NIMCET 2018 — Mathematics PYQ

Calculation:

Let y=f(x) is the equation of the curve such that y=0 for x=0.
A rectangle is made with length and breadth as x and y respectively.
This rectangle divides the curve y=f(x) in two halves, area of one half is double the area of other half.
Area of the rectangle is xy. So area of the one half is $xy-\underset{0}{x}ydx$
Area of another half is $\underset{0}{x}ydx$

Following the condition gven in the question.

$$\begin{array}{rl}& amp;xy-\underset{0}{\overset{x}{\int }}ydx=2\underset{0}{\overset{x}{\int }}ydx\\ & amp;xy=3\underset{0}{\overset{x}{\int }}ydx\end{array}$$

Now, differentiating the equation with respect to x as follows:

$y+x\frac{dy}{dx}=3y$

$\frac{dy}{dx}=\frac{2y}{x}$

On further calculation,

$\frac{1}{2}\frac{dy}{y}=\frac{dx}{x}$

On integrating both sides, we get,

$\frac{1}{2}\ln y=\ln x+\ln c$

$\ln y=2\ln x+\ln c$

Or, $y=c{x}^2$, here $c$ is some arbitrary constant.

This equation represents the family of parabolas passing through origin.