21). If \( \alpha \) is a root of \( \Large 25 \cos^{2} \theta + 5 \cos \theta - 12 = 0,\) \( \Large \frac{\pi}{2} < \alpha < \pi \), then \( \Large sin 2\alpha \) is equal to
| A). \( \Large \frac{24}{25} \) |
| B). \( \Large -\frac{24}{25} \) |
| C). \( \Large \frac{13}{18} \) |
| D). \( \Large -\frac{13}{18} \) |
|
22). The value of \( \Large \cos \frac{ \pi }{65} \cos \frac{2 \pi }{65} \cos \frac{4 \pi }{65}.....\cos \frac{32 \pi }{65} \) is
| A). \( \Large \frac{1}{32} \) |
| B). \( \Large \frac{1}{64} \) |
| C). \( \Large -\frac{1}{32} \) |
| D). \( \Large -\frac{1}{64} \) |
|
23). If \( \Large \tan x + \cot x = 2,\ then\ \sin^{2n}x + \cos^{2n}x \) is equal to:
| A). \( \Large 2^{n} \) |
| B). \( \Large -\frac{1}{2} \) |
| C). \( \Large \frac{1}{2} \) |
| D). \( \Large \frac{1}{2^{n-1}} \) |
|
24). If \( \Large \sin A + \cos B = a\ and\ \sin B + \cos A = b,\) then \( \Large \sin \left(A+B\right) \) is equal to:
| A). \( \Large \frac{a^{2}+b^{2}}{2} \) |
| B). \( \Large \frac{a^{2}-b^{2}+2}{2} \) |
| C). \( \Large \frac{a^{2}+b^{2}-2}{2} \) |
| D). none of these |
|
25). \( \Large \frac{2 \sin \alpha }{1+ \cos\ \alpha + \sin \alpha } = x \), then \( \Large \frac{1- \cos \alpha - \sin \alpha }{\cos \alpha } \) equal to
| A). \( \Large \frac{1}{x} \) |
| B). x |
| C). 1-x |
| D). none of these |
|
26). If \( \Large \sec \theta + \tan \theta = 1 \), then root of the equation \( \Large \left(a-2b+c\right)x^{2} + \left(b-2c+a\right)x + \left(c-2a+b\right) = 0 \) is:
| A). \( \Large \sec \theta \) |
| B). \( \Large \tan \theta \) |
| C). \( \Large \sin \theta \) |
| D). \( \Large \cos \theta \) |
|
27). Which of the following relations is possible?
| A). \( \Large \sin \theta = \frac{5}{3} \) |
| B). \( \Large \tan \theta = 100^{2} \) |
| C). \( \Large \cos \theta = \frac{1+p^{2}}{1-p^{2}},\ \left(p \ne \pm 1\right) \) |
| D). \( \Large \sec \theta = \frac{1}{2} \) |
|
28). The value of \( \Large \sin 20 ^{\circ} \sin 40 ^{\circ} \sin 60 ^{\circ} \sin80 ^{\circ} \) is equal to:
| A). \( \Large -\frac{3}{16} \) |
| B). \( \Large \frac{5}{16} \) |
| C). \( \Large \frac{3}{16} \) |
| D). \( \Large -\frac{5}{16} \) |
|
29). The value of \( \Large \tan \alpha + 2 \tan 2 \alpha + 4 \tan + 8 \cot 8 \alpha \) is equal to:
| A). \( \Large \tan \alpha \) |
| B). \( \Large 2 \tan \alpha \) |
| C). \( \Large \cot \alpha \) |
| D). \( \Large \cot 2 \alpha \) |
|
30). If \( \Large \cos 2 B = \frac{\cos \left(A+C\right) }{\cos \left(A-C\right) } \), then \( \Large \tan A \), \( \Large \tan B \), \( \Large \tan C \) are in
| A). AP |
| B). GP |
| C). HP |
| D). none of these |
|
