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

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CDS Exam (I)- 2025 - Elementary Mathematics

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

31.
If \( \sin \alpha + \cos \alpha = \sqrt{2} \), where \( 0 < \alpha < \frac{\pi}{2} \), then what is \( \sin^{3} \alpha - \cos^{3} \alpha \) equal to?
A.
1
B.
\( \frac{1}{2} \)
C.
\( \frac{1}{4} \)
D.
0
😑 View Answer Error
ANSWER :
D. 0
32.
What is \( (1 + \cot \alpha - \mathrm{cosec} \alpha)(1 + \tan \alpha + \sec \alpha) \) equal to?
A.
\( \frac{1}{2} \)
B.
1
C.
2
D.
4
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ANSWER :
C. 2
33.
If \( \tan \theta = \frac{\sin \alpha - \cos \alpha}{\sin \alpha + \cos \alpha} \), where \( \theta \) and \( \alpha \) \( \left(\alpha \neq \frac{\pi}{4}\right) \) are acute angles, then what is \( \sqrt{2} \sin \theta \) equal to?
A.
\( \sin \alpha - \cos \alpha \)
B.
\( \sin \alpha + \cos \alpha \)
C.
\( \cos \alpha - \sin \alpha \)
D.
\( \pm (\sin \alpha - \cos \alpha) \)
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ANSWER :
A. \( \sin \alpha - \cos \alpha \)
34.
For how many values of \( \alpha \) does the expression \( (\sin \alpha + 2)(\sin \alpha + 4)(\sin \alpha - 2)(\sin \alpha - 4) \) become zero?
A.
No value
B.
One
C.
Two
D.
Four
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ANSWER :
A. No value
35.
What is the value of x, where \( 0^{\circ} \leq x < 30^{\circ} \), satisfying \( \tan 3x \tan 6x = 1 \)?
A.
\( 0^{\circ} \)
B.
\( 10^{\circ} \)
C.
\( 12^{\circ} \)
D.
\( 15^{\circ} \)
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ANSWER :
B. \( 10^{\circ} \)
36.
What is
\( \frac{(a - b)^{2}}{(b - c)(c - a)} + \frac{(b - c)^{2}}{(c - a)(a - b)} + \frac{(c - a)^{2}}{(a - b)(b - c)} - 3 \)
equal to, where \( a \neq b \neq c \)?
A.
0
B.
3
C.
a + b + c
D.
3(a - b)(b - c)(c - a)
😑 View Answer Error
ANSWER :
A. 0
37.
Given that \( \frac{100 \times 99 \times 98 \times \ldots \times 3 \times 2 \times 1}{100^{n}} \) is an integer. What is the largest value of n for which this is true?
A.
20
B.
21
C.
24
D.
None of the above
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ANSWER :
D. None of the above
38.
A man starting from a place P went x metre (\( x > 120 \) m) East before turning South. He went 40 m straight before turning to West. He went 60 m to reach a place Q. From Q he went 200 m North and reached a place R. If PR = 200 m, then what is x equal to?
A.
150 m
B.
180 m
C.
200 m
D.
240 m
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ANSWER :
B. 180 m
39.
If \( x^{2} + y^{2} + z^{2} = 3 \), where x, y and z are integers, then how many values can (xy + yz + zx) have?
A.
One
B.
Two
C.
Three
D.
Four
😑 View Answer Error
ANSWER :
B. Two
40.
If x, y, z are real numbers such that x + y + z = 10 and xy + yz + zx = 18, then what is the value of \( x^{3} + y^{3} + z^{3} - 3xyz \)?
A.
400
B.
440
C.
460
D.
500
😑 View Answer Error
ANSWER :
C. 460
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