Question

On the curve $x^3=12y$, the abscissa changes at a faster rate than the ordinate. Then $x$ belongs to the interval :

• A. $(-2,2)$
• B. $(-3,3)$
• C. $(0,2)$
• D. None of these

Answer 1

The correct option is A

$(-2,2)$

Explanation for the correct option:

Finding the interval in which x belongs:

We have,

$$\begin{gathered} \left|dx\right|=\left|dy\right| \\ \Rightarrow \left|\frac{dx}{dy}>1\right| \end{gathered}$$

Also,

$$\begin{gathered} \Rightarrow x^3=12y \\ \Rightarrow 3x^2\frac{dx}{dy}=12 \\ \Rightarrow \frac{dx}{dy}=\frac{12}{3x^2} \\ \Rightarrow \frac{dx}{dy}=\frac{4}{x^2} \end{gathered}$$

As $\left|\frac{dx}{dy}\right|>1$

So,

$$\begin{gathered} \Rightarrow \left|\frac{4}{x^2}\right|>1 \\ \Rightarrow \left|x^2\right|<4 \\ \Rightarrow \left|x\right|\le 2 \\ \Rightarrow -2<x<2,x\ne 0 \end{gathered}$$

Therefore, the correct answer is option (A).