CDS Exam (II)- 2025 - Elementary Mathematics Defence Question Paper

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CDS Exam (II)- 2025 - Elementary Mathematics

CDS Exam (II)- 2025 - Elementary Mathematics Defence Questions

31.
If \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}} = \mathrm{cosec} \theta \), then what is \( \sin \theta \) equal to?
A.
1
B.
\( \frac{\sqrt{3}}{2} \)
C.
\( \frac{1}{\sqrt{2}} \)
D.
\( \frac{1}{2} \)
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ANSWER :
D. \( \frac{1}{2} \)
32.
If \( 8 \sin \theta - \cos \theta = 4 \), where \( 0 < \theta < \pi/2 \), then what is \( \mathrm{cosec} \theta \) equal to?
A.
1
B.
3/2
C.
5/3
D.
2
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ANSWER :
C. 5/3
33.
If \( 2 \tan \theta = \sec^{2} \theta - 2 \), where \( 0 < \theta < \pi/2 \), then what is \( \cot \theta \) equal to?
A.
\( \sqrt{2} - 1 \)
B.
\( \sqrt{2} + 1 \)
C.
\( \sqrt{3} - 1 \)
D.
\( \sqrt{3} + 2 \)
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ANSWER :
A. \( \sqrt{2} - 1 \)
34.
What is \( (\sec \theta - \tan \theta) - \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \) equal to?
A.
0
B.
\( 2 \tan \theta \)
C.
\( 2 \sec \theta \)
D.
\( \sin \theta + \cos \theta \)
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ANSWER :
A. 0
35.
If \( \cot \theta = \sqrt{7} \), then what is \( \frac{\mathrm{cosec}^{2} \theta - \sec^{2} \theta}{\mathrm{cosec}^{2} \theta + \sec^{2} \theta} \) equal to?
A.
1/2
B.
1/3
C.
2/3
D.
3/4
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ANSWER :
D. 3/4
36.
The difference between the two acute angles in a right-angled triangle is \( \frac{\pi}{12} \) radian. One of the acute angles of the triangle is
A.
\( 60^{\circ} \)
B.
\( 57.5^{\circ} \)
C.
\( 52.5^{\circ} \)
D.
\( 47.5^{\circ} \)
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ANSWER :
C. \( 52.5^{\circ} \)
37.
If \( \alpha \) and \( \beta \) are the roots of the equation \( \log_{10} [ 998 + \sqrt{x^{2} - 18x + 76} ] = 3 \), then what is \( (\alpha - \beta)^{2} \) equal to?
A.
16
B.
25
C.
36
D.
49
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ANSWER :
C. 36
38.
If \( x^{4} + y^{4} = 14x^{2}y^{2} \), then consider the following:
I. \( \log_{10} (x^{2} + y^{2}) = \log_{10} x + \log_{10} y + 2 \log_{10} 2 \)
II. \( \log_{10} (x^{2} - y^{2}) = \log_{10} x + \log_{10} y + \log_{10} 2 + 0.5 \log_{10} 3 \)
Which of the above is/are correct?
A.
I only
B.
II only
C.
Both I and II
D.
Neither I nor II
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ANSWER :
C. Both I and II
39.
Which of the following is/are the factor(s) of \( (3x + y)^{2} + (3x + y)(x + 5y) - 20(x + 5y)^{2} \)?
I. \( (4x + 13y) \)
II. \( (x + 19y) \)
Select the correct answer using the code given below.
A.
I only
B.
II only
C.
Both I and II
D.
Neither I nor II
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ANSWER :
C. Both I and II
40.
What is \( \frac{\frac{x}{x - y} + \frac{y}{y - z} + \frac{z}{z - x}}{\frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x} + 3} \) equal to?
A.
1
B.
1/2
C.
1/3
D.
1/4
😑 View Answer Error
ANSWER :
B. 1/2
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