Embibe Experts Solutions for Chapter: Area Under Curves, Exercise 1: JEE Advanced Paper 1 - 2013

Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$ . Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0, 0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

Area of the region $\{\left(x,y\right)\in R^2: y\ge \sqrt{\left|x+3\right|},$ $5y\le x+9\le 15\}$ is equal to

The area of the region
$\left\{(x,y):0\le x\le \frac{9}{4}, 0\le y\le 1, x\ge 3y, x+y\ge 2\right\}$ is

Let the functions $f:R\to R$ and $g:R\to R$ be defined by $f\left(x\right)=e^{x-1}-e^{-\left|x-1\right|}$ and $g\left(x\right)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area (in sq. units) of the region in the first quadrant bounded by the curves $y=f\left(x\right), y=g\left(x\right)$ and $x=0$ is

The area of the region $\left\{\left(x,y\right):xy\le 8, 1\le y\le x^2\right\}$ is

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P\left(\mathrm{0,\; 0}\right),Q\left(\mathrm{1,\; 1}\right)$ and $R\left(\mathrm{2,\; 0}\right).$ From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n, n>1$. If the area of the region taken away by the farmer $F_2$ is exactly $\mathrm{30\%}$ of the area of $∆PQR,$ then the value of $n$ is

The area enclosed by the curves $y=\sin x+\cos x$ and $y=\left|\cos \mathit{x}-\sin \mathit{x}\right|$ over the interval $\left[0,\frac{\pi }{2}\right]$ is