Let the number 2,b,c be in an A.P. and

Question:

Let the number 2,b,c be in an A.P. and

$\mathrm{A}=\left[\begin{array}{ccc}1 & 1 & 1 \\ 2 & \mathrm{~b} & \mathrm{c} \\ 4 & \mathrm{~b}^{2} & \mathrm{c}^{2}\end{array}\right] .$ If $\operatorname{det}(\mathrm{A}) \in[2,16]$, then $\mathrm{c}$

lies in the interval :

  1. $[2,3)$

  2. $\left(2+2^{3 / 4}, 4\right)$

  3. $\left[3,2+2^{3 / 4}\right]$

  4. $[4,6]$


Correct Option: , 4

Solution:

put $\mathrm{b}=\frac{2+\mathrm{c}}{2}$ in determinant of $\mathrm{A}$

$|A|=\frac{c^{3}-6 c^{2}+12 c-8}{4} \in[2,16]$

$\Rightarrow(\mathrm{c}-2)^{3} \in[8,64]$

$\Rightarrow \mathrm{c} \in[4,6]$

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