NDA & NA Exam (II)- 2025 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2025 - Mathematics

NDA & NA Exam (II)- 2025 - Mathematics Defence Questions

21.
For the following two (02) items:
Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation
\( x^{2} + (\log_{0.5}(a^{2}))x - 2(\log_{0.5}(a^{2}))^{2} = 0 \)
where \( a^{2} \neq 1 \) and \( \log_{0.5}(a^{2}) > 0 \). Further, \( \beta > 0 \).

What is \( \beta \) equal to?
A.
\( \log_{a^{2}}(0.5) \)
B.
\( \log_{0.5}(a^{2}) \)
C.
\( 2(\log_{a^{2}}(0.5)) \)
D.
\( 2\log_{0.5}(a^{2}) \)
😑 View Answer Error
ANSWER :
B. \( \log_{0.5}(a^{2}) \)
22.
What is the relation between \( \alpha \) and \( \beta \)?
A.
\( \alpha = 2\beta \)
B.
\( 2\alpha = \beta \)
C.
\( \alpha = -2\beta \)
D.
\( 2\alpha = -\beta \)
😑 View Answer Error
ANSWER :
C. \( \alpha = -2\beta \)
23.
For the following two (02) items:
Let \( p = \sum_{j=1}^{n} \log_{10} 2^{j} \) and \( q = \sum_{j=1}^{n} \log_{10} 5^{j} \).

If \( p + q = 66 \), then which one of the following is correct?
A.
\( n < 7 \)
B.
\( 7 < n < 9 \)
C.
\( 9 < n < 12 \)
D.
\( n > 12 \)
😑 View Answer Error
ANSWER :
C. \( 9 < n < 12 \)
24.
If \( p + q = 15 \), then what is \( q - p \) equal to?
A.
\( \log_{10} 2.5 \)
B.
\( 5\log_{10} 2.5 \)
C.
\( 10\log_{10} 2.5 \)
D.
\( 15\log_{10} 2.5 \)
😑 View Answer Error
ANSWER :
D. \( 15\log_{10} 2.5 \)
25.
For the following two (02) items:
Let \( \sin A + \sin B = p \) and \( \cos A + \cos B = q \).

What is \( \frac{p}{q} \) equal to?
A.
\( \tan\left(\frac{A-B}{2}\right) \)
B.
\( \cot\left(\frac{A-B}{2}\right) \)
C.
\( \tan\left(\frac{A+B}{2}\right) \)
D.
\( \cot\left(\frac{A+B}{2}\right) \)
😑 View Answer Error
ANSWER :
C. \( \tan\left(\frac{A+B}{2}\right) \)
26.
What is \( \frac{p^{2} - q^{2}}{p^{2} + q^{2}} \) equal to?
A.
\( \cos(A + B) \)
B.
\( \cos(A - B) \)
C.
\( \cos\left(\frac{\pi}{2} - A - B\right) \)
D.
\( \cos(\pi - A - B) \)
😑 View Answer Error
ANSWER :
D. \( \cos(\pi - A - B) \)
27.
For the following two (02) items:
Let \( p = \mathrm{cosec} 20^{\circ} \) and \( q = \mathrm{cosec} 70^{\circ} \).

What is \( \left(\frac{\sqrt{3}p}{4} - \frac{q}{4}\right) \) equal to?
A.
-1
B.
0
C.
1
D.
2
😑 View Answer Error
ANSWER :
C. 1
28.
What is \( \frac{p^{2} + q^{2}}{p^{2}q^{2}} \) equal to?
A.
\( \frac{1}{2} \)
B.
1
C.
\( \frac{3}{2} \)
D.
2
😑 View Answer Error
ANSWER :
B. 1
29.
For the following two (02) items:
Let \( \cos(2x + 3y) = \frac{1}{2} \) and \( \cos(3x + 2y) = \frac{\sqrt{3}}{2} \), where \( -\pi < (2x + 3y) < \pi \) and \( -\pi < (3x + 2y) < \pi \).

How many values does \( (x + y) \) have?
A.
Two
B.
Three
C.
Four
D.
More than four
😑 View Answer Error
ANSWER :
C. Four
30.
How many values does \( (y - x) \) have?
A.
Two
B.
Three
C.
Four
D.
More than four
😑 View Answer Error
ANSWER :
C. Four
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