Question:
Let the number $2, b, c$ be in an A.P. and $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & b & c \\ 4 & b^{2} & c^{2}\end{array}\right] .$ If $\operatorname{det}(A) \in[2,16]$, then $c$
lies in the interval :
Correct Option: , 4
Solution:
put $\mathrm{b}=\frac{2+\mathrm{c}}{2}$ in determinant of $\mathrm{A}$
$|A|=\frac{c^{3}-6 c^{2}+12 c-8}{4} \in[2,16]$
$\Rightarrow(\mathrm{c}-2)^{3} \in[8,64]$
$\Rightarrow \mathrm{c} \in[4,6]$