NDA & NA Exam (II)- 2021 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2021 - Mathematics

NDA & NA Exam (II)- 2021 - Mathematics Defence Questions

41.
Let \( ABC \) be a triangle. If
\( \cos 2A + \cos 2B + \cos 2C = -1 \)
then which one of the following is correct?
A.
\( \sin A \sin B \sin C = 0 \)
B.
\( \sin A \sin B \cos C = 0 \)
C.
\( \cos A \sin B \sin C = 0 \)
D.
\( \cos A \cos B \cos C = 0 \)
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ANSWER :
D. \( \cos A \cos B \cos C = 0 \)
42.
What is the value of the following determinant?
\( \begin{vmatrix} \cos C & \tan A & 0 \\ \sin B & 0 & -\tan A \\ 0 & \sin B & \cos C \end{vmatrix} \)
A.
−1
B.
0
C.
\( 2 \tan A \sin B \sin C \)
D.
\( -2 \tan A \sin B \sin C \)
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ANSWER :
B. 0
43.
Suppose set \( A \) consists of first 250 natural numbers that are multiples of 3 and set \( B \) consists of first 200 even natural numbers. How many elements does \( A \cup B \) have?
A.
324
B.
364
C.
384
D.
400
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ANSWER :
C. 384
44.
Let \( S_{k} \) denote the sum of first \( k \) terms of an AP. What is \( \frac{S_{30}}{S_{20} - S_{10}} \) equal to?
A.
1
B.
2
C.
3
D.
4
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ANSWER :
C. 3
45.
If the roots of the equation
\( 4x^{2} - (5k + 1)x + 5k = 0 \)
differ by unity, then which one of the following is a possible value of \( k \)?
A.
−3
B.
−1
C.
\( -\frac{1}{5} \)
D.
\( -\frac{3}{5} \)
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ANSWER :
C. \( -\frac{1}{5} \)
46.
Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?
A.
240
B.
180
C.
120
D.
60
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ANSWER :
A. 240
47.
How many distinct matrices exist with all four entries taken from \( \{1, 2\} \)?
A.
8
B.
24
C.
32
D.
48
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ANSWER :
D. 48
48.
If \( i = \sqrt{-1} \), then how many values does \( i^{-2n} \) have for different \( n \in \mathbb{Z} \)?
A.
One
B.
Two
C.
Four
D.
Infinite
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ANSWER :
B. Two
49.
If
\( x = \frac{a}{b-c}, y = \frac{b}{c-a}, z = \frac{c}{a-b} \)
then what is the value of the following?
\( \begin{vmatrix} 1 & -x & x \\ 1 & 1 & -y \\ 1 & z & 1 \end{vmatrix} \)
A.
0
B.
1
C.
\( abc \)
D.
\( ab + bc + ca \)
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ANSWER :
A. 0
50.
Consider the following in respect of the matrix
\( A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} \):
1. Inverse of \( A \) does not exist
2. \( A^{T} = A \)
3. \( 3A = A^{2} \)
Which of the above are correct?
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
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ANSWER :
D. 1, 2 and 3
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