Question

Let the numbers $2,b,c$ 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 $\text{det}\left(A\right)\in [2,16]$, then $c$ lies in the interval :

• A. $[3,2+2^{3/4}]$
• B. $[4,6]$
• C. $[2,3]$
• D. $[2+2^{3/4},4]$

Answer 1

The correct option is B $[4,6]$

$|A|=\left|\begin{array}{ccc}1& 1& 1\\ 2& b& c\\ 4& b^2& c^2\end{array}\right|$

$C_2\to C_2-C_1,C_3\to C_3-C_1$
$|A|=\left|\begin{array}{ccc}1& 0& 0\\ 2& b-2& c-2\\ 4& b^2-4& c^2-4\end{array}\right|$

$\Rightarrow |A|=(b-2)(c-b)(c-2)$

$2,b,c$ are in A.P.
Let the common difference be $k$.
Then $|A|=k\cdot k\cdot 2k=2k^3$
$\Rightarrow 2\le 2k^3\le 16$
$\Rightarrow 1\le k\le 2$
$\Rightarrow 1\le \frac{c-2}{2}\le 2$
$\Rightarrow 4\le c\le 6$