Q 51 :

Let $A=\left\{\right(x,y)\in {\mathrm{ℝ}}^2:y\ge 0, 2x\le y\le \sqrt{ 4-{(x-1)}^2 }\}$

and $B=\left\{\right(x,y)\in \mathrm{ℝ}\times \mathrm{ℝ}:0\le y\le \min \{2x,\sqrt{ 4-{(x-1)}^2 }\left\}\right\}.$

Then the ratio of the area of $A$ to the area of $B$ is                             [2023]

• $\frac{\pi }{\pi +1}$

   

• $\frac{\pi -1}{\pi +1}$

   

• $\frac{\pi +1}{\pi -1}$

   

• $\frac{\pi }{\pi -1}$

   

Q 52 :

$\text{Let }\Delta \text{ be the area of the region }\left\{\right(x,y)\in {\mathrm{ℝ}}^2:x^2+y^2\le 21, y^2\le 4x, x\ge 1\}.$ $\text{Then }\frac{1}{2}(\mathrm{\Delta }-21{\sin }^{-1}\frac{{\displaystyle 2}}{{\displaystyle \sqrt{7}}})\text{ is equal to}$          [2023]

• $\sqrt{3}-\frac{2}{3}$

   

• $2\sqrt{3}-\frac{2}{3}$

   

• $2\sqrt{3}-\frac{1}{3}$

   

• $\sqrt{3}-\frac{4}{3}$

   

Q 53 :

The area of the region $A=\left\{\right(x,y):|\cos x-\sin x|\le y\le \sin x, 0\le x\le \frac{{\displaystyle \pi }}{{\displaystyle 2}}\}$ is               [2023]

• $\sqrt{5}-2\sqrt{2}+1$

   

• $\sqrt{5}+2\sqrt{2}-4.5$

   

• $1-\frac{3}{\sqrt{2}}+\frac{4}{\sqrt{5}}$

   

• $\frac{3}{\sqrt{5}}-\frac{3}{\sqrt{2}}+1$

   

Q 54 :

$\text{Let }q\text{ be the maximum integral value of p in [0,10] for which the roots of the equation }{x}^2-px+\frac{5}{4}p=0\text{ are rational.}$ Then the area of the region 
$\left\{\right(x,y):0\le y\le {(x-q)}^2, 0\le x\le q\}$ is            [2023]

• $164$

   

• $243$

   

• $\frac{125}{3}$

   

• $25$

   

Q 55 :

If the area of region $S=\left\{\right(x,y): 2y-y^2\le x^2\le 2y, x\ge y\}$ is equal to $\frac{n+2}{n+1}-\frac{\pi }{n-1},$ then the natural number $n$ is equal to _______ .    [2023]

Q 56 :

Let the area enclosed by the lines $x+y=2, y=0, x=0$ and the curve $f\left(x\right)=\min \{x^2+\frac{{\displaystyle 3}}{{\displaystyle 4}}, 1+[x\left]\right\},$ where $\left[x\right]$ denotes the greatest integer $\le x$, be $A$. Then the value of $12A$ is____________ .             [2023]

Q 57 :

Let $y=p\left(x\right)$ be the parabola passing through the points $(-1,0),(0,1)$ and $(1,0)$. If the area of the region

 $\left\{\right(x,y):{(x+1)}^2+{(y-1)}^2\le 1, y\le p(x\left)\right\}$ is $A$, then $12(\pi -4A)$ is equal to ______ .         [2023]

Q 58 :

If the area of the region $\left\{\right(x,y):|x^2-2|\le y\le x\}$ is A, then $6A+16\sqrt{2}$ is equal to _______ .           [2023]

 

Q 59 :

If A is the area in the first quadrant enclosed by the curve $C: 2x^2-y+1=0$, the tangent to C at the point (1, 3) and the line $x+y=1$, then the value of 60A is ______ .    [2023]

 

Q 60 :

If the area bounded by the curve $2y^2=3x,$ lines $x+y=3, y=0$ and outside the circle ${(x-3)}^2+y^2=2$ is A, then $4(\pi +4A)$ is equal to _____ .       [2023]