Matrices & Determinants · NDA previous year questions

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

If $M^T$ is the transpose of a $2\times 2$ matrix $M$, then which of the following is/are correct ? I. $|M+M^T|=|M|+|M^T|$ if $M$ is symmetric. II. $|M+M^T|=0$ if $M$ is anti-symmetric. Select the answer using the code given below :

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

If $M_k=\left[\begin{array}{cc}k& k-1\\ k-1& k\end{array}\right]$ where $k$ is a natural number, then what is $|M_1|+|M_2|+|M_3|+\dots +|M_{50}|$ equal to ?

1. A.50
2. B.1250
3. C.2500
4. D.5000

Consider the following statements : I. If $n\times n$ ($n>1$) matrix is symmetric, then its inverse is also a symmetric matrix. II. If $n\times n$ ($n>1$) matrix is singular, then its adjoint is also a singular matrix. Which of the statements given above is/are correct ?

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

Let $p=(x+y+z)$ and $q=xyz$. If $\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is positive, then which one of the following is correct ?

1. A.$q>p$
2. B.$q+1>p$
3. C.$q+2>p$
4. D.$q+2\ge p$

1. A.8
2. B.64
3. C.256
4. D.512

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

1. A.$-1$
2. B.0
3. C.$\frac{1}{2}$
4. D.1

1. A.$-1$
2. B.0
3. C.1
4. D.$q$

What is the value of the determinant of the inverse of the matrix $\left[\begin{array}{cc}-4& -5\\ 2& 2\end{array}\right]$?

1. A.1/2
2. B.1
3. C.2
4. D.4

In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

1. A.45
2. B.30
3. C.25
4. D.15

The system of equations 15y - 10x + 50 = 0, 2x - 3y - 5 = 0

1. A.has a unique solution
2. B.has infinitely many solutions
3. C.is inconsistent
4. D.is consistent and has exactly two solutions

If $(\frac{1-i}{1+i}{)}^{2m}(\frac{1+i}{1-i}{)}^{2n}=1$ where $i=\sqrt{-1}$ then what is the smallest positive value of (m - n)?

1. A.1
2. B.2
3. C.4
4. D.8

In obtaining the solution of the system of equations x + y + z = 7, x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer's rule, the value of y is obtained by dividing D by $D_2$ where $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 4\end{array}\right|$ What is the value of the determinant $D_2$ ?

1. A.-13
2. B.-3
3. C.3
4. D.13

Consider the following in respect of non-singular matrices A and B: I. $(AB{)}^{-1}=A^{-1}{B}^{-1}$ II. $(BA)(AB{)}^{-1}=I$ where I is the identity matrix III. $(AB{)}^T=A^T{B}^T$ How many of the above are correct?

1. A.None
2. B.One
3. C.Two
4. D.All three

1. A.$\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ l& m& n\end{array}\right|$
2. B.$\left|\begin{array}{ccc}l& m& n\\ a& b& c\\ p& q& r\end{array}\right|$
3. C.$\left|\begin{array}{ccc}p& q& r\\ a& b& c\\ l& m& n\end{array}\right|$
4. D.$\left|\begin{array}{ccc}a& p& l\\ b& q& m\end{array}\right|$

For the following two (02) items: Let $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$ What is the value of the determinant of the matrix $A^4$

1. A.0
2. B.1
3. C.$\cos 4\theta -\sin 4\theta$
4. D.${\cos }^{2}4\theta -{\sin }^{2}4\theta$

For the following two (02) items: Let $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$ What is $[adjA{]}^{-1}$ equal to?

1. A.-A
2. B.$-A^T$
3. C.A
4. D.$A^T$

Consider the following statements: Statement-I: If X is an nn matrix, then $\det (mX)=m^n\det (X)$, where m is a scalar. Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then $\det (Y)=m\det (X)$. Which one of the following is correct in respect of the above statements?

1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-1
3. C.Statement-I is correct but Statement-II is not correct
4. D.Statement-I is not correct but Statement-II is correct

Consider the following statements about the matrix $M=\left[\begin{array}{ccc}71& 23& 48\\ 57& 28& 29\\ 65& 17& 48\end{array}\right]$ Statement-I: The inverse of M does not exist. Statement-II: M is non-singular. Which one of the following is correct in respect of the above statements?

1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
3. C.Statement-I is correct but Statement-II is not correct
4. D.Statement-I is not correct but Statement-II is correct

What is ${\cot }^{-1}9+{\mathrm{cosec}}^{-1}\left(\frac{\sqrt{41}}{4}\right)$ equal to?

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{2}$
4. D.$\pi$

How many values of $\theta$, where $-\pi <\theta <\pi$ satisfy both the equations $\cot \theta =-\sqrt{3}$ and $\csc \theta =-2$ simultaneously?

1. A.4
2. B.2
3. C.1
4. D.None

If $x+\frac{1}{x}=2\cos \theta$ then what is $x^3+\frac{1}{x^3}$ equal to?

1. A.${\cos }^3\theta$
2. B.$\cos 3\theta$
3. C.$2\cos 3\theta$
4. D.$3\cos 3\theta$

If $0\le x\le \frac{\pi }{2}$ then what is the number of values of x satisfying the equation $\tan x+\sec x=2\cos x$

1. A.0
2. B.1
3. C.2
4. D.3

What is the value of $\tan \left[\frac{1}{2}{\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)\right]$?

1. A.$2-\sqrt{3}$
2. B.$2+\sqrt{3}$
3. C.$\sqrt{3}-1$
4. D.$\sqrt{3}+1$