Mathematics - Determinant Question with Solution | TestHub

Given, $A=\left[\begin{array}{ccc}[x+1]& [x+2]& [x+3]\\ \left[x\right]& [x+3]& [x+3]\\ \left[x\right]& [x+2]& [x+4]\end{array}\right]$

$A=\left[\begin{array}{ccc}\left[x\right]+1& \left[x\right]+2& \left[x\right]+3\\ \left[x\right]& \left[x\right]+3& \left[x\right]+3\\ \left[x\right]& \left[x\right]+2& \left[x\right]+4\end{array}\right]$

$R_1\to R_1-R_3,R_2\to R_2-R_3$

$A=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& -1\\ \left[x\right]& \left[x\right]+2& \left[x\right]+4\end{array}\right]$

$\det \left(A\right)=1\left(\right[x]+4+[x]+2)-1(-[x\left]\right)$

$=3\left[x\right]+6$

Given, $\mathrm{de}t\left(A\right)=192$

$192=3\left[x\right]+6$

$3\left[x\right]=186$

$\left[x\right]=62$

$x\in [62,63)$