NDA & NA Exam (I)- 2019 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2019 - Mathematics

NDA & NA Exam (I)- 2019 - Mathematics Defence Questions

41.
If \( \tan \theta = \frac{1}{2} \) and \( \tan \phi = \frac{1}{3} \), then what is the value of \( (\theta + \phi) \)?
A.
0
B.
\( \frac{\pi}{6} \)
C.
\( \frac{\pi}{4} \)
D.
\( \frac{\pi}{3} \)
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ANSWER :
C. \( \frac{\pi}{4} \)
42.
If \( \cos A = \frac{3}{4} \), then what is the value of \( \sin \left(\frac{A}{2}\right) \sin \left(\frac{3A}{2}\right) \)?
A.
\( \frac{5}{8} \)
B.
\( \frac{5}{16} \)
C.
\( \frac{5}{24} \)
D.
\( \frac{7}{32} \)
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ANSWER :
B. \( \frac{5}{16} \)
43.
What is the value of \( \tan 75^{\circ} + \cot 75^{\circ} \)?
A.
2
B.
4
C.
\( 2\sqrt{3} \)
D.
\( 4\sqrt{3} \)
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ANSWER :
B. 4
44.
What is the value of \( \cos 46^{\circ} \cos 47^{\circ} \cos 48^{\circ} \cos 49^{\circ} \cos 50^{\circ} \ldots \cos 135^{\circ} \)?
A.
−1
B.
0
C.
1
D.
Greater than 1
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ANSWER :
B. 0
45.
If \( \sin 2\theta = \cos 3\theta \), where \( 0 < \theta < \frac{\pi}{2} \), then what is \( \sin \theta \) equal to?
A.
\( \frac{\sqrt{5} + 1}{4} \)
B.
\( \frac{\sqrt{5} - 1}{4} \)
C.
\( \frac{\sqrt{5} + 1}{16} \)
D.
\( \frac{\sqrt{5} - 1}{16} \)
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ANSWER :
B. \( \frac{\sqrt{5} - 1}{4} \)
46.
If the roots of the equation \( x^{2} + px + q = 0 \) are \( \tan 19^{\circ} \) and \( \tan 26^{\circ} \), then which one of the following is correct?
A.
\( q - p = 1 \)
B.
\( p - q = 1 \)
C.
\( p + q = 2 \)
D.
\( p + q = 3 \)
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ANSWER :
A. \( q - p = 1 \)
47.
What is the fourth term of an AP of n terms whose sum is \( n(n + 1) \)?
A.
6
B.
8
C.
12
D.
20
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ANSWER :
B. 8
48.
What is \( \frac{(1 + \tan \alpha \tan \beta)^{2} + (\tan \alpha - \tan \beta)^{2}}{\sec^{2} \alpha \sec^{2} \beta} \) equal to?
A.
0
B.
1
C.
2
D.
4
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ANSWER :
B. 1
49.
If \( p = \csc \theta - \cot \theta \) and \( q = (\csc \theta + \cot \theta)^{-1} \), then which one of the following is correct?
A.
\( pq = 1 \)
B.
\( p = q \)
C.
\( pq = -1 \)
D.
\( p + q = 0 \)
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ANSWER :
B. \( p = q \)
50.
If the angles of a triangle ABC are in the ratio 1 : 2 : 3, then the corresponding sides are in the ratio
A.
1 : 2 : 3
B.
3 : 2 : 1
C.
\( 1 : \sqrt{3} : 2 \)
D.
\( 1 : \sqrt{3} : \sqrt{2} \)
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ANSWER :
C. \( 1 : \sqrt{3} : 2 \)
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