JEE Main 2025 (Online) 4th April Morning ShiftQuestion 1
Consider the sets $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}+y^{2}=25\}$, $B=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x^{2}+9y^{2}=144\}$, $C=\{(x,y)\in\mathbb{Z}\times\mathbb{Z}:x^{2}+y^{2}\le4\}$ and $D=A\cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:
Sets, Relations and Function
Solution
$x^2+y^2=25$ and $x^2+9y^2=144\Rightarrow 8y^2=119$, $y^2=\tfrac{119}{8}$, $x^2=\tfrac{81}{8}$: $4$ points, so $|D|=4$. $C$: integer points with $x^2+y^2\le4$: origin $1$; $(\pm1,0),(0,\pm1)$: $4$; $(\pm1,\pm1)$: $4$; $(\pm2,0),(0,\pm2)$: $4$; total $13$. One-one functions: $13\cdot12\cdot11\cdot10=17160$. Option B.