1. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \tan\theta - \cot\theta \) is —

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

2. If \(\tan\theta + \cot\theta = 2\), then the value of \(\tan\theta - \cot\theta\) is —

(a) 1 (b) 2 (c) -1 (d) None of the above

3. 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) \(\cfrac{\pi^c}{4}\)

4. If \(a = \frac{\sqrt{3}}{2}\), then the value of \(\sqrt{1+a} + \sqrt{1-a}\) will be —

(a) \(a\) (b) \(3a\) (c) \(2a\) (d) \(4a\)

5. If \(x(2 + \sqrt{3}) = y(2 - \sqrt{3}) = 1\), then the value of \(\frac{1}{x + 1} + \frac{1}{y + 1}\) will be—

(a) \(1\) (b) \(\sqrt{3}\) (c) \(2\sqrt{3}\) (d) \(2\)

6. If \( \tan^2\theta + \cot^2\theta = \cfrac{10}{3} \), then find the values of \( \tan\theta + \cot\theta \) and \( \tan\theta - \cot\theta \). From there, determine the value of \( \tan\theta \).

7. If \( \sin^2 \theta + 2x \cos^2 \theta = 1 \), then the value of \( x \) will be _____

8. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0

9. If \( \tan A \tan B = 1\), then the value of \( \tan \cfrac{(A+B)}{2} \) will be –

(a) 1 (b) √3 (c) \(\cfrac{1}{√3}\) (d) None of these

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

(a) \(\sqrt3+\sqrt2\) (b) \(\sqrt2-3\) (c) \(\sqrt3-2\) (d) \(2-\sqrt3\)

11. If \(a+\cfrac{1}{a}=\sqrt{3}\), then the value of \(a^3+\cfrac{1}{a^3}\) will be —

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

12. 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

13. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

14. If \(5\cos\theta + 12\sin\theta = 13\), then what is the value of \(\tan\theta\)?

(a) \(\cfrac{13}{15}\) (b) \(\cfrac{12}{5}\) (c) \(\cfrac{5}{13}\) (d) \(\cfrac{5}{12}\)

15. If \(x = 9 + 4\sqrt{5}\), then the value of \(\sqrt{x} - \frac{1}{\sqrt{x}}\) will be —

(a) 4 (b) 3 (c) 2 (d) 1

16. If \(\cos^2\theta - \sin^2\theta = \frac{1}{2}\), then the value of \(\tan\theta\) is —

(a) \(-\cfrac{1}{\sqrt3} (b) \(\cfrac{1}{3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{2}{3}\)

17. If the mean of the numbers \(x_1, x_2, x_3, x_4, ..., x_n\) is \(\bar{x}\), then the value of \((x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + ... + (x_n - \bar{x})\) will be —

(a) 0 (b) 1 (c) 3 (d) 5

18. If \(\sum\limits_{i=1}^5 x_i = 5\) and \(\sum\limits_{i=1}^5 x_i^2 = 14\), then the value of \(\sum\limits_{i=1}^5 2x_i(x_i - 3)\) will be —

(a) 2 (b) -2 (c) 0 (d) 4

19. If \(x = 2 + \sqrt{3}\), then the value of \(x + \frac{1}{x}\) will be \(2\sqrt{3}\).

20. If \( \cos\theta = \frac{p}{\sqrt{p^2 + q^2}} \), then find the value of \( \tan\theta \).

(a) \(\cfrac{q}{p}\) (b) \(\cfrac{p}{q}\) (c) \(\cfrac{\sqrt{p}}{q}\) (d) None of the above

21. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

22. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2 + 2x - 3 = 0\), then the value of \(\alpha^2 + \beta^2\) will be \(\cfrac{32}{25}\).

23. If \((a+b) : \sqrt{ab} = 2:1\), then the value of \(a:b\) will be 1:1.

24. If \(\tan \theta \cos 60° = \cfrac{\sqrt{3}}{2}\), then the value of \(\sin (\theta - 15°)\) will be _____ .

25. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

26. If \( \tan\theta - \cot\theta = 0 \), then find the value of \( \sin(\theta - 15^\circ) + \cos(\theta + 15^\circ) \).

27. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

28. If \( \tan \theta \cdot \tan \phi = 1 \), then the value of \( \sin\left(\frac{\theta + \phi}{3}\right) \) will be –

(a) \(1\) (b) \(\cfrac{1}{\sqrt2}\) (c) \(\cfrac{\sqrt3}{2}\) (d) \(\cfrac{1}{2}\)

29. If \(x (4 - \sqrt{3}) = y (4 + \sqrt{3}) = 1\), then the value of \(x^2 + y^2\) will be _____

30. If the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\), then the value of \(k\) will be \(12\).