1. 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) \(-r^2\)

2. If \(x = 3 \cos \theta\) and \(y = 3 \sin \theta\), then what is the value of \(x^2 + y^2\)?

3. If \(x = a \cos\theta + b \sin\theta\) and \(y = a \sin\theta - b \cos\theta\), then prove that \(x^2 + y^2 = a^2 + b^2\).

4. If \(\cos\theta = \frac{x}{\sqrt{x^2 + y^2}}\), then prove that \(x \sin \theta = y \cos \theta\).

5. Eliminate \(\theta\) from the given equations \(2x = 3\sin\theta\) and \(5y = 3\cos\theta\), and express the relationship between \(x\) and \(y\).

6. If \( \sin \theta - \cos \theta = 0 \) (where \( 0^\circ \lt \theta \lt 90^\circ \)) and \( \sec \theta + \csc \theta = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) √2 (d) 2√2

7. Write the relation between \(x\) and \(y\) by eliminating \(\theta\) from the equations \(2x = 3\sin\theta\) and \(5y = 3\cos\theta\).

8. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

9. Given: \(2x = 3\sin\theta\) ⇒ \(\sin\theta = \frac{2x}{3}\) \(5y = 3\cos\theta\) ⇒ \(\cos\theta = \frac{5y}{3}\) Now using identity: \(\sin^2\theta + \cos^2\theta = 1\) \(\left(\frac{2x}{3}\right)^2 + \left(\frac{5y}{3}\right)^2 = 1\) \(\frac{4x^2 + 25y^2}{9} = 1\) ⇒ \(4x^2 + 25y^2 = 9\)

10. If \(3x = 5\sin\theta\) and \(4y = 5\cos\theta\), then which of the following relations is correct?

(a) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=1\) (b) \(\cfrac{9x^2}{25}+\cfrac{16y^2}{25}=0\) (c) \(\cfrac{9x^2}{25}-\cfrac{16y^2}{25}=1\) (d) \(\cfrac{16x^2}{25}+\cfrac{9y^2}{25}=1\)

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

12. 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) \(\cfrac{5}{12}\)

13. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

14. If \(x = a\sec\theta\) and \(y = b\tan\theta\), then find the relation between \(x\) and \(y\) eliminating \(\theta\).

15. 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) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above

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

17. If \(x = \frac{1 + \sin\theta}{\cos\theta}\), then show that \(x^2 - 2x\tan\theta - 1 = 0\).

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

19. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?

(a) 30° (b) 25° (c) 60° (d) 90°

20. If \(\tan\theta = 3\), then what is the value of \(a\sin\theta + b\cos\theta\)?

(a) \(\sqrt{a^2+b^2}\) (b) \(\sqrt{a^2-b^2}\) (c) \(\sqrt{a^3+b^3}\) (d) \(\sqrt{a^3-b^3}\)

21. If \(\cfrac{\sin \theta}{x} = \cfrac{\cos \theta}{y}\), then prove that \( \sin \theta - \cos \theta = \cfrac{x - y}{\sqrt{x^2 + y^2}} \).

22. If \(\cfrac{\sin\theta}{x} = \cfrac{\cos\theta}{y}\), then prove that \(\sin\theta - \cos\theta = \cfrac{x - y}{\sqrt{x^2 + y^2}}\).

23. If \( \sin\theta - \cos\theta = 0 \), then \( \sec\theta + \cosec\theta = ----------- \)

24. If \[ \sin\theta + \cos\theta = \frac{7}{5} \quad \text{and} \quad \sin\theta \cos\theta = \frac{12}{25}, \] then show that \( \sin\theta = ? \)