Question

The angle at which the curve $y=k{e}^{kx}$ intersects the y - axis is

• A. ${\tan }^{-1}{k}^2$
• B. ${\cot }^{-1}\left(k^2\right)$
• C. ${\sin }^{-1\left(\frac{1}{\sqrt{1+k^4}}\right)}$
• D. $\sec -1\left(\sqrt{1+k^4}\right)$

Answer 1

Answer: (B), (C)

The correct options are
B ${\cot }^{-1}\left(k^2\right)$
C ${\sin }^{-1\left(\frac{1}{\sqrt{1+k^4}}\right)}$

$\frac{dy}{dx}=K^2{e}^{kx}$

$\frac{dy}{dx}{|}_{x=0}=K^2=\tan \theta$

(where $\varphi$ is angle made by x-axis)

Let $\varphi$ be the gngle made by y-axis

$\tan \theta =\tan (\frac{3\pi }{2}-\varphi )=\cot \varphi$

$\cot \varphi =K^2$

$\varphi ={\cot }^{-1}\left(K^2\right)$

$\Rightarrow \varphi ={\sin }^{-1}\left(\frac{1}{\sqrt{1+K^4}}\right)$