NDA & NA Exam (I)- 2024 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2024 - Mathematics

NDA & NA Exam (I)- 2024 - Mathematics Defence Questions

41.
ABC is a triangle such that angle \( C = 60^{\circ} \), then what is \( \frac{\cos A + \cos B}{\cos\left(\frac{A - B}{2}\right)} \) equal to?
A.
\( \sqrt{2} \)
B.
2
C.
1
D.
\( \frac{1}{2} \)
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ANSWER :
C. 1
42.
What is \( \sqrt{15 + \cot^{2}\left(\frac{\pi}{4} - 2\cot^{-1} 3\right)} \) equal to?
A.
1
B.
7
C.
8
D.
16
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ANSWER :
C. 8
43.
What is the value of \( \sin 10^{\circ} \cdot \sin 50^{\circ} + \sin 50^{\circ} \cdot \sin 250^{\circ} + \sin 250^{\circ} \cdot \sin 10^{\circ} \) equal to?
A.
\( -\frac{1}{4} \)
B.
\( -\frac{3}{4} \)
C.
\( \frac{3\sin 10^{\circ}}{4} \)
D.
\( -\frac{3\cos 10^{\circ}}{4} \)
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ANSWER :
B. \( -\frac{3}{4} \)
44.
What is \( \tan^{-1}\left(\frac{a}{b}\right) - \tan^{-1}\left(\frac{a - b}{a + b}\right) \) equal to?
A.
\( -\frac{\pi}{4} \)
B.
\( \frac{\pi}{4} \)
C.
\( \tan^{-1}\left(\frac{a^{2} - b^{2}}{a^{2} + b^{2}}\right) \)
D.
\( \tan^{-1}\left(\frac{2ab}{a^{2} + b^{2}}\right) \)
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ANSWER :
B. \( \frac{\pi}{4} \)
45.
Which one of the following conditions on \( \beta \in [0, \pi] \) is both necessary and sufficient for the equation \( (\cos\beta - 1)x^{2} + (\cos\beta)x + \sin\beta = 0 \) in x to have a real root?
A.
\( 1 - \cos\beta < 0 \)
B.
\( 1 - \cos\beta \leq 0 \)
C.
\( 1 - \cos\beta > 0 \)
D.
\( 1 - \cos\beta \geq 0 \)
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ANSWER :
D. \( 1 - \cos\beta \geq 0 \)
46.
In a triangle ABC, AB = 16 cm, BC = 63 cm and AC = 65 cm. What is the value of cos 2A + cos 2B + cos 2C?
A.
−1
B.
0
C.
1
D.
\( \frac{76}{65} \)
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ANSWER :
A. −1
47.
If \( f(\theta) = \frac{1}{1 + \tan\theta} \) and \( \alpha + \beta = \frac{5\pi}{4} \), then what is the value of \( f(\alpha)f(\beta) \)?
A.
\( -\frac{1}{2} \)
B.
\( \frac{1}{2} \)
C.
1
D.
2
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ANSWER :
B. \( \frac{1}{2} \)
48.
If \( \tan\alpha \) and \( \tan\beta \) are the roots of the equation \( x^{2} - 6x + 8 = 0 \), then what is the value of \( \cos(2\alpha + 2\beta) \)?
A.
\( \frac{13}{75} \)
B.
\( \frac{13}{85} \)
C.
\( \frac{17}{85} \)
D.
\( \frac{19}{85} \)
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ANSWER :
B. \( \frac{13}{85} \)
49.
What is the value of \( \tan 65^{\circ} + 2\tan 45^{\circ} - 2\tan 40^{\circ} - \tan 25^{\circ} \)?
A.
0
B.
1
C.
2
D.
4
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ANSWER :
C. 2
50.
Consider the following statements:
1. In a triangle ABC, if \( \cot A \cdot \cot B \cdot \cot C > 0 \), then the triangle is an acute angled triangle.
2. In a triangle ABC, if \( \tan A \cdot \tan B \cdot \tan C > 0 \), then the triangle is an obtuse angled triangle.
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
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ANSWER :
A. 1 only
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