1. If \(a \propto b\) and when \(a = 2\), \(b = 14\), then what is the value of \(b\) when \(a = 5\)?

2. If \(x \propto \cfrac{1}{y}\), then the value of \(x+y\) will be the smallest when \(x=y\).

3. Given \(x \propto y\), and \(y = 8\) when \(x = 2\); then when \(y = 16\), find the value of \(x\).

(a) 2 (b) 4 (c) 6 (d) 8

4. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

5. If \(r \cos \theta = 2\), what should be the value of \(r\) when \(\theta = 60^\circ\)?

6. \(x \propto y\) and \(y = 8\) when \(x = 2\); if \(y = 16\), what is the value of \(x\)?

(a) 2 (b) 4 (c) 6 (d) 8

7. \(a \propto b^2\) and \(1+b \propto 6\), and if \(a = 1\) when \(c = 9\) and \(b = 5\), determine the value of \(c\).

8. \(x ∝ y^2\) and \(y = 4\) when \(x = 8\); when \(x = 32\), find the positive value of \(y\).

(a) 4 (b) 8 (c) 16 (d) 32

9. \(x \propto \sqrt{y}\) and \(y = a^2\), if \(x = 2a\), then find the value of \(x^2 : y\).

10. If \(\cfrac{p}{q} = \cfrac{5}{7}\) and \(p - q = -2\), then the value of \(p + q\) is –

(a) 12 (b) 13 (c) 14 (d) 15

11. If \(∑f_i d_i = 400\), \(∑f_i = 50\), and \(a =\) assumed mean \(= 52\), then the value of the combined mean is –

(a) 52 (b) 60 (c) 80 (d) 55

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

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

13. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^6}\)?

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

14. If \((a^2 + b^2)(x^2 + y^2) = (ax + by)^2\), then what are the values of \(x\) and \(y\)?

(a) \(b:a\) (b) \(b^2:a^2\) (c) \(a^2:b^2\) (d) \(a:b\)

15. If \(\sum \limits_{i=1}^n (x_i - 7) = -8\) and \(\sum \limits_{i=1}^n (x_i + 3) = 72\), then what are the values of \(\bar{x}\) (the mean of \(x_i\)) and \(n\) (the number of terms)?

(a) \(\bar{x}=5, n=8\) (b) \(\bar{x}=6, n=8\) (c) \(\bar{x}=4, n=7\) (d) \(\bar{x}=8, n=6\)

16. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

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

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

19. 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

20. If \(x + \cfrac{1}{x} = -2\), then what is the value of \(x^7 + \cfrac{1}{x^7}\)?

21. If \(b + c = a^2\), \(c + a = b^2\), and \(a + b = c^2\), then find the value of \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c}\).

(a) 2 (b) \(\infty\) (c) 0 (d) 1

22. If \(\sin A + \sin B = 2\), where \(0^\circ \leq A \leq 90^\circ\) and \(0^\circ \leq B \leq 90^\circ\), then find the value of \(\cos A + \cos B\).

23. If \(\log_{10} (7x - 5) = 2\), then the value of \(x\) is:

(a) 10 (b) 12 (c) 15 (d) 18

24. \(x\) and \(y\) vary directly with the square of each other, and \(z\) varies inversely with the cube root. Given: \(y = 8\), \(z = 8\), \(x = 16\) When \(x = 24\), \(z = 27\), find the value of \(y =\) ?

(a) \(\pm{16}\) (b) \(\pm{14}\) (c) \(\pm{12}\) (d) \(\pm{10}\)

25. If \(x + \frac{1}{x} = 2\), then what is the value of \(x^{119} + \frac{1}{x^{119}}\)?

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

26. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

27. If \(x = 2\), \(y = 3\), and \(z = 6\), then calculate the value of \[ \frac{3\sqrt{x}}{\sqrt{y} + \sqrt{z}} - \frac{4\sqrt{y}}{\sqrt{z} + \sqrt{x}} + \frac{\sqrt{z}}{\sqrt{x} + \sqrt{y}} \]

28. \(y\) is directly proportional to the square root of \(x\), and \(y = 9\) when \(x = 9\). Determine the value of \(x\) when \(y = 6\).

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

29. \(y\) varies directly with the square root of \(x\), and \(y = 9\) when \(x = 9\). Determine the value of \(x\) when \(y = 6\).

30. \(x ∝ y\), and \(y = 8\) when \(x = 2\). Determine the value of \(x\) when \(y = 16\).

(a) 2 (b) 8 (c) 6 (d) 4