NDA & NA Exam (I)- 2023 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2023 - Mathematics

NDA & NA Exam (I)- 2023 - Mathematics Defence Questions

31.
Directions (Qn. 31-32): Consider the following for the next two (02) items that follow:
Let \( \sin \beta \) be the GM of \( \sin \alpha \) and \( \cos \alpha \); \( \tan \gamma \) be the AM of \( \sin \alpha \) and \( \cos \alpha \).

What is \( \cos 2\beta \) equal to?
A.
\( (\cos \alpha - \sin \alpha)^{2} \)
B.
\( (\cos \alpha + \sin \alpha)^{2} \)
C.
\( (\cos \alpha - \sin \alpha)^{3} \)
D.
\( \frac{(\cos \alpha + \sin \alpha)^{2}}{2} \)
😑 View Answer Error
ANSWER :
A. \( (\cos \alpha - \sin \alpha)^{2} \)
32.
What is the value of \( \sec 2\gamma \)?
A.
\( \frac{3 - \sin 2\alpha}{5 + 2 \sin 2\alpha} \)
B.
\( \frac{5 + \sin 2\alpha}{3 - \sin 2\alpha} \)
C.
\( \frac{3 - 2 \sin 2\alpha}{4 + \sin 2\alpha} \)
D.
\( \frac{3 - \sin 2\alpha}{4 + 3 \sin 2\alpha} \)
😑 View Answer Error
ANSWER :
B. \( \frac{5 + \sin 2\alpha}{3 - \sin 2\alpha} \)
33.
Directions (Qn. 33-34): Consider the following for the next two (02) items that follow:
A flagstaff 20 m long standing on a pillar 10 m high subtends an angle \( \tan^{-1}(0.5) \) at a point P on the ground. Let \( \theta \) be the angle subtended by the pillar at this point P.

If \( x \) is the distance of P from bottom of the pillar, then consider the following statements:
1. \( x \) can take two values which are in the ratio 1 : 3
2. \( x \) can be equal to height of the flagstaff
Which of the statements given above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
A. 1 only
34.
What is a possible value of \( \tan \theta \)?
A.
\( \frac{3}{2} \)
B.
\( \frac{2}{3} \)
C.
\( \frac{1}{3} \)
D.
\( \frac{1}{4} \)
😑 View Answer Error
ANSWER :
C. \( \frac{1}{3} \)
35.
Directions (Qn. 35-36): Consider the following for the next two (02) items that follow:
The perimeter of a triangle ABC is 6 times the AM of sine of angles of the triangle. Further \( BC = \sqrt{3} \), \( CA = 1 \) and angle A is acute.

What is the perimeter of the triangle?
A.
\( \sqrt{3} + 1 \)
B.
\( \sqrt{3} + 2 \)
C.
\( \sqrt{3} + 3 \)
D.
\( 2\sqrt{3} + 1 \)
😑 View Answer Error
ANSWER :
C. \( \sqrt{3} + 3 \)
36.
Consider the following statements:
1. ABC is right angled triangle
2. The angles of the triangle are in AP
Which of the statements given 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
37.
Directions (Qn. 37-38): Consider the following for the next two (02) items that follow:
Let \( x = \frac{\sin^{2} A + \sin A + 1}{\sin A} \), where \( 0 < A \leq \frac{\pi}{2} \).

What is the minimum value of \( x \)?
A.
1
B.
2
C.
3
D.
4
😑 View Answer Error
ANSWER :
C. 3
38.
At what value of \( A \) does \( x \) attain the minimum value?
A.
\( \frac{\pi}{6} \)
B.
\( \frac{\pi}{4} \)
C.
\( \frac{\pi}{3} \)
D.
\( \frac{\pi}{2} \)
😑 View Answer Error
ANSWER :
D. \( \frac{\pi}{2} \)
39.
Directions (Qn. 39-40): Consider the following for the next two (02) items that follow:
In the triangle ABC, \( a^{2} + b^{2} + c^{2} = ac + \sqrt{3}bc \).

What is the nature of the triangle?
A.
Equilateral
B.
Isosceles
C.
Right angled triangle
D.
Scalene but not right angled
😑 View Answer Error
ANSWER :
C. Right angled triangle
40.
If \( c = 8 \), what is the area of the triangle?
A.
\( 4\sqrt{3} \)
B.
\( 6\sqrt{3} \)
C.
\( 8\sqrt{3} \)
D.
\( 12\sqrt{3} \)
😑 View Answer Error
ANSWER :
C. \( 8\sqrt{3} \)
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