If the system of equations
$ 2 x+7 y+\lambda z=3 $
$ 3 x+2 y+5 z=4 $
$ x+\mu y+32 z=-1$
has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$
$38$
$39$
$34$
$15$
$\left| {\,\begin{array}{*{20}{c}}{{{({a^x} + {a^{ - x}})}^2}}&{{{({a^x} - {a^{ - x}})}^2}}&1\\{{{({b^x} + {b^{ - x}})}^2}}&{{{({b^x} - {b^{ - x}})}^2}}&1\\{{{({c^x} + {c^{ - x}})}^2}}&{{{({c^x} - {c^{ - x}})}^2}}&1\end{array}\,} \right| = $
If $|A|$ denotes the value of the determinant of the square matrix $A$ of order $3$ , then $ |-2A|=$
For how many diff erent values of $a$ does the following system have at least two distinct solutions?
$a x+y=0$
$x+(a+10) y=0$
For the system of linear equations
$2 x-y+3 z=5$
$3 x+2 y-z=7$
$4 x+5 y+\alpha z=\beta$
Which of the following is NOT correct ?