1. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

2. If \(r \cos θ = 2\sqrt{3}\), \(r \sin θ = 2\), and \(0^\circ < θ < 90^\circ\), then find the values of \(r\) and \(θ\).

3. If \(a + \cfrac{1}{b} = 1\) and \(b + \cfrac{1}{c} = 1\), then find the values of \((c + \cfrac{1}{a})\) and \((abc + 1)\).

(a) 1 and 0 (b) 0 and 1 (c) 0 and 0 (d) 1 and 1

4. If \(\csc θ + \cot θ = \sqrt{3}\), then find the values of both \(\csc θ\) and \(\cot θ\).

5. If \(\sum_{i=1}^n (x_i - 3) = 0\) and \(\sum_{i=1}^n (x_i + 3) = 66\), then find the values of \(\bar{x}\) (the mean) and \(n\).

6. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).

7. ABCD is a cyclic quadrilateral and O is the center of the circle. If ∠COD = 120° and ∠BAC = 30°, then find the values of ∠BOC and ∠BCD.

8. If \(r \cos \theta = \frac{1}{2}\) and \(r \sin \theta = \frac{\sqrt{3}}{2}\), then find the value of \(r\), where \(0^\circ < \theta < 90^\circ\).

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

10. If \(\sum(x_i - 3) = 0\) and \(\sum(x_i + 3) = 66\), then find the values of \(\bar{x}\) (mean) and \(n\) (number of observations).

11. If \(\sec \theta - \tan \theta = \frac{1}{2}\), then find the values of \(\sec \theta\) and \(\tan \theta\).

12. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

13. If \(x=\cfrac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}\) and \(xy=1\), then find the value of \(x^2 - xy + y^2\).

14. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).

15. If \(x=\cfrac{\sqrt5+1}{\sqrt5-1}\) and \(xy=1\), then find the value of \(\cfrac{3x^2+5xy+3y^2}{3x^2-5xy+3y^2}\).

16. If \( r \cos\theta = 2\sqrt{3} \), \( r \sin\theta = 2 \), and \( 0^\circ < \theta < 90^\circ \), then find the values of \( r \) and \( \theta \).

17. If \( r\cosθ = 2\sqrt{3} \), \( r\sinθ = 2 \), and \( 0^\circ < θ < 90^\circ \), then find the values of \( r \) and \( θ \).

18. If \(a =\cfrac{\sqrt{5}+1}{\sqrt{5}-1}\) and \(ab = 1\), then find the value of \(\left(\cfrac{a}{b}+\cfrac{b}{a}\right)\).

19. If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

20. If the quadratic equation \(ax^2 + 7x + b = 0\) has two roots \(\frac{2}{3}\) and \(-3\), then let us find the values of \(a\) and \(b\).

21. If \(\sec θ - \tan θ = \frac{1}{\sqrt{3}}\), then find the values of both \(\sec θ\) and \(\tan θ\).

22. If \(\tan^2 θ + \cot^2 θ = \frac{10}{3}\), then find the values of \(\tan θ + \cot θ\) and \(\tan θ - \cot θ\), and from there calculate the value of \(\tan θ\).

23. If \( \tan A = \cfrac{n}{m} \), then find the values of \( \sin A \) and \( \sec A \).

24. If \( r \cos θ = 2\sqrt{3} \), \( r \sin θ = 2 \), and \( 0° < θ < 90° \), then find the values of both \( r \) and \( θ \).

25. If \(x \tan 30^\circ + y \cot 60^\circ = 0\) and \(2x - y \tan 45^\circ = 1\), then calculate and write the values of \(x\) and \(y\).

26. If triangle ABC is a right-angled triangle with ∠C = 90°, BC = 21 units, and AB = 29 units, then find the values of sin A, cos A, sin B, and cos B.

27. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

28. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]

29. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x - \frac{1}{x} \]

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