1. 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

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

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

3. 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

4. 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) \(a:b\)

5. 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) \(\bar{x}=8, n=6\)

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

7. 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

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

9. 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

10. 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 \(\cos A + \cos B\).

11. 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

12. 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 \(q\).

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

14. If \(a = 3 + 2\sqrt2\), then find the value of \(\cfrac{a^6 + a^4 + a^2 + 1}{a^3}\).

15. 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

16. If \((\sqrt3 - \sqrt2)^x = (\sqrt3 + \sqrt2)^2\), then find the value of \(x\).

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

17. If \(x=\sqrt3+\sqrt2\), then determine the simplified value of \(x^3-\cfrac{1}{x^3}\).

18. If \(x=\sqrt3+\sqrt2\), then find the simplest value of \(x-\cfrac{1}{x}\).

19. 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

20. If \((5x - 3y) : (2x + 4y) = 11 : 12\), then determine the value of \(x : y\).

21. If \(a : b = 3 : 2\) and \(b : c = 3 : 2\), then find the value of the ratio \((a + b) : (b + c)\).

22. If \(\sec θ + \tan θ = 2\), then find the value of \(\sec θ - \tan θ\).

23. If \(\tan θ + \cot θ = 2\), then find the value of \(\tan θ - \cot θ\).

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

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

26. If \(\cos\theta + \sec\theta = 2\), then what is the value of \(\cos^{11}\theta + \sec^{11}\theta\)?

(a) 0 (b) 1 (c) 2 (d) 22

27. If sinA+sinB=2, then what is the value of cosA-cosB

(a) 0 (b) 1 (c) 2 (d) 3

28. 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) -\(\cfrac{1}{3}\)

29. 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

30. 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