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

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NDA & NA Exam (I)- 2022 - Mathematics

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

1.
If
\( \Delta_{1} = \begin{vmatrix} 1 & p & q \\ 1 & q & r \\ 1 & r & p \end{vmatrix} \) and \( \Delta_{2} = \begin{vmatrix} 1 & 1 & 1 \\ q & r & p \\ r & p & q \end{vmatrix} \)
where \( p \neq q \neq r \), then \( \Delta_{1} + \Delta_{2} \) is
A.
0
B.
always positive
C.
always negative
D.
positive if \( p, q, r \) are positive else negative
😑 View Answer Error
ANSWER :
C. always negative
2.
If \( (a - b)(b - c)(c - a) = 2 \) and \( abc = 6 \), then what is the value of
\( \begin{vmatrix} a & b & c \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{vmatrix} \)?
A.
3
B.
12
C.
14
D.
15
😑 View Answer Error
ANSWER :
B. 12
3.
Under which of the following conditions does the determinant
\( \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \)
vanish?
1. \( a + b + c = 0 \)
2. \( a^{3} + b^{3} + c^{3} = 3abc \)
3. \( a^{2} + b^{2} + c^{2} - ab - bc - ca = 0 \)
Select the correct answer using the code given below:
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
😑 View Answer Error
ANSWER :
D. 1, 2 and 3
4.
Consider the following in respect of the matrices:
\( A = \begin{bmatrix} m & n \end{bmatrix}, B = \begin{bmatrix} -n & -m \end{bmatrix} \) and \( C = \begin{bmatrix} m \\ -m \end{bmatrix} \)
1. \( CA = CB \)
2. \( AC = BC \)
3. \( C(A + B) = CA + CB \)
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
2 and 3
D.
1 and 2
😑 View Answer Error
ANSWER :
C. 2 and 3
5.
If
\( A = \begin{bmatrix} 2 \sin \theta & \cos \theta & 0 \\ -2 \cos \theta & \sin \theta & 0 \\ -1 & 1 & 1 \end{bmatrix} \)
then what is \( A(adj A) \) equal to, where \( I \) is the identity matrix?
A.
Null matrix
B.
\( -I \)
C.
\( I \)
D.
\( 2I \)
😑 View Answer Error
ANSWER :
D. \( 2I \)
6.
For what value of \( k \) is the matrix
\( \begin{bmatrix} 2 \cos 2\theta & 2 \cos 2\theta & 6 \\ 1 - 2 \sin^{2} \theta & 2 \cos^{2} \theta - 1 & 3 \\ k & 2k & 1 \end{bmatrix} \)
singular?
A.
0 only
B.
1 only
C.
2 only
D.
Any real value
😑 View Answer Error
ANSWER :
D. Any real value
7.
Let \( A \) be a non-singular matrix and \( B = adj A \). Which of the following statements is/are correct?
1. \( AB = BA \)
2. \( AB \) is a scalar matrix
3. \( AB \) can be a null matrix
Select the correct answer using the code given below:
A.
1 only
B.
1 and 2 only
C.
2 only
D.
1, 2 and 3
😑 View Answer Error
ANSWER :
B. 1 and 2 only
8.
Consider the following statements in respect of square matrices \( A \) and \( B \) of same order:
1. If \( AB \) is a null matrix, then at least one of \( A \) and \( B \) is a null matrix.
2. If \( AB \) is an identity matrix, then \( BA = AB \).
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
B. 2 only
9.
If \( A \) is the identity matrix of order 3 and \( B \) is its transpose, then what is the value of the determinant of the matrix \( C = A + B \)?
A.
1
B.
2
C.
4
D.
8
😑 View Answer Error
ANSWER :
D. 8
10.
Let \( A \) and \( B \) be non-singular matrices of the same order such that \( AB = A \) and \( BA = B \). Which of the following statements is/are correct?
1. \( A^{2} = A \)
2. \( AB^{2} = A^{2}B \)
Select the correct answer using the code given below:
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
C. Both 1 and 2
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