1. 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

2. 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

3. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then what is the value of \( \cos\theta + \sin\theta \)?

(a) \(2 cos\theta\) (b) \(\sqrt2 sin \theta\) (c) \(2 sin\theta\) (d) \(\sqrt2 cos \theta\)

4. If \( \cos\theta - \sin\theta = \sqrt{2} \sin\theta \), then the value of \( \cos\theta + \sin\theta - \sqrt{2} \cos\theta \) is —

(a) \(0\) (b) \(1\) (c) \(\sqrt2\) (d) None of the above

5. If \( \cos\theta + \sin\theta = \sqrt{2} \cos\theta \), then what is the value of \( \sin\theta - \cos\theta \)?

(a) \(\sqrt2 sin\theta\) (b) \(-\sqrt2 sin\theta\) (c) \(-\sqrt2 cos\theta\) (d) None of the above

6. If \(x = \sqrt{3} + \sqrt{2}\), then the value of \(x^3 + \frac{1}{x^3}\) is —

(a) \(18\sqrt2\) (b) \(18\sqrt3\) (c) \(18\sqrt5\) (d) \(18\sqrt6\)

7. If \( \sqrt{2} \sin(2x + 5^\circ) = \cot 45^\circ \), then what is the value of \( \sec 3x \)?

8. If \(\sin \theta + \cos \theta = \sqrt{2}\), then what is the value of \(\theta\)?

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

9. If \(5 + \sqrt{x} = \sqrt{2} - y\), then the value of \((x + y)\) is _________.

10. If \(x = 3 + 2\sqrt{2}\), then what is the value of \(x + \cfrac{1}{x} + 2\)?

11. Find the simplest value of the following expression: \[ \cfrac{\sqrt{5}}{\sqrt{3}+\sqrt{2}} - \cfrac{3\sqrt{3}}{\sqrt{2}+\sqrt{5}} + \cfrac{2\sqrt{2}}{\sqrt{3}+\sqrt{5}} \]

12. Write the simplest value of \(\Big[\cfrac{1}{\sqrt{2} + 1} + \cfrac{1}{\sqrt{3} + \sqrt{2}} + \cfrac{1}{\sqrt{4} + \sqrt{3}}\Big]\).

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

14. Determine the simplest value of \[ \left(\frac{1}{\sqrt{2}+1} + \frac{1}{\sqrt{3}+\sqrt{2}} + \frac{1}{\sqrt{4}+\sqrt{3}}\right) \]

15. If \(x = 3 + 2\sqrt{2}\) and \(xy = 1\), then what is the value of \(\cfrac{x^2 + y^2}{xy}\)?

16. In triangle ABC, if AB\(^2\) + BC\(^2\) = AC\(^2\) and AC = \(\sqrt{2}\) × BC, then what type of triangle is it?

(a) Right-angled (b) Right-angled isosceles (c) Equilateral (d) Isosceles

17. What is the simplest value of \(\cfrac{\sqrt{8} + \sqrt{12}}{\sqrt{32} + \sqrt{48}}\)?

(a) \(\cfrac{1}{3}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{1}{2}\) (d) \(\cfrac{1}{\sqrt2}\)

18. The simplest value of \(\sin\theta \times \tan\theta + \cos\theta\) is —

(a) \(cos\theta\) (b) \(tan\theta\) (c) \(cosec\theta\) (d) \(sec\theta\)

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

20. If \(x = \sqrt{3} + \sqrt{2}\), find the value of \(x^3 + \cfrac{1}{x^3}\).

21. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \cfrac{1}{\sqrt{3} + \sqrt{2}}\), then find the value of \((x + y)^2 + (x - y)^2\).

22. If \(x = \sqrt{2} + 1\), find the values of \(x^4 + \frac{1}{x^4}\) and \(x^4 - \frac{1}{x^4}\).

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

24. If \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

(a) \(\cfrac{\pi^c}{2}\) (b) \(\cfrac{\pi^c}{3}\) (c) \(\pi^c\) (d) \(\cfrac{\pi^c}{4}\)

25. In \(\triangle\)ABC, if AB = \((2a-1)\) cm, AC = \(2\sqrt{2}a\) cm, and BC = \((2a+1)\) cm, then write the value of \(\angle\)BAC.

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

27. Let us evaluate the value of: \(\cfrac{8+3\sqrt{2}}{3+\sqrt{5}}-\cfrac{8-3\sqrt{2}}{3-\sqrt{5}}\)

28. If \(\frac{\sqrt{5}+\sqrt{2}}{\sqrt{7}} = \frac{1}{7}(\sqrt{35} + a)\), then find the value of \(a\).

29. If \(\csc θ - \cot θ = \sqrt{2} - 1\), then calculate the value of \(\csc θ + \cot θ\).

30. In triangle ABC, if AB = (2a−1) cm, AC = \(2\sqrt{2}a\) cm, and BC = (2a+1) cm, then write the value of ∠BAC.