Question

The tangent to the curve $y=x^3+1$ at $(1,2)$ makes an angle $\theta$ with $y$-axis, then the value of tan $\theta$ is

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

Answer 1

Answer: (C)

Since, the tangent to the curve $y=x^3+1$ at point $(1,2)$ makes an angle $\theta$ to the $y$ -axis. Then, the tangent line makes an angle from $x$ -axis is $90^{\circ }-\theta$.
Now, $Y=x^3+1\Rightarrow \frac{dy}{dx}=3x^2$
At point $(1,2)$,
$\frac{dy}{dx}=3(1{)}^2=3$
$\therefore \tan \left(90^{\circ }-\theta \right)=3$
$\Rightarrow -\cot \theta =3$
$\Rightarrow \cot \theta =-3$
$\Rightarrow \tan \theta =-\frac{1}{3}$