1. If \( \sin\theta - \cos\theta = \frac{7}{13} \), then the value of \( \sin\theta + \cos\theta \) is —

(a) \(\cfrac{13}{17}\) (b) \(\cfrac{17}{13}\) (c) \(\cfrac{13}{7}\) (d) None of the above

2. 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

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

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

4. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)?

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

5. If \( \sin\theta \cos\theta = \frac{1}{2} \), then what is the value of \( (\sin\theta + \cos\theta)^2 \)?

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

6. 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

7. If \(\cot \theta = \cfrac{x}{y}\), then the value of \(\cfrac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta}\) is —

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

8. If \(\tan \theta = \cfrac{8}{15}\), then the value of \(\sqrt{\cfrac{1 - \sin \theta}{1 + \sin \theta}}\) is —

(a) \(\cfrac{2}{5}\) (b) \(\cfrac{3}{5}\) (c) \(\cfrac{1}{5}\) (d) None of the above

9. If \(\cos\theta = \frac{4}{5}\), then what is the value of \(\csc\theta \div (1 + \cot\theta)\)?

(a) \(\cfrac{2}{4}\) (b) \(\cfrac{5}{7}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{1}{3}\)

10. If the product of the roots of the equation \(x^2 - 2x + k = 8\) is \(-2\), then the value of \(k\) is—

(a) -2 (b) 6 (c) 1 (d) -6

11. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then the value of \(k\) is —

(a) -2 (b) -8 (c) 8 (d) 12

12. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), then the value of \(\left(\cfrac{1}{α} + \cfrac{1}{β}\right)\) is —

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

13. If \(2\cos(3\theta) = 1\), then what is the value of \(\theta\)?

(a) \(10^\circ\) (b) \(15^\circ\) (c) \(20^\circ\) (d) \(30^\circ\)

14. If \(\sin 3\theta = \frac{1}{2}\), then what is the value of \(\cos(80^\circ + \theta)\)?

15. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

16. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

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

17. 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

18. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –

(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

19. If \( r\cos\theta = 1 \) and \( r\sin\theta = \sqrt{3} \), then the value of \( \theta \) is—

(a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{6}\)

20. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—

(a) 2 (b) 18 (c) 9 (d) 8

21. If \(tanα + cotα = 2\), then the value of \(tan^{13}α + cot^{13}α\) is—?

(a) 13 (b) 2 (c) 1 (d) 0

22. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

(a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

23. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?

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

24. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

25. 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) \(\sqrt2 cos \theta\)

26. 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) \(\cfrac{x^2-y^2}{x^2+y^2}\)

27. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct?

(a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)

28. 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) \(2-\sqrt3\)

29. If \(x = r\cos\theta\cos\phi\), \(y = r\cos\theta\sin\phi\), and \(z = r\sin\theta\), then what is the value of \(x^2 + y^2 + z^2\)?

(a) \(r\) (b) \(1\) (c) \(r^2\) (d) \(-r^2\)

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

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