JEE Main - 25 July 2021 Shift 1

The area of the region $A=\left\{\left(x,y\right):\left|\cos x-\sin x\right|\le y\le \sin x,0\le x\le \frac{\pi }{2}\right\}$

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

Let $\alpha$ be the area of the larger region bounded by the curve $y^2=8x$ and the lines $y=x$ and $x=2$, which lies in the first quadrant. Then the value of $3\alpha$ is equal to

Let $q$ be the maximum integral value of $p$ in $\left[0,10\right]$ for which the roots of the equation $x^2-px+\frac{5}{4}p=0$ are rational. Then the area of the region $\left\{\left(x,y\right):0\le y\le (x-q{)}^2,0\le x\le q\right\}$ is

Let $A$ be the area of the region $\left\{(x,y):y\ge x^2, y\ge (1-x{)}^2, y\le 2x(1-x)\right\}$. Then $540A$ is equal to

Let for $x\in R$, $f\left(x\right)=\frac{x+\left|x\right|}{2}$ and $g\left(x\right)=\left\{\begin{array}{l}x,\\ x^2,\end{array}\begin{array}{c}x<0\\ x\ge 0\end{array}\right.$. Then area bounded by the curve $y=\left(fog\right)\left(x\right)$ and the lines $y=0,2y-x=15$ is equal to _____sq. units.

Let the area of the region $\left\{\left(x,y\right):\left|2x-1\right|\le y\le \left|x^2-x\right|,0\le x\le 1\right\}$ be $A$. Then ${\left(6A+11\right)}^2$ is equal to _____ .

Let $A$ be the area bounded by the curve $y=x\left|x-3\right|$, the $x$-axis and the ordinates $x=-1$ and $x=2$. Then $12 A$ is equal to _____ .