CDS Exam (II)- 2023 - Elementary Mathematics Defence Question Paper

CDS Previous Year Question Papers Question Papers
CDS Exam (II)- 2023 - Elementary Mathematics

CDS Exam (II)- 2023 - Elementary Mathematics Defence Questions

31.
Consider the following for the next two (02) items that follow:
A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be \( \alpha \). Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be \( \beta \) and that of the top of the tower is observed to be \( \gamma \). Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower.

Which one of the following is correct?
A.
\( H\tan \gamma - h\tan \beta = 0 \)
B.
\( h\tan \gamma - H\tan \beta = 0 \)
C.
\( H\tan \gamma - h\tan \alpha = 0 \)
D.
\( h\tan \gamma - H\tan \alpha = 0 \)
😑 View Answer Error
ANSWER :
A. \( H\tan \gamma - h\tan \beta = 0 \)
32.
Which one of the following is correct?
A.
\( d = \frac{H(\cot \alpha - \cot \beta)}{2} \)
B.
\( d = \frac{H(\tan \alpha - \tan \beta)}{2} \)
C.
\( d = H(\cot \alpha - \cot \beta) \)
D.
\( d = H(\tan \alpha - \tan \beta) \)
😑 View Answer Error
ANSWER :
C. \( d = H(\cot \alpha - \cot \beta) \)
33.
Consider the following for the next two (02) items that follow:
AB is a straight road leading to the foot P of a tower of height h. Q is at distance x from P and R is at a distance y from Q (R is farther from P than Q; R, Q are on the same side). The angle of elevation of the top of the tower at Q is twice of that at R. (Use the formula \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^{2} \theta} \))

Which one of the following is correct?
A.
\( x = y \)
B.
\( x < y \)
C.
\( x > y \)
D.
Cannot be concluded due to insufficient data
😑 View Answer Error
ANSWER :
B. \( x < y \)
34.
Which one of the following is correct?
A.
\( h^{2} = x^{2} - y^{2} \)
B.
\( h^{2} = x^{2} + y^{2} \)
C.
\( h^{2} = 2(y^{2} - x^{2}) \)
D.
\( h^{2} = y^{2} - x^{2} \)
😑 View Answer Error
ANSWER :
D. \( h^{2} = y^{2} - x^{2} \)
35.
Consider the following for the next two (02) items that follow:
Two parallel chords AB and CD of a circle are of lengths 60 cm and 80 cm respectively. They are on the same side of the centre O and 10 cm apart.

What is the diameter of the circle?
A.
120 cm
B.
110 cm
C.
100 cm
D.
90 cm
😑 View Answer Error
ANSWER :
C. 100 cm
36.
If the chord AB subtends an angle \( \alpha \) and chord CD subtends an angle \( \beta \) at the centre O, then what is the value of \( \tan \left(\frac{\beta}{2}\right) - \tan \left(\frac{\alpha}{2}\right) \)?
A.
\( \frac{3}{4} \)
B.
\( \frac{5}{12} \)
C.
\( \frac{1}{2} \)
D.
\( \frac{7}{12} \)
😑 View Answer Error
ANSWER :
D. \( \frac{7}{12} \)
37.
Consider the following for the next two (02) items that follow:
Let \( p = x^{4} - y^{2}z^{2} \), \( q = y^{4} - z^{2}x^{2} \), \( r = z^{4} - x^{2}y^{2} \).

What is \( px^{2} + qy^{2} + rz^{2} \) equal to?
A.
\( (x^{2} + y^{2} + z^{2})(p + q + r) \)
B.
\( -(x^{2} + y^{2} + z^{2})(p + q + r) \)
C.
\( (y^{2} + z^{2} - x^{2})(r - q - p) \)
D.
\( (x^{2} + y^{2} - z^{2})(p - q - r) \)
😑 View Answer Error
ANSWER :
A. \( (x^{2} + y^{2} + z^{2})(p + q + r) \)
38.
What is \( x^{2}(px^{2} + qy^{2} + rz^{2}) + qr - p^{2} \) equal to?
A.
0
B.
1
C.
\( p + q + r \)
D.
\( x^{2} + y^{2} + z^{2} \)
😑 View Answer Error
ANSWER :
A. 0
39.
Consider the following for the next two (02) items that follow:
A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let \( 2\theta \) be the vertical angle of the cone.

What is the height of the cone?
A.
\( \frac{2r^{2}}{R - r} \)
B.
\( \frac{2R^{2}}{R - r} \)
C.
\( \frac{2(r^{2} + R^{2})}{R - r} \)
D.
\( \frac{r^{2} + R^{2}}{R - r} \)
😑 View Answer Error
ANSWER :
B. \( \frac{2R^{2}}{R - r} \)
40.
What is the radius of the base of the cone?
A.
\( \frac{2r^{2}\tan \theta}{R - r} \)
B.
\( \frac{2R^{2}\tan \theta}{R - r} \)
C.
\( \frac{2(r^{2} + R^{2})\tan \theta}{R - r} \)
D.
\( \frac{(r^{2} + R^{2})\tan \theta}{R - r} \)
😑 View Answer Error
ANSWER :
B. \( \frac{2R^{2}\tan \theta}{R - r} \)
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