + n^2} \]

30. If \(bcx = cay = abz\), then prove that \[ \frac{ax + by}{a^2 + b^2} = \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} \]