Question

Let $\theta$ be the acute angle between the curves $y = x ^{ x } \ln x$ and $y =\frac{2^{ x }-2}{\ln 2}$ at their point of intersection on the line $y=0$. The value of $\tan \theta$ is equal to

• A. $\frac{1}{2}$
• B. 2
• C. $\frac{1}{3}$
• D. 3

Answer 1

Answer: (C)

$ y = x ^{ x } \ln x ; y =\frac{2^{ x }-2}{\ln 2}$
At $x =1, m _1=\frac{ dy }{ dx }= x ^{ x } \cdot \frac{1}{ x }+ x ^{ x }(\ln x +1) \ln x =1$
At $x =1, m _2=\frac{ dy }{ dx }=2^{ x } \frac{\ln 2}{\ln 2}=2$
$\therefore \tan \theta=\left|\frac{2-1}{1+2}\right|=\frac{1}{3} $