MCA Entrance Previous Year Question Bank

Let $M=\left[\begin{array}{cc}0& -\alpha \\ \alpha & 0\end{array}\right]$, where $\alpha$ is a non-zero real number and $N={\sum }_{k=1}^{49}{M}^{2k}$. If $(I-M^2)N=-2I$, then the positive integral value of $\alpha$ is ________.

Let $A=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$. If $B=I-{}^5{C}_1(\text{adj }A)+{}^5{C}_2(\text{adj }A{)}^2-\cdots -{}^5{C}_5(\text{adj }A{)}^5$, then the sum of all elements of the matrix $B$ is:

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\cdots +A^{10}$ is equal to:

If the system of linear equations

$2x+y-z=7$

$x-3y+2z=1$

$x+4y+\delta z=k$,

where $\delta ,k\in R$ has infinitely many solutions, then $\delta +k$ is equal to:

For the system of linear equations $\alpha x+y+z=1$, $x+\alpha y+z=1$, $x+y+\alpha z=\beta$, which one of the following statements is NOT correct?

If $A=\frac{1}{2}\left[\begin{array}{cc}1& \sqrt{3}\\ -\sqrt{3}& 1\end{array}\right]$ then:

Let $f(x)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& 1+{\cos }^{2}x& \sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\sin 2x\end{array}\right|,x\in \left[\frac{\pi }{6},\frac{\pi }{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then:

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations

is inconsistent, then ${\sum }_{\lambda \in S}(|\lambda {|}^2+|\lambda |)$ is equal to:

Let $A$ be an $n\times n$ matrix such that $|A|=2$. If the determinant of the matrix $\text{Adj}(2\cdot \text{Adj}(2A^{-1}))$ is $2^{84}$ then $n$ is equal to

Let $A=[a_{ij}],a_{ij}\in \mathbb{Z}\cap [0,4],1\le i,j\le 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p\in (2,13)$ is