1. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]

2. Solve: \[ 3x + 2 + \frac{3}{3x + 2} = -4 \]

3. If \[ \cfrac{3x - 5y}{3x + 5y} = \cfrac{1}{2}, \] then find the value of \[ \cfrac{3x^2 - 5y^2}{3x^2 + 5y^2}. \]

4. 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

5. 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) \(\cfrac{\sqrt3}{2}\)

6. 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) \(\sqrt5\)

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

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

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

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

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

12. 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 \(-1\).

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

14. 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**.

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

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

17. If \[ \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} = \frac{ax + by}{a^2 + b^2} \] then prove that \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c} \]

18. If \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

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

20. Solve: \[ \frac{1}{x} - \frac{1}{3} = \frac{1}{x + 2} - \frac{1}{5} \]

21. In triangle \(\triangle ABC\), prove that: \[ \sin\left(\frac{A + B}{2}\right) + \cos\left(\frac{B + C}{2}\right) = \cos\left(\frac{C}{2}\right) + \sin\left(\frac{A}{2}\right) \]

22. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

23. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

24. Solve: \[ \frac{x - 2}{x + 2} + 6\left(\frac{x - 2}{x - 6}\right) = 1 \]

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

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

27. If \[ \frac{a}{y + z} = \frac{b}{z + x} = \frac{c}{x + y} \] then prove that \[ \frac{a(b - c)}{y^2 - z^2} = \frac{b(c - a)}{z^2 - x^2} = \frac{c(a - b)}{x^2 - y^2} \]

28. For what value of \(k\) will the system of equations \[ 3x + y = \frac{1}{k - 4}, \quad 2x + 3y = 5 \] have no solution?

(a) 3 (b) 2 (c) 4 (d) None of the above

29. If \( a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \) and \( b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} \), then find the value of \[ \frac{a^2 + ab + b^2}{a^2 - ab + b^2} \]

30. 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) \(2\sqrt{3}\) (b) \(\cfrac{1}{\sqrt{3}}\) (c) \(\sqrt{3}\) (d) \(\sqrt{3}\)