31). The value of \( \Large \sin 36 ^{\circ} \sin 72 ^{\circ} \sin 108 ^{\circ} \sin 144 ^{\circ} \) is equal to:
A). \( \Large \frac{1}{4} \) |
B). \( \Large \frac{1}{16} \) |
C). \( \Large \frac{3}{4} \) |
D). \( \Large \frac{5}{16} \) |
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32). If \( \Large \cos \left( \theta - \alpha \right),\ \cos \theta\ and\ \cos \left( \theta + \alpha \right) \) are in HP, the: \( \Large \cos \theta \sec \frac{ \alpha }{2} \) is equal to:
A). \( \Large \pm \sqrt{2} \) |
B). \( \Large \pm \sqrt{3} \) |
C). \( \Large \pm \frac{1}{\sqrt{2}} \) |
D). none of these |
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33). If \( \Large \cos x + \cos y + \cos \alpha = 0 \) and \( \Large \sin x + \sin y + \sin \alpha = 0,\) then \( \Large \cot \left(\frac{x+y}{2}\right) \) is equal to:
A). \( \Large \sin \alpha \) |
B). \( \Large \cos \alpha \) |
C). \( \Large \cot \alpha \) |
D). \( \Large \sin \left(\frac{x+y}{2}\right) \) |
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34). If \( \Large \tan \alpha = \left(1+2^{-x}\right)^{-1},\ \tan \beta = \left(1-2^{x+1}\right)^{-1},\ then\ \alpha - \beta \) equal
A). \( \Large \frac{ \pi }{6} \) |
B). \( \Large \frac{ \pi }{4} \) |
C). \( \Large \frac{ \pi }{3} \) |
D). \( \Large \frac{ \pi }{2} \) |
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35). If \( \Large \tan A + \sin A = m\ and\ \tan A - \sin A = n,\ then\ \frac{ \left(m^{2}-n^{2}\right)^{2} }{mn} \) is equal to:
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36). \( \Large 3 \left(\sin x - \cos x\right)^{4} + 6 \left(\sin x + \cos x\right)^{2} + 4 \left(\sin^{6}x + \cos^{6}x\right) \) is equal to:
A). 11 |
B). 12 |
C). 13 |
D). 14 |
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37). If \( \Large \cos \theta = \cos \alpha \cos \beta .\ then\ \tan \frac{ \theta + \alpha }{2} \tan \frac{ \theta - \alpha }{2} \) is equal to:
A). \( \Large \tan^{2}\frac{ \alpha }{2} \) |
B). \( \Large \tan^{2}\frac{ \beta }{2} \) |
C). \( \Large \tan^{2}\frac{ \theta }{2} \) |
D). \( \Large \cot^{2}\frac{ \beta }{2} \) |
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38). The greatest value of \( \Large \cos \theta \) for which \( \Large \cos 5 \theta = 0 \), is:
A). 0 |
B). \( \Large \frac{1+\sqrt{5}}{4} \) |
C). \( \Large \sqrt{\frac{5+\sqrt{5}}{8}} \) |
D). \( \Large \sqrt{\frac{5+\sqrt{1}}{4}} \) |
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39). A solution \( \Large \left(x, y\right)\ of\ x^{2}+2x \sin xy + 1 = 0 \) is:
A). (1,0) |
B). \( \Large \left(1,\ \frac{7 \pi }{2}\right) \) |
C). \( \Large \left(-1, \frac{7 \pi }{2}\right) \) |
D). (-1, 0) |
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40). If \( \Large \tan \theta ,\ 2 \tan \theta + 2,\ 3 \tan \theta + 3 \) are in GP, then the value of \( \Large \frac{7-5 \cot \theta }{9-4\sqrt{sec ^{2} \theta - 1}} \) is:
A). \( \Large \frac{12}{5} \) |
B). \( \Large -\frac{33}{28} \) |
C). \( \Large \frac{33}{100} \) |
D). \( \Large \frac{12}{13} \) |
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