Mathematics - Determinant Question with Solution | TestHub

$$\begin{gathered} \left|\begin{array}{ccc}1& 1& 1\\ 2& b& c\\ 4& b^2& c^2\end{array}\right|\stackrel{C_2\to C_2-C_1, C_3\to C_3-C_1}{\to }\left|\begin{array}{ccc}1& 0& 0\\ 2& b-2& c-2\\ 4& b^2-4& c^2-4\end{array}\right| \\ =\left(b-2\right)\left(c-2\right)\left(c-b\right) \end{gathered}$$
Let common difference of$AP=d$
$\therefore b=2+d, c=2+2d$
$\left|A\right|=d\times 2d\times d=2d^3\in \left[2, 16\right]$
$\Rightarrow d^3\in \left[1, 8\right]\Rightarrow d\in \left[1, 2\right]\Rightarrow 2d\in \left[2, 4\right]$
$\Rightarrow 2+2d\in \left[4, 6\right]$
$\therefore c\in \left[4, 6\right]$