JEE Main - 4th April 2015

Let the system of linear equations

$x+y+kz=2$

$2x+3y-z=1$

$3x+4y+2z=k$

have infinitely many solutions. Then the system

$\left(k+1\right) x+\left(2k-1\right) y=7$

$\left(2k+1\right)x+\left|k+5\right|y=10$ has :

For $\alpha ,\beta \in \mathrm{ℝ}$, suppose the system of linear equations
$x-y+z=5$ 
$2x+2y+\alpha z=8$
$3x-y+4z=\beta$

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of 

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

$\lambda x+y+z=1$

$x+\lambda y+z=1$

$x+y+\lambda z=1$
is inconsistent, then $\sum _{\lambda \in S}\left({\left|\lambda \right|}^2+\left|\lambda \right|\right)$ is equal to

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

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

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right]$. Then the sum of the diagonal elements of the matrix ${\left(A+I\right)}^{11}$ is equal to:

Let $A$ be a $n\times n$ matrix such that $\left|A\right|=2$. If the determinant of the matrix $Adj\left(2.Adj\left(2 {\mathrm{A}}^{-1}\right)\right)$ is $2^{84}$, then $n$ is equal to _____ .

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