JEE 2023 — Mathematics PYQ

\begin{aligned}
 & \mathrm{Now,~mean~}(\overline{x})=9 \\
 & \Rightarrow\frac{x+y+52}{8}=9 \\
 & \Rightarrow x+y=20 \\
 & \mathrm{Also,variance}=9.25 \\
 & \Rightarrow\frac{\left(x-9\right)^{2}+\left(y-9\right)^{2}+54}{8}=9.25 \\
 & \Rightarrow x^{2}+y^{2}+81+81-2\times9(x+y)=20 \\
 & \Rightarrow x^{2}+y^{2}-18\times20=-142 \\
 & \Rightarrow x^{2}+y^{2}=218 \\
 & \Rightarrow x^{2}+(20-x)^{2}=218 \\
 & \Rightarrow x^{2}+400+x^{2}-40x=218 \\
 & \Rightarrow2x^{2}-40x+182=0 \\
 & \Rightarrow x=\frac{40\pm12}{4} \\
 & \Rightarrow x=13\mathrm{or}x=7\Rightarrow y=7\mathrm{or}y=13 \\
 & \operatorname{But}x>y \\
 & \therefore x=13\mathrm{and}y=7 \\
 & \mathrm{So},3x-2y=39-14=25
\end{aligned}