Question

The arrangment of the slopes of the normals to the curve $y=e^{\log \left(cosx\right)}$ in the ascending order at the points given below.
$A)x=\frac{\pi }{6},B)x=\frac{7\pi }{4},C)x=\frac{11\pi }{6},D)x=\frac{\pi }{3}$

• A. $C,B,D,A$
• B. $B,C,A,D$
• C. $A,D,C,B$
• D. $D,A,C,B$

Answer 1

Answer: (A)

$C,B,D,A$

We have, $y=e^{\log \cos x}=\left(\cos x{)}^{\log e}\right[\because a^{{\log }_{b}c}=c^{{\log }_{b}a}]=\cos x$
$\Rightarrow \frac{dy}{dx}=-\sin x=m$(say)
Thus slope of normal to the given curve at any point is,
$m^{\prime }=-\frac{1}{m}=\frac{1}{\sin x}$
Now, $m_{\frac{\pi }{6}}^{\prime }=2,m_{\frac{7\pi }{4}}^{\prime }=-\sqrt{2},m_{\frac{\pi }{6}}^{\prime }=-2,m_{\frac{\pi }{6}}^{\prime }=\frac{\sqrt{3}}{2}$
Clearly ascending order is, $C,B,D,A$
Note: Given function is not defined where $\cos x<0$