NDA & NA Exam (II)- 2017 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2017 - Mathematics

NDA & NA Exam (II)- 2017 - Mathematics Defence Questions

31.
The angle of elevation of a stationary cloud from a point 25 m above a lake is \( 15^{\circ} \) and the angle of depression of its image in the lake is \( 45^{\circ} \). The height of the cloud above the lake level is
A.
25 m
B.
\( 25\sqrt{3} \) m
C.
50 m
D.
\( 50\sqrt{3} \) m
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ANSWER :
B. \( 25\sqrt{3} \) m
32.
The value of \( \tan 9^{\circ} - \tan 27^{\circ} - \tan 63^{\circ} + \tan 81^{\circ} \) is equal to
A.
−1
B.
0
C.
1
D.
4
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ANSWER :
D. 4
33.
The value of \( \sqrt{3} cosec 20^{\circ} - \sec 20^{\circ} \) is equal to
A.
4
B.
2
C.
1
D.
−4
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ANSWER :
A. 4
34.
Angle \( \alpha \) is divided into two parts A and B such that \( A - B = x \) and \( \tan A : \tan B = p : q \). The value of \( \sin x \) is equal to
A.
\( \frac{(p+q)\sin\alpha}{p-q} \)
B.
\( \frac{p\sin\alpha}{p+q} \)
C.
\( \frac{p\sin\alpha}{p-q} \)
D.
\( \frac{(p-q)\sin\alpha}{p+q} \)
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ANSWER :
D. \( \frac{(p-q)\sin\alpha}{p+q} \)
35.
The value of \( \sin^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) \) is equal to
A.
0
B.
\( \frac{\pi}{4} \)
C.
\( \frac{\pi}{3} \)
D.
\( \frac{\pi}{2} \)
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ANSWER :
B. \( \frac{\pi}{4} \)
36.
The angles of elevation of the top of a tower from the top and foot of a pole are respectively \( 30^{\circ} \) and \( 45^{\circ} \). If \( h_{T} \) is the height of the tower and \( h_{P} \) is the height of the pole, then which of the following are correct?
1. \( \frac{2h_{P}h_{T}}{3 + \sqrt{3}} = h_{P}^{2} \)
2. \( \frac{h_{T} - h_{P}}{\sqrt{3} + 1} = \frac{h_{P}}{2} \)
3. \( \frac{2(h_{P} + h_{T})}{h_{P}} = 4 + \sqrt{3} \)
Select the correct answer using the code given below.
A.
1 and 3 only
B.
2 and 3 only
C.
1 and 2 only
D.
1, 2 and 3
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ANSWER :
C. 1 and 2 only
37.
In a triangle ABC, \( a - 2b + c = 0 \). The value of \( \cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right) \) is
A.
\( \frac{9}{2} \)
B.
3
C.
\( \frac{3}{2} \)
D.
1
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ANSWER :
B. 3
38.
\( \sqrt{1 + \sin A} = -\left(\sin\frac{A}{2} + \cos\frac{A}{2}\right) \) is true, if
A.
\( \frac{3\pi}{2} < A < \frac{5\pi}{2} \) only
B.
\( \frac{\pi}{2} < A < \frac{3\pi}{2} \) only
C.
\( \frac{3\pi}{2} < A < \frac{7\pi}{2} \)
D.
\( 0 < A < \frac{3\pi}{2} \)
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ANSWER :
C. \( \frac{3\pi}{2} < A < \frac{7\pi}{2} \)
39.
In triangle ABC, if \( \frac{\sin^{2}A + \sin^{2}B + \sin^{2}C}{\cos^{2}A + \cos^{2}B + \cos^{2}C} = 2 \), then the triangle is
A.
right-angled
B.
equilateral
C.
isosceles
D.
obtuse-angled
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ANSWER :
A. right-angled
40.
The principal value of \( \sin^{-1} x \) lies in the interval
A.
\( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
B.
\( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
C.
\( \left[0, \frac{\pi}{2}\right] \)
D.
\( [0, \pi] \)
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ANSWER :
B. \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
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