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, $Y=x^{3}+1 \Rightarrow \frac{d y}{d x}=3 x^{2}$
At point $(1,2)$,
$\frac{d y}{d x} =3(1)^{2}= 3$
$\therefore \tan \left(90^{\circ}-\theta\right) =3$
$\Rightarrow -\cot \theta=3$
$\Rightarrow \cot \theta=-3$
$\Rightarrow \tan \theta=-\frac{1}{3}$