1. If \(a^2 b^2 + 1 = 2ab\), then which of the following is correct?

(a) \(a-b \propto b\) (b) \(a-b \propto a\) (c) \(a \propto b\) (d) \(a \propto \cfrac{1}{b}\)

2. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

3. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct?

(a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)

4. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

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

5. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

6. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), then the value of \(\left(\cfrac{1}{α} + \cfrac{1}{β}\right)\) is —

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

7. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is—

(a) 2 (b) 18 (c) 9 (d) 8

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

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

9. If \(9x^2 - 13x + 9 = 0\), then what is the value of \(x + \cfrac{1}{x}\)?

(a) \(\cfrac{9}{4}\) (b) \(\cfrac{4}{9}\) (c) \(\cfrac{13}{9}\) (d) 1

10. If \( \sin\theta - \cos\theta = \frac{7}{13} \), then the value of \( \sin\theta + \cos\theta \) is —

(a) \(\cfrac{13}{17}\) (b) \(\cfrac{17}{13}\) (c) \(\cfrac{13}{7}\) (d) None of the above

11. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is —

(a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

12. If \(a+\cfrac{1}{a}=\sqrt{3}\), then the value of \(a^3+\cfrac{1}{a^3}\) will be —

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

13. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), the value of \( \theta \) is —

(a) 30° (b) 60° (c) 45° (d) None of the above

14. If \(x - \cfrac{1}{x} = \sqrt{5}\), then the value of \(x^4 + \cfrac{1}{x^4}\) is —

(a) 45 (b) 46 (c) 47 (d) 48

15. If \(x = 9 + 4\sqrt{5}\), then the value of \(\sqrt{x} - \frac{1}{\sqrt{x}}\) will be —

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

16. If \(a = \frac{\sqrt{3}}{2}\), then the value of \(\sqrt{1+a} + \sqrt{1-a}\) will be —

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

17. If \(x = \sqrt{3} + \sqrt{2}\), then the value of \(x^3 + \frac{1}{x^3}\) is —

(a) \(18\sqrt2\) (b) \(18\sqrt3\) (c) \(18\sqrt5\) (d) \(18\sqrt6\)

18. If \(a = \frac{1}{5 + 2\sqrt{6}}\), then what is the value of \(a^2 - \frac{1}{a^2}\)?

(a) \(4\sqrt6\) (b) -\(4\sqrt6\) (c) \(40\sqrt6\) (d) -\(40\sqrt6\)

19. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?

20. If \(x^2 + y^2 - 4x - 6y + 13 = 0\), then what is the value of \((x + y) : (y - x)\)?

21. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

22. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

23. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

24. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?

25. If \(b^2 = 4ac\) for the quadratic equation \(ax^2 + bx + c = 0\), then the roots are real and ——.

26. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

27. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

28. If \(\tan \theta = \cfrac{8}{15}\), then the value of \(\sqrt{\cfrac{1 - \sin \theta}{1 + \sin \theta}}\) is —

(a) \(\cfrac{2}{5}\) (b) \(\cfrac{3}{5}\) (c) \(\cfrac{1}{5}\) (d) None of the above

29. If \(x(2 + \sqrt{3}) = y(2 - \sqrt{3}) = 1\), then the value of \(\frac{1}{x + 1} + \frac{1}{y + 1}\) will be—

(a) \(1\) (b) \(\sqrt{3}\) (c) \(2\sqrt{3}\) (d) \(2\)

30. If \(a + b + c = 0\), then prove that \[ \frac{bc}{2a^2 + bc} + \frac{ca}{2b^2 + ca} + \frac{ab}{2c^2 + ab} = 1 \]