Question

The number of distinct real roots of
$\Delta =\left|\begin{array}{ccc}\tan x& \cot x& \cot x\\ \cot x& \tan x& \cot x\\ \cot x& \cot x& \tan x\end{array}\right|$
in the interval $-\pi /4\le x\le \pi /4$ is

• A. 0
• B. 2
• C. 1
• D. 3

Answer 1

The correct option is B 2

$\Delta =\left|\begin{array}{ccc}\tan x& \cot x& \cot x\\ \cot x& \tan x& \cot x\\ \cot x& \cot x& \tan x\end{array}\right|$

Applying $C_1\to C_1+C_2+C_3$

$\Delta =\left|\begin{array}{ccc}\tan x+2\cot x& \cot x& \cot x\\ \tan x+2\cot x& \tan x& \cot x\\ \tan x+2\cot x& \cot x& \tan x\end{array}\right|=(\tan x+2\cot x)\left|\begin{array}{ccc}1& \cot x& \cot x\\ 1& \tan x& \cot x\\ 1& \cot x& \tan x\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Delta =(\tan x+2\cot x)\left|\begin{array}{ccc}1& \cot x& \cot x\\ 0& \tan x-\cot x& 0\\ 0& 0& \tan x-\cot x\end{array}\right|$
$=(\tan x+2\cot x){(\tan x-\cot x)}^2$

For $\Delta =0$
Gives $\tan x+2\cot x=0$ or $\tan x-\cot x=0$
${\tan }^{3}x=-2$ or ${\tan }^{2}x=1$