1. 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**.

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

3. 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).

4. Given: \[ r\cos\theta = 2\sqrt{3}, \quad r\sin\theta = 2 \] and \(\theta\) is an acute angle. Find the values of \(r\) and \(\theta\).

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

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

7. 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).

8. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

9. Let \(a\) be a positive number, and given: \[ a : \frac{27}{64} = \frac{3}{4} : a \] Find the value of \(a\).

10. 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.

11. For what value of \(k\) will the system of equations \[ x + 5y = 8 \quad \text{and} \quad 2x - ky = 13 \] have no solution?

(a) 1 (b) 5 (c) 10 (d) -10

12. For what value of \(k\) will the system of equations \[ x - ky = k \quad \text{and} \quad x + (k - 2)y = 2 \] have no solution?

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

13. For what value of \(k\) will the system of equations \[ 3x + y = \frac{1}{k - 4}, \quad 2x + 3y = 5 \] have no solution?

(a) 3 (b) 2 (c) 4 (d) None of the above

14. If \((a+b) : \sqrt{ab} = 2:1\), then the value of \(a:b\) will be 1:1.

15. 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) \(\cfrac{16}{9}\)

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

17. 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, \[ AB \cdot AC = AE \cdot AD \quad \text{(proved)} \]

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

19. 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

20. Find the minimum value of: \[ 9\csc^2 \alpha + 25\sin^2 \alpha \quad \text{when} \quad 0^\circ \leq \alpha \leq 90^\circ \]

21. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

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

22. If \(a = 3 + 2\sqrt{2}\), then what is the value of \[ \frac{a^6 + a^4 + a^2 + 1}{a^3}? \]

(a) 100 (b) 200 (c) 204 (d) 250

23. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

24. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

25. Given: \[ \theta + \phi = \frac{7\pi}{12},\quad \tan\theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3} \] \[ \phi = \frac{7\pi}{12} - \frac{\pi}{3} = \frac{7\pi}{12} - \frac{4\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4} \] \[ \tan\phi = \tan\left(\frac{\pi}{4}\right) = 1 \]

(a) \(\cfrac{1}{2}\) (b) 1 (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)

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

27. Evaluate the value of: \[ \frac{5 \cos^2 \left( \frac{\pi}{3} \right) + 4 \sec^2 \left( \frac{\pi}{6} \right) - \tan^2 \left( \frac{\pi}{6} \right)}{\sin^2 \left( \frac{\pi}{6} \right) + \cos^2 \left( \frac{\pi}{6} \right)} \]

28. Evaluate the value of: \[ \cfrac{5\cos^2\left(\cfrac{\pi}{3}\right) + 4\sec^2\left(\cfrac{\pi}{6}\right) - \tan^2\left(\cfrac{\pi}{4}\right)}{\sin^2\left(\cfrac{\pi}{6}\right) + \cos^2\left(\cfrac{\pi}{6}\right)} \]

29. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

30. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]