Question

2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\\ 2&b&c\\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is

• A. [6, 8]
• B. [0, 4]
• C. [8, 12]
• D. [4, 6]

Answer 1

Answer: (D)

[4, 6]

Answer 2

$\begin{vmatrix}1& 2 & 4 \\\ 1&b&b^2 \\\ 1 & c &c^{2}\end{vmatrix} = (2 - b) (b - c) (c - 2)$
Given that 2, b, c are in AP.
So, $2b = c + 2$
$2 \le\left(2 - \frac{c+2}{2}\right)\left(\frac{c+2}{2} -c\right) \left(c-2\right) \le 16$
$2 \le\frac{\left(c-2\right)^{3}}{4} \le16$
$2\le c-2\le4$
$4 \le c \le 6$

So, the correct option is (D): $[4, 6]$.