NDA & NA Exam (II)- 2026 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2026 - Mathematics

NDA & NA Exam (II)- 2026 - Mathematics Defence Questions

1.
Directions (Next 2 Questions):
From the top of a building, the angles of depression of the top and bottom of a tower of height h are \( \alpha \) and \( \beta \) respectively. Let H be the height of the building and D be the horizontal distance between the building and the tower.

What is \( H^{2} + D^{2} \) equal to?
A.
\( \frac{h^{2} \cos^{2} \alpha}{\sin^{2}(\beta - \alpha)} \)
B.
\( \frac{h^{2} \cos^{2} \beta}{\sin^{2}(\beta - \alpha)} \)
C.
\( \frac{h^{2} \cos^{2} \alpha}{\sin^{2}(\beta + \alpha)} \)
D.
\( \frac{h^{2} \cos^{2} \beta}{\sin^{2}(\beta + \alpha)} \)
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ANSWER :
A. \( \frac{h^{2} \cos^{2} \alpha}{\sin^{2}(\beta - \alpha)} \)
2.
If \( \alpha = 15^{\circ} \) and \( \beta = 45^{\circ} \), then which one of the following is correct?
A.
\( H = 2D \)
B.
\( D = 2H \)
C.
\( 2H = \sqrt{3}D \)
D.
\( H = D \)
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ANSWER :
D. \( H = D \)
3.
Directions (Next 2 Questions):
The angles A, B and C of a triangle ABC are in the ratio 1 : 2 : 7.

What is the ratio of the side of greatest length to the side of least length?
A.
7 : 1
B.
\( \sqrt{7} : 1 \)
C.
\( (3 + \sqrt{5}) : 2 \)
D.
\( (3 + \sqrt{5}) : 3 \)
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ANSWER :
C. \( (3 + \sqrt{5}) : 2 \)
4.
What is \( \sin A \cdot \cos B \) equal to?
A.
1/4
B.
1/2
C.
1
D.
2
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ANSWER :
A. 1/4
5.
Directions (Next 2 Questions):
In a triangle ABC, \( \frac{a + b}{13} = \frac{b + c}{11} = \frac{c + a}{12} \).

What is sin A : sin B : sin C equal to?
A.
5 : 6 : 7
B.
7 : 6 : 5
C.
13 : 11 : 12
D.
12 : 11 : 13
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ANSWER :
B. 7 : 6 : 5
6.
What is cos A : cos B : cos C equal to?
A.
7 : 19 : 25
B.
11 : 12 : 13
C.
7 : 11 : 13
D.
12 : 11 : 13
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ANSWER :
A. 7 : 19 : 25
7.
Directions (Next 2 Questions):
Given that \( \tan\left(\frac{A}{2} + \frac{B}{2}\right) = p \) and \( \tan\left(\frac{A}{2} - \frac{B}{2}\right) = q \), where \( pq \neq \pm 1 \).

What is tan A equal to?
A.
\( \frac{p - q}{1 + pq} \)
B.
\( \frac{p + q}{1 - pq} \)
C.
\( \frac{p + q}{1 + pq} \)
D.
\( \frac{p - q}{1 - pq} \)
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ANSWER :
B. \( \frac{p + q}{1 - pq} \)
8.
What is tan B equal to?
A.
\( \frac{p - q}{1 + pq} \)
B.
\( \frac{p + q}{1 - pq} \)
C.
\( \frac{p + q}{1 + pq} \)
D.
\( \frac{p - q}{1 - pq} \)
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ANSWER :
A. \( \frac{p - q}{1 + pq} \)
9.
Directions (Next 2 Questions):
Let \( 4(A + B) = \pi \).

What is \( (1 + \tan A)(1 + \tan B) \) equal to?
A.
1
B.
2
C.
4
D.
6
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ANSWER :
B. 2
10.
What is \( (\cot A - 1)(\cot B - 1) \) equal to?
A.
1/2
B.
1
C.
3/2
D.
2
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ANSWER :
D. 2
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