1. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?
(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1
2. What is the value of \( \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 89^\circ \)?
3. The modal class of the above frequency distribution is 15â20. 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: \[ = 15 + \left(\frac{28 - 18}{2 \times 28 - 18 - 17}\right) \times 5 = 15 + \frac{10}{21} \times 5 = 15 + \frac{50}{21} = 15 + 2.38 = 17.38 \quad \text{(approx)} \] â Therefore, the mode is approximately **17.38**.
4. The value of \((0.243)^{0.2} \times (10)^{0.6}\)
(a) 0.3 (b) 3 (c) 0.9 (d) 9
5. The value of \(25^3 - 75^3 + 50^3 + 3 \times 25 \times 75 \times 50\)
(a) \(150\) (b) \(0\) (c) \(25\) (d) \(50\)
6. Find the value of: \( \cfrac{1 - \sin^2 30^\circ}{1 + \sin^2 45^\circ} \times \cfrac{\cos^2 60^\circ + \cos^2 30^\circ}{\csc^2 90^\circ - \cot^2 90^\circ} \div (\sin 60^\circ \cdot \tan 30^\circ) \)
7. If \(\sin C = \frac{2}{3}\), then calculate and write the value of \(\cos C \times \csc C\).
8. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?
(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm
9. The simplest value of \(\sin\theta \times \tan\theta + \cos\theta\) is â
(a) \(cos\theta\) (b) \(tan\theta\) (c) \(cosec\theta\) (d) \(sec\theta\)
10. Write the simplest value of \(\tan 70^\circ \times \tan 20^\circ\).
11. If the ratio of cost price to selling price is 12:13, Profit percentage = \( \frac{13 - 12}{12} \times 100 = \frac{1}{12} \times 100 \approx 8.33\% \)
(a) \(7\cfrac{1}{3}\)% (b) \(7\cfrac{2}{3}\)% (c) 8% (d) \(8\cfrac{1}{3}\)%
12. The value of \(2^{1/2} \times 2^{-1/2} \times \sqrt{16}\)
(a) 1 (b) 2 (c) 4 (d) \(\frac{1}{2}\)
13. If \(4 \times 5^x = 500\), then the value of \(x^x\) is:
(a) 8 (b) 1 (c) 64 (d) 27
14. If \( \tan 4Îļ \times \tan 6Îļ = 1 \) and \( 6Îļ \) is a positive acute angle, then find the value of \( Îļ \).
(a) \(5°\) (b) \(10°\) (c) \(9°\) (d) \(4°\)
15. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters. The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\). Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters. From \( \triangle \)ABO, we get: \(\cfrac{AB}{BO} = \tan \theta\) Or, \(\cfrac{x}{60} = \tan\theta\) --------(i) From \( \triangle \)COD, we get: \(\cfrac{CD}{OD} = \tan(90^o - \theta)\) Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii) Multiplying equations (i) and (ii), we get: \(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\) Or, \(\cfrac{2x^2}{60 \times 60} = 1\) Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\) Or, \(x = 30\sqrt2\) \(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters. And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters. (Proved).
16. If \( \tan 4\theta \times \tan 6\theta = 1 \) and \( 6\theta \) is an acute positive angle, find the value of \( \theta \).
17. The radius of the first sphere \( = \cfrac{21}{2} \) cm And the radius of the second sphere \( = \cfrac{17.5}{2} \) cm \( = \cfrac{175}{2 \times 10} \) cm \( = \cfrac{35}{4} \) cm The ratio of the amount of metal used to make the two spheres = Ratio of their surface areas \( = 4Ï\left(\cfrac{21}{2}\right)^2 : 4Ï\left(\cfrac{35}{4}\right)^2 = \cfrac{21 \times 21}{4} : \cfrac{35 \times 35}{16} \) \( = 9 : \cfrac{25}{4} = 36 : 25 \) (Answer)
18. If \(\sqrt{6} \times \sqrt{15} = x\sqrt{10}\), write the value of \(x\).
19. For the following data of handicraft scores awarded to 45 girl students in our village, calculate the **median** of the frequency distribution:
20.
21. What is the value of \[ \tan 15^\circ \times \tan 45^\circ \times \tan 60^\circ \times \tan 75^\circ\] ?
22. What is the value of \[ \sin 12^\circ \times \cos 18^\circ \times \sec 78^\circ \times \csc 72^\circ \]
23. If \(\tan 4\theta \times \tan 6\theta = 1\) and \(6\theta\) is a positive acute angle, find the value of \(\theta\).
24. Got it â sticking strictly to translation. Here's the English version without any extra commentary: If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 5x^2 - 3x + 6 = 0 \), then \( \alpha + \beta = -\frac{-3}{5} = \frac{3}{5} \) and \( \alpha\beta = \frac{6}{5} \) \(\therefore \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \) \( = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2} \) (Answer)
