1. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

2. \(a\) is a positive number, and given \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), determine the value of \(a\).

3. Given: \(\sin 5A = \csc (A + 36^\circ)\) and \(5A\) is a positive acute angle. Find the value of \(A\).

4. \(a\) is a positive number, and in the proportion \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), what is the value of \(a\)?

(a) \(\cfrac{9}{11}\) (b) \(\cfrac{3}{4}\) (c) \(\cfrac{9}{20}\) (d) \(\cfrac{9}{16}\)

5. 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

6. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

7. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]

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 a planet's gravitational force on its satellite is directly proportional to the planet's mass (M) and inversely proportional to the square of their distance (D), and the square of the satellite's orbital period (T) is directly proportional to the cube of the distance and inversely proportional to the gravitational force, then for two sets of values \(m_1, d_1, t_1\) and \(m_2, d_2, t_2\) corresponding to M, D, and T respectively, prove that: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]

10. If \[ a : 2 = b : 5 \] then, what percentage of \(a\) and \(b\) will be equal?

(a) 20 (b) 40 (c) 30 (d) 50

11. The modal class of the above frequency distribution is 15–20. So, the mode is calculated as: \[ \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \] Where: - \(l = 15\) (lower boundary of modal class) - \(f_1 = 28\) (frequency of modal class) - \(f_0 = 18\) (frequency of class before modal class) - \(f_2 = 17\) (frequency of class after modal class) - \(h = 5\) (class width) Substituting the values: \[ = 15 + \left(\frac{28 - 18}{2 \times 28 - 18 - 17}\right) \times 5 = 15 + \frac{10}{21} \times 5 = 15 + \frac{50}{21} = 15 + 2.38 = 17.38 \quad \text{(approx)} \] ✅ Therefore, the mode is approximately **17.38**.

12. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

13. Given: \[ x \cos \theta = 3 \quad \text{and} \quad 4 \tan \theta = y \] Find the relation between \(x\) and \(y\) eliminating \(\theta\).

14. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

15. For which value or values of \(k\) will the following equation have real and equal roots: \[ (3k + 1)x^2 + 2(k + 1)x + k = 0 \]

16. \(x\) 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) \(\pm{10}\)

17. Given: \[ u_i = \frac{x_i - 25}{10}, \quad \sum f_i u_i = 20, \quad \sum f_i = 100 \] Find the value of \(\bar{x}\) (the mean).

18. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

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

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

21. Let ∆ABC be inscribed in a circle. AD and AE are diameters of the circle and perpendicular to side BC, intersecting BC at point E. To prove: ∆AEB and ∆ACD are similar Proof: ∠AEB = ∠ACD (both are right angles) ∠BAE = ∠DAC (angles in the same segment) So, ∆AEB ∼ ∆ACD (by AA similarity) Now, since the triangles are similar: \[ \frac{AB}{AE} = \frac{AC}{AD} \] Cross-multiplying: \[ AB \cdot AD = AC \cdot AE \] So, \[ AB \cdot AC = AE \cdot AD \quad \text{(proved)} \]

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

23. Given: \[ x = 3 + \sqrt{3}, \quad xy = 6 \] Find the value of \((x + y)\).

24. If the current population of a village is **p**, and the annual growth rate is **2r%**, then after **n years**, the population will be: \[ \text{Population after } n \text{ years} = p \left(1 + \frac{2r}{100} \right)^n \]

(a) \(p\left(1+\cfrac{r}{100}\right)^n\) (b) \(p\left(1+\cfrac{r}{50}\right)^n\) (c) \(p\left(1+\cfrac{r}{100}\right)^{2n}\) (d) \(p\left(1-\cfrac{r}{100}\right)^n\)

25. In a partnership business, three friends invested capital amounts of ₹4,500, ₹6,000, and ₹9,000 respectively. After a certain period, their profit shares will be in the same ratio as their investments: \[ \text{Profit Ratio} = 4500 : 6000 : 9000 = 3 : 4 : 6 \] So, their respective shares of profit will be in the ratio 3 : 4 : 6.

26. \[ \sum\limits_{i=1}^5 x_i = 5 \quad \text{and} \quad \sum\limits_{i=1}^5 x_i^2 = 14 \] Find the value of: \[ \sum\limits_{i=1}^5 (x_i - 3) \cdot 2x_i \]

27. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:

(a) >0 (b) <0 (c) 0 (d) None of these

28. 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).

29. If \( \tan 2A = \cot (A - 18^\circ) \) and \(2A\) is a positive acute angle, then find the value of \(A\).

30. Given: \(a \cos θ = 3\) and \(b \tan θ = 4\), Find the relation between \(a\) and \(b\) that eliminates \(θ\).