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

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

3. If \( x \cos\theta = 3 \) and \( y \cot\theta = 4 \), then find the relationship between \( x \) and \( y \) eliminating \( \theta \).

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

5. Given: \(x = a \sec \theta\), \(y = b \tan \theta\); eliminate \(\theta\) and establish a relation between \(x\) and \(y\).

6. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

7. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero

8. Given: \[ r\cos\theta = 2\sqrt{3}, \quad r\sin\theta = 2 \] and \(\theta\) is an acute angle. Find the values of \(r\) and \(\theta\).

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