Question

If $xϵR$ and $nϵI$, then the determinant $\Delta =\left|\begin{array}{ccc}\sin \left(n\pi \right)& \sin x-\cos x& \log \left(\tan x\right)\\ \cos x-\sin x& \cos \left[\right(2n+1\left)\frac{\pi }{2}\right]& \log \left(\cot x\right)\\ \log \left(\cot x\right)& \log \left(\tan x\right)& \tan \left(n\pi \right)\end{array}\right|$

• A. $\sin (\frac{\pi }{4}-x)$
• B. $\tan (\frac{\pi }{4}-x)$
• C. $\log \left(\tan x\right)-\log \left(\cot x\right)$
• D. 0

Answer 1

The correct option is D

0

We can write $\Delta$ as

$\Delta =\left|\begin{array}{ccc}0& \sin x-\cos x& \log \left(\tan x\right)\\ -(\sin x-\cos x)& 0& -\log \left(\tan x\right)\\ -\log \left(\tan x\right)& \log \left(\tan x\right)& 0\end{array}\right|$

$=(-1{)}^3\left|\begin{array}{ccc}0& -(\sin x-\cos x)& -\log \left(\tan x\right)\\ \sin x-\cos x& 0& \log \left(\tan x\right)\\ \log \left(\tan x\right)& -\log \left(\tan x\right)& 0\end{array}\right|$ [taking $-1$ common from $R_1,R_2$ and $R_3$]

$=-\Delta$

$\Rightarrow 2\Delta =0\Rightarrow \Delta =0$