1. If \((5 + √3)(5 - √3) = 25 - x^2\), find the value of \(x\).

(a) 3 (b) √3 (c) -√3 (d) \(\pm\)√3

2. If \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 25 - x^2\) is an equation, find the value of \(x\).

3. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

4. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

5. If \(\cfrac{(5 + \sqrt{3})}{(5 - \sqrt{3})} = x - \sqrt{15}y\), then find the value of \(x + y\).

6. 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 \(x\).

7. 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 \(x\).

8. If \(x+\cfrac{1}{x}=\cfrac{13}{6}\), find the value of \(x\).

(a) \(\cfrac{2}{5}\) (b) \(1\cfrac{3}{4}\) (c) \(\cfrac{3}{2}\) (d) \(\cfrac{1}{4}\)

9. If \((\sqrt3 - \sqrt2)^x = (\sqrt3 + \sqrt2)^2\), then find the value of \(x\).

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

10. If \(\cfrac{x − x\tan^2 30°}{1 + \tan^2 30°} = \sin^2 30° + 4\cos^2 45° - \sec^2 60°\), find the value of \(x\). Let me know if you'd like it solved as well.

11. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\). Let me know if you'd like a step-by-step solution next. I'm ready when you are.

12. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

13. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

14. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

15. In the adjacent figure, if \(LM \parallel AB\), and \(AL = (x - 3)\) units, \(AC = 2x\) units, \(BM = (x - 2)\) units, and \(BC = (2x + 3)\) units, then find the value of \(x\).

16. If \[ 1\frac{1}{2} \cdot \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} + \frac{x - \sqrt{x^2 - 1}}{x + \sqrt{x^2 - 1}} = 14 \] then find the value of \(x\).

17. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

18. If \( (\sqrt{3} - \sqrt{2})^x = (\sqrt{3} + \sqrt{2})^2 \), then find the value of \(x\).

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

19. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\).

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

20. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

21. If \(x = \sqrt{7 + 4√3}\), find the value of \(x - \cfrac{1}{x}\).

(a) 2 (b) 2√3 (c) 4 (d) 2-√3

22. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\).

(a) 60° (b) 80° (c) 120° (d) None of these

23. If \(\cfrac{\sinθ + \cosθ}{\sinθ - \cosθ} = \cfrac{3}{2}\), find the value of \(\cosθ\)

(a) \(\cfrac{1}{5}\) (b) \(\cfrac{3}{2}\) (c) \(\cfrac{1}{\sqrt{26}}\) (d) None of these

24. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{x^3}{x^6 + 7x^3 + 1}\).

(a) \(\cfrac{1}{2737}\) (b) \(\cfrac{1}{2730}\) (c) \(\cfrac{1}{2710}\) (d) \(\cfrac{1}{2709}\)

25. If \(x = 3 + \sqrt{8}\) and \(y = 3 - \sqrt{8}\), then find the value of \(x^{-3} + y^{-3}\).

(a) 199 (b) 195 (c) 198 (d) 201

26. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

27. If \(a : b : c = 2 : 3 : 5\), then find the value of \(\frac{2a + 3b - 3c}{c}\).

(a) \(=-\cfrac{2}{5}\) (b) \(=-\cfrac{3}{5}\) (c) \(=\cfrac{2}{5}\) (d) \(=\cfrac{3}{5}\)

28. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

29. If \(x = 3 + 2\sqrt{2}\), then find the value of \(\left(\sqrt{x} + \cfrac{1}{\sqrt{x}}\right)\).

30. In triangle ∆ABC, ∠B = 90°, AC = √13 cm, and AB + BC = 5 cm. Find the value of (cos A + cos C).