NDA & NA Exam (I) & (II)- 2020 - Mathematics Defence Question Paper

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NDA & NA Exam (I) & (II)- 2020 - Mathematics

NDA & NA Exam (I) & (II)- 2020 - Mathematics Defence Questions

21.
Directions for the following two (02) items:
Read the following information and answer the two items that follow:
ABCD is a trapezium such that AB and CD are parallel and BC is perpendicular to them. Let \( \angle ADB = \theta \), \( \angle ABD = \alpha \), \( BC = p \) and \( CD = q \).

Consider the following:
1. \( AD \sin \theta = AB \sin \alpha \)
2. \( BD \sin \theta = AB \sin(\theta + \alpha) \)
Which of the above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
C. Both 1 and 2
22.
What is AB equal to?
A.
\( \frac{(p^{2} + q^{2}) \sin \theta}{p \cos \theta + q \sin \theta} \)
B.
\( \frac{p^{2} + q^{2}}{p^{2} \cos \theta + q^{2} \sin \theta} \)
C.
\( \frac{(p^{2} + q^{2}) \sin \theta}{q \cos \theta + p \sin \theta} \)
D.
\( \frac{(p^{2} - q^{2}) \cos \theta}{q \cos \theta + p \sin \theta} \)
😑 View Answer Error
ANSWER :
A. \( \frac{(p^{2} + q^{2}) \sin \theta}{p \cos \theta + q \sin \theta} \)
23.
If \( \tan \theta = \frac{\cos 17^{\circ} - \sin 17^{\circ}}{\cos 17^{\circ} + \sin 17^{\circ}} \), then what is the value of \( \theta \)?
A.
\( 0^{\circ} \)
B.
\( 28^{\circ} \)
C.
\( 38^{\circ} \)
D.
\( 52^{\circ} \)
😑 View Answer Error
ANSWER :
B. \( 28^{\circ} \)
24.
A and B are positive acute angles such that \( \cos 2B = 3 \sin^{2} A \) and \( 3 \sin 2A = 2 \sin 2B \). What is the value of \( (A + 2B) \)?
A.
\( \frac{\pi}{6} \)
B.
\( \frac{\pi}{4} \)
C.
\( \frac{\pi}{3} \)
D.
\( \frac{\pi}{2} \)
😑 View Answer Error
ANSWER :
D. \( \frac{\pi}{2} \)
25.
What is \( \sin 3x + \cos 3x + 4 \sin^{3} x - 3 \sin x + 3 \cos x - 4 \cos^{3} x \) equal to?
A.
0
B.
1
C.
\( 2 \sin 2x \)
D.
\( 4 \cos 4x \)
😑 View Answer Error
ANSWER :
A. 0
26.
The value of ordinate of the graph of \( y = 2 + \cos x \) lies in the interval
A.
\( [0, 1] \)
B.
\( [0, 3] \)
C.
\( [-1, 1] \)
D.
\( [1, 3] \)
😑 View Answer Error
ANSWER :
D. \( [1, 3] \)
27.
What is the value of \( 8 \cos 10^{\circ} \cos 20^{\circ} \cos 40^{\circ} \)?
A.
\( \tan 10^{\circ} \)
B.
\( \cot 10^{\circ} \)
C.
\( \csc 10^{\circ} \)
D.
\( \sec 10^{\circ} \)
😑 View Answer Error
ANSWER :
B. \( \cot 10^{\circ} \)
28.
What is the value of \( \cos 48^{\circ} - \cos 12^{\circ} \)?
A.
\( \frac{\sqrt{5} - 1}{4} \)
B.
\( \frac{1 - \sqrt{5}}{4} \)
C.
\( \frac{\sqrt{5} + 1}{8} \)
D.
\( \frac{1 - \sqrt{5}}{8} \)
😑 View Answer Error
ANSWER :
B. \( \frac{1 - \sqrt{5}}{4} \)
29.
Consider the following statements:
1. If ABC is a right-angled triangle, right-angled at A and if \( \sin B = \frac{1}{3} \), then \( \csc C = 3 \).
2. If \( b \cos B = c \cos C \) and if the triangle ABC is not right-angled, then ABC must be isosceles.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
B. 2 only
30.
Consider the following statements:
1. If in a triangle ABC, \( A = 2B \) and \( b = c \), then it must be an obtuse-angled triangle.
2. There exists no triangle ABC with \( A = 40^{\circ} \), \( B = 65^{\circ} \) and \( \frac{a}{c} = \sin 40^{\circ} \csc 15^{\circ} \).
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
B. 2 only
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