NDA & NA Exam (I)- 2018 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2018 - Mathematics

NDA & NA Exam (I)- 2018 - Mathematics Defence Questions

41.
If x, x − y and x + y are the angles of a triangle (not an equilateral triangle) such that tan(x − y), tan x and tan(x + y) are in GP, then what is x equal to?
A.
\( \frac{\pi}{4} \)
B.
\( \frac{\pi}{3} \)
C.
\( \frac{\pi}{6} \)
D.
\( \frac{\pi}{2} \)
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ANSWER :
B. \( \frac{\pi}{3} \)
42.
ABC is a triangle inscribed in a circle with centre O. Let \( \alpha = \angle BAC \), where \( 45^{\circ} < \alpha < 90^{\circ} \). Let \( \beta = \angle BOC \). Which one of the following is correct?
A.
\( \cos\beta = \frac{1 - \tan^{2}\alpha}{1 + \tan^{2}\alpha} \)
B.
\( \cos\beta = \frac{1 + \tan^{2}\alpha}{1 - \tan^{2}\alpha} \)
C.
\( \cos\beta = \frac{2\tan\alpha}{1 + \tan^{2}\alpha} \)
D.
\( \sin\beta = 2\sin^{2}\alpha \)
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ANSWER :
A. \( \cos\beta = \frac{1 - \tan^{2}\alpha}{1 + \tan^{2}\alpha} \)
43.
If a flag-staff of 6 m height placed on the top of a tower throws a shadow of \( 2\sqrt{3} \) m along the ground, then what is the angle that the sun makes with the ground?
A.
\( 60^{\circ} \)
B.
\( 45^{\circ} \)
C.
\( 30^{\circ} \)
D.
\( 15^{\circ} \)
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ANSWER :
A. \( 60^{\circ} \)
44.
What is \( \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{3}{5}\right) \) equal to?
A.
0
B.
\( \frac{\pi}{4} \)
C.
\( \frac{\pi}{3} \)
D.
\( \frac{\pi}{2} \)
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ANSWER :
B. \( \frac{\pi}{4} \)
45.
A spherical balloon of radius r subtends an angle \( \alpha \) at the eye of an observer, while the angle of elevation of its centre is \( \beta \). What is the height of the centre of the balloon (neglecting the height of the observer)?
A.
\( \frac{r\sin\beta}{\sin\left(\frac{\alpha}{2}\right)} \)
B.
\( \frac{r\sin\beta}{\sin\left(\frac{\alpha}{4}\right)} \)
C.
\( \frac{r\sin\left(\frac{\beta}{2}\right)}{\sin\alpha} \)
D.
\( \frac{r\sin\alpha}{\sin\left(\frac{\beta}{2}\right)} \)
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ANSWER :
A. \( \frac{r\sin\beta}{\sin\left(\frac{\alpha}{2}\right)} \)
46.
If \( \frac{\sin(x + y)}{\sin(x - y)} = \frac{a + b}{a - b} \), then what is \( \frac{\tan x}{\tan y} \) equal to?
A.
\( \frac{a}{b} \)
B.
\( \frac{b}{a} \)
C.
\( \frac{a + b}{a - b} \)
D.
\( \frac{a - b}{a + b} \)
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ANSWER :
A. \( \frac{a}{b} \)
47.
If \( \sin\alpha + \sin\beta = 0 = \cos\alpha + \cos\beta \), where \( 0 < \beta < \alpha < 2\pi \), then which one of the following is correct?
A.
\( \alpha = \pi - \beta \)
B.
\( \alpha = \pi + \beta \)
C.
\( \alpha = 2\pi - \beta \)
D.
\( 2\alpha = \pi + 2\beta \)
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ANSWER :
B. \( \alpha = \pi + \beta \)
48.
Suppose cos A is given. If only one value of \( \cos\left(\frac{A}{2}\right) \) is possible, then A must be
A.
an odd multiple of \( 90^{\circ} \)
B.
a multiple of \( 90^{\circ} \)
C.
an odd multiple of \( 180^{\circ} \)
D.
a multiple of \( 180^{\circ} \)
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ANSWER :
C. an odd multiple of \( 180^{\circ} \)
49.
If \( \cos\alpha + \cos\beta + \cos\gamma = 0 \), where \( 0 < \alpha \leq \frac{\pi}{2} \), \( 0 < \beta \leq \frac{\pi}{2} \), \( 0 < \gamma \leq \frac{\pi}{2} \), then what is the value of \( \sin\alpha + \sin\beta + \sin\gamma \)?
A.
0
B.
3
C.
\( \frac{5\sqrt{2}}{2} \)
D.
\( \frac{3\sqrt{2}}{2} \)
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ANSWER :
B. 3
50.
The maximum value of \( \sin\left(x + \frac{\pi}{5}\right) + \cos\left(x + \frac{\pi}{5}\right) \), where \( x \in \left(0, \frac{\pi}{2}\right) \), is attained at
A.
\( \frac{\pi}{20} \)
B.
\( \frac{\pi}{15} \)
C.
\( \frac{\pi}{10} \)
D.
\( \frac{\pi}{2} \)
😑 View Answer Error
ANSWER :
A. \( \frac{\pi}{20} \)
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