Question

Cosine of the angle of intersection of curves $y=3^{x-1}\ln x$ and $y=x^x-1$ is

• A. $1$
• B. $1/2$
• C. $0$
• D. $1/3$

Answer 1

Answer: (A)

$1$

The curves intersect at the point $(1,0)$

Let $m_1$ and $m_2$ be the slope of tangent to the curves at the point of intersection

$y=3^{x-1}logx$

$\frac{dy}{dx}=3^{x-1}log3\left(logx\right)+\left(3^{x-1}\right)\frac{1}{x}$

$\therefore m_1=1$

$y=x^x-1$

$\frac{dy}{dx}=x^x+x^{x}logx$

$\therefore m_2=1$

The tangents are parallel. So angle between them is $\theta =0$

$\therefore cos\theta =cos0=1$

So, the answer is option A