1. Evaluate the value of: \[ \frac{5 \cos^2 \left( \frac{\pi}{3} \right) + 4 \sec^2 \left( \frac{\pi}{6} \right) - \tan^2 \left( \frac{\pi}{6} \right)}{\sin^2 \left( \frac{\pi}{6} \right) + \cos^2 \left( \frac{\pi}{6} \right)} \]

2. Evaluate the value of: \[ \cfrac{5\cos^2\left(\cfrac{\pi}{3}\right) + 4\sec^2\left(\cfrac{\pi}{6}\right) - \tan^2\left(\cfrac{\pi}{4}\right)}{\sin^2\left(\cfrac{\pi}{6}\right) + \cos^2\left(\cfrac{\pi}{6}\right)} \]

3. Evaluate the value of: \[ \sin^2 \left(\frac{\pi}{3}\right) - \sec^2 \left(\frac{\pi}{4}\right) - \csc^2 \left(\frac{\pi}{4}\right) + \cot^2 \left(\frac{\pi}{4}\right) \]

4. Find the value of: \[ \frac{2\tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ - \sec 60^\circ \]

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

6. Evaluate the value of: \[ \cot\left(\frac{\pi}{20}\right) \cdot \cot\left(\frac{3\pi}{20}\right) \cdot \cot\left(\frac{5\pi}{20}\right) \cdot \cot\left(\frac{7\pi}{20}\right) \cdot \cot\left(\frac{9\pi}{20}\right) \]

(a) 3 (b) 5 (c) 1 (d) 2

7. Find the value of: \[ \cos^2 12^\circ + \cos^2 78^\circ + \tan^2 78^\circ - \sec^2 12^\circ \cdot \tan^2 78^\circ \]

(a) 0 (b) 1 (c) 2 (d) 3

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

9. Find the simplest value of: \[ \cfrac{2\sqrt{7}}{\sqrt{5} + \sqrt{3}} + \cfrac{2\sqrt{3}}{\sqrt{7} + \sqrt{5}} - \cfrac{4\sqrt{5}}{\sqrt{7} + \sqrt{3}} \]

10. Evaluate the value of: \[ 3\tan^2 45^{\circ} - \sin^2 60^{\circ} - \cfrac{1}{3} \cos^2 30^{\circ} - \cfrac{1}{8} \sec^2 45^{\circ} \]

11. Evaluate the value of: \[ \frac{\sin 43^\circ \cdot \sin 65^\circ \cdot \tan 25^\circ + \sin 65^\circ \cdot \cot 65^\circ \cdot \cos 47^\circ}{\cos 65^\circ \cdot \sqrt{1 - \cos^2 43^\circ}} \]

12. Find the minimum value of: \[ 9\csc^2 \alpha + 25\sin^2 \alpha \quad \text{when} \quad 0^\circ \leq \alpha \leq 90^\circ \]

13. Find the simplest value of: \[ \frac{3\sqrt{20} + 2\sqrt{28} + \sqrt{12}}{5\sqrt{45} + 2\sqrt{175} + \sqrt{75}} \]

14. Evaluate the value of: \[ \sqrt{x^3\sqrt{y\sqrt{x^3\sqrt{y\sqrt{x^3\sqrt{y \cdots \infty}}}}}} \]

15. Evaluate the value of: \[ \frac{\tan{60^\circ} - \tan{30^\circ}}{1 + \tan{60^\circ} \cdot \tan{30^\circ}} + \cos{60^\circ} \cos{30^\circ} + \sin{60^\circ} \sin{30^\circ} \]

16. \[ \sum\limits_{i=1}^5 x_i = 5 \quad \text{and} \quad \sum\limits_{i=1}^5 x_i^2 = 14 \] Find the value of: \[ \sum\limits_{i=1}^5 (x_i - 3) \cdot 2x_i \]

17. 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

18. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

19. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

20. What is the value of \[ \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}} \]

(a) 3 (b) 4 (c) 5 (d) 6

21. The simplest value of \[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \ldots + \frac{1}{\sqrt{99} + \sqrt{100}} \] is —

(a) 100 (b) 99 (c) 9 (d) 1

22. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

23. What is the value of \[ \frac{4}{\sec^2θ} + \frac{1}{1 + \cot^2θ} + 3\sin^2θ? \]

24. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

25. If \(x, y, z\) are in continued proportion, then what is the value of \[ x^2y^2z^2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)? \]

(a) \(x+y+z\) (b) \(x^2+y^2+z^2\) (c) \(x^3+y^3+z^3\) (d) None of the above

26. 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) \(\sqrt5\)

27. What is the value of \[ \frac{2\tan 30^\circ}{1 - \tan^2 30^\circ}? \]

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

28. The numerical value of the volume and the lateral surface area of a right circular cone are equal. If the height and radius of the cone are \( h \) units and \( r \) units respectively, find the value of \[ \left( \frac{1}{h^2} + \frac{1}{r^2} \right) \]

29. 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**.

30. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]