NDA & NA Exam (II)- 2022 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2022 - Mathematics

NDA & NA Exam (II)- 2022 - Mathematics Defence Questions

31.
Directions (Qn. 31-33): Consider the following for the next three (03) items that follow:
Let \( f(x) \) be a function satisfying \( f(x + y) = f(x) f(y) \) for all \( x, y \in \mathbb{N} \) such that \( f(1) = 2 \).

If \( \sum_{x=2}^{n} f(x) = 2044 \), then what is the value of \( n \)?
A.
8
B.
9
C.
10
D.
11
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ANSWER :
C. 10
32.
What is \( \sum_{x=1}^{5} f(2x - 1) \) equal to?
A.
341
B.
682
C.
1023
D.
1364
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ANSWER :
B. 682
33.
What is \( \sum_{x=1}^{6} 2^{x} f(x) \) equal to?
A.
1365
B.
2730
C.
4024
D.
5460
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ANSWER :
D. 5460
34.
Directions (Qn. 34-36): Consider the following for the next three (03) items that follow:
A university awarded medals in basketball, football and volleyball. Only \( x \) students \( (x < 6) \) got medal in all the three sports and the medals went to a total of \( 15x \) students. It awarded \( 5x \) medals in basketball, \( (4x + 15) \) medals in football and \( (x + 25) \) medals in volleyball.

How many received medals in exactly two of the three sports?
A.
\( 30 - 4x \)
B.
\( 35 - 7x \)
C.
\( 40 - 7x \)
D.
\( 45 - 5x \)
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ANSWER :
C. \( 40 - 7x \)
35.
How many received medals in at least two of three sports?
A.
\( 30 - 6x \)
B.
\( 35 - 6x \)
C.
\( 40 - 5x \)
D.
\( 40 - 6x \)
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ANSWER :
D. \( 40 - 6x \)
36.
How many received medals in exactly one of three sports?
A.
\( 21x - 40 \)
B.
\( 21x - 35 \)
C.
\( 20x - 35 \)
D.
\( 20x - 25 \)
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ANSWER :
A. \( 21x - 40 \)
37.
Directions (Qn. 37-39): Consider the following for the next three (03) items that follow:
Let
\( A = \begin{bmatrix} 0 & \sin^{2} \theta & \cos^{2} \theta \\ \cos^{2} \theta & 0 & \sin^{2} \theta \\ \sin^{2} \theta & \cos^{2} \theta & 0 \end{bmatrix} \)
and \( A = P + Q \) where \( P \) is symmetric matrix and \( Q \) is skew-symmetric matrix.

What is \( P \) equal to?
A.
\( \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
B.
\( \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} \)
C.
\( \cos 2\theta \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix} \)
D.
\( \cos 2\theta \begin{bmatrix} 0 & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
38.
What is \( Q \) equal to?
A.
\( \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
B.
\( \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} \)
C.
\( \cos 2\theta \begin{bmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{bmatrix} \)
D.
\( \cos 2\theta \begin{bmatrix} 0 & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
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ANSWER :
D. \( \cos 2\theta \begin{bmatrix} 0 & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \)
39.
What is the minimum value of determinant of \( A \)?
A.
\( \frac{1}{4} \)
B.
\( \frac{1}{2} \)
C.
2
D.
1
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ANSWER :
A. \( \frac{1}{4} \)
40.
Directions (Qn. 40-42): Consider the following for the next three (03) items that follow:
ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle \( 45^{\circ} \) at the vertex B.

What is the height of the lamp post?
A.
6 m
B.
7 m
C.
8 m
D.
9 m
😑 View Answer Error
ANSWER :
C. 8 m
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