NDA & NA Exam (II)- 2016 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2016 - Mathematics

NDA & NA Exam (II)- 2016 - Mathematics Defence Questions

41.
\( \sin A + 2\sin 2A + \sin 3A \) is equal to which of the following?
1. \( 4\sin 2A \cos^{2}\left(\frac{A}{2}\right) \)
2. \( 2\sin 2A \left(\sin\frac{A}{2} + \cos\frac{A}{2}\right)^{2} \)
3. \( 8\sin A \cos A \cos^{2}\left(\frac{A}{2}\right) \)
Select the correct answer using the code given below.
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
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ANSWER :
C. 1 and 3 only
42.
If \( x = \sin 70^{\circ} \sin 50^{\circ} \) and \( y = \cos 60^{\circ} \cos 80^{\circ} \), then what is xy equal to?
A.
\( \frac{1}{16} \)
B.
\( \frac{1}{8} \)
C.
\( \frac{1}{4} \)
D.
\( \frac{1}{2} \)
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ANSWER :
A. \( \frac{1}{16} \)
43.
If \( \sin\theta_{1} + \sin\theta_{2} + \sin\theta_{3} + \sin\theta_{4} = 4 \), then what is the value of \( \cos\theta_{1} + \cos\theta_{2} + \cos\theta_{3} + \cos\theta_{4} \)?
A.
0
B.
1
C.
2
D.
4
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ANSWER :
A. 0
44.
What is the value of \( \left(1 + \cos\frac{\pi}{8}\right)\left(1 + \cos\frac{3\pi}{8}\right)\left(1 + \cos\frac{5\pi}{8}\right)\left(1 + \cos\frac{7\pi}{8}\right) \)?
A.
\( \frac{1}{2} \)
B.
\( \frac{1}{2} + \frac{1}{2\sqrt{2}} \)
C.
\( \frac{1}{2} - \frac{1}{2\sqrt{2}} \)
D.
\( \frac{1}{8} \)
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ANSWER :
D. \( \frac{1}{8} \)
45.
If \( x\cos\theta + y\sin\theta = z \), then what is the value of \( (x\sin\theta - y\cos\theta)^{2} \)?
A.
\( x^{2} + y^{2} - z^{2} \)
B.
\( x^{2} - y^{2} - z^{2} \)
C.
\( x^{2} - y^{2} + z^{2} \)
D.
\( x^{2} + y^{2} + z^{2} \)
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ANSWER :
A. \( x^{2} + y^{2} - z^{2} \)
46.
What is the value of \( \cos(2\cos^{-1} 0.8) \)?
A.
0.81
B.
0.56
C.
0.48
D.
0.28
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ANSWER :
D. 0.28
47.
The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?
A.
\( \frac{h \cot q}{\cot q - \cot p} \)
B.
\( \frac{h \cot p}{\cot p - \cot q} \)
C.
\( \frac{2h \tan p}{\tan p - \tan q} \)
D.
\( \frac{2h \tan q}{\tan q - \tan p} \)
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ANSWER :
B. \( \frac{h \cot p}{\cot p - \cot q} \)
48.
If \( \sin 18^{\circ} = \frac{\sqrt{5} - 1}{4} \), then what is the value of \( \sin 81^{\circ} \)?
A.
\( \frac{\sqrt{3+\sqrt{5}} + \sqrt{5-\sqrt{5}}}{4} \)
B.
\( \frac{\sqrt{3+\sqrt{5}} + \sqrt{5+\sqrt{5}}}{4} \)
C.
\( \frac{\sqrt{3-\sqrt{5}} + \sqrt{5-\sqrt{5}}}{4} \)
D.
\( \frac{\sqrt{3+\sqrt{5}} - \sqrt{5-\sqrt{5}}}{4} \)
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ANSWER :
A. \( \frac{\sqrt{3+\sqrt{5}} + \sqrt{5-\sqrt{5}}}{4} \)
49.
A moving boat is observed from the top of a cliff of 150 m height. The angle of depression of the boat changes from 60° to 45° in 2 minutes. What is the speed of the boat in metres per hour?
A.
\( \frac{4500}{\sqrt{3}} \)
B.
\( \frac{4500(\sqrt{3} - 1)}{\sqrt{3}} \)
C.
\( 4500\sqrt{3} \)
D.
\( \frac{4500(\sqrt{3} + 1)}{\sqrt{3}} \)
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ANSWER :
B. \( \frac{4500(\sqrt{3} - 1)}{\sqrt{3}} \)
50.
What is \( \frac{1 - \tan 2^{\circ} \cot 62^{\circ}}{\tan 152^{\circ} - \cot 88^{\circ}} \) equal to?
A.
\( \sqrt{3} \)
B.
\( -\sqrt{3} \)
C.
\( \sqrt{2} - 1 \)
D.
\( 1 - \sqrt{2} \)
😑 View Answer Error
ANSWER :
B. \( -\sqrt{3} \)
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