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

21.
If \( 2 \sin^{4} \alpha + 2 \cos^{4} \alpha - 1 = 0 \), where \( 0 \leq \alpha < \frac{\pi}{2} \), then what is \( \sin 2\alpha + \cos 2\alpha \) equal to?
A.
0
B.
1
C.
\( \frac{\sqrt{3} + 1}{2} \)
D.
\( \frac{\sqrt{3} - 1}{2} \)
😑 View Answer Error
ANSWER :
B. 1
22.
Consider the following:
I. \( 1 - \sin^{6} \alpha = \cos^{2} \alpha (\cos^{4} \alpha - 3 \cos^{2} \alpha + 3) \)
II. \( \cos^{8} \alpha - \sin^{8} \alpha = 2 \sin^{2} \alpha (1 - \cos^{4} \alpha + \sin^{2} \alpha \cos^{2} \alpha) \)
Which of the above is/are identities?
A.
I only
B.
II only
C.
Both I and II
D.
Neither I nor II
😑 View Answer Error
ANSWER :
A. I only
23.
If \( p = \frac{1}{\mathrm{cosec} \theta + \cot \theta} \) and \( q = \mathrm{cosec} \theta \), then what is \( p^{2} - 2pq \) equal to?
A.
-1
B.
0
C.
1
D.
2
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ANSWER :
A. -1
24.
Consider the following statements:
I. \( (\mathrm{cosec} \alpha - \sec \alpha) \) is always positive in the first quadrant.
II. \( (\tan \alpha - \cot \alpha) \) is always negative in the first quadrant.
Which of the statements given above is/are correct?
A.
I only
B.
II only
C.
Both I and II
D.
Neither I nor II
😑 View Answer Error
ANSWER :
D. Neither I nor II
25.
A tower subtends an angle \( 60^{\circ} \) at a point A on the same level as the foot of the tower. B is a point vertically above A and AB = h. The angle of depression of the foot of the tower, measured from B is \( 30^{\circ} \). What is the height of the tower?
A.
2h
B.
2.5h
C.
3h
D.
3.5h
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ANSWER :
C. 3h
26.
What is \( \frac{\sin \theta}{1 - \cot \theta} + \frac{\cos \theta}{1 - \tan \theta} \) \( \left(\theta \neq \frac{\pi}{4}\right) \) equal to?
A.
\( \sin \theta + \cos \theta \)
B.
\( \sin \theta - \cos \theta \)
C.
\( \cos \theta - \sin \theta \)
D.
\( -(\sin \theta + \cos \theta) \)
😑 View Answer Error
ANSWER :
A. \( \sin \theta + \cos \theta \)
27.
The length of an arc of a circle of radius 4 cm is \( \pi \) cm. What is the magnitude of the angle subtended by the arc at the centre?
A.
\( \pi \)
B.
\( \frac{\pi}{2} \)
C.
\( \frac{\pi}{3} \)
D.
\( \frac{\pi}{4} \)
😑 View Answer Error
ANSWER :
D. \( \frac{\pi}{4} \)
28.
If \( \cot^{2} \theta - 3\sqrt{3} \cot \theta + 6 = 0 \), where \( \frac{\pi}{6} \leq \theta < \frac{\pi}{2} \), then what is a value of \( \sin \theta + \cos 2\theta \)?
A.
0
B.
1
C.
\( \sqrt{3} \)
D.
\( 1 + \sqrt{2} \)
😑 View Answer Error
ANSWER :
B. 1
29.
Which of the following equations is/are possible?
I. \( \sin^{2} \theta = \frac{(x + y)^{2}}{4xy} \), where x, y are positive unequal real quantities.
II. \( \sin \theta + \cos \theta = x + \frac{1}{x} \), where x is a positive real quantity.
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 :
D. Neither I nor II
30.
If \( m^{2} (\sin \theta - 1) + n^{2} (\sin \theta + 1) = 0 \), where \( 0 < \theta < \frac{\pi}{2} \), then what is \( (m^{2} + n^{2}) \cos \theta - (m^{2} - n^{2}) \cot \theta \) equal to?
A.
4mn
B.
2mn
C.
1
D.
0
😑 View Answer Error
ANSWER :
D. 0
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