If the mean and variance of eight numbers $3,7,9$, $12,13,20, x$ and $y$ be 10 and 25 respectively,

Question

If the mean and variance of eight numbers $3,7,9$, $12,13,20, x$ and $y$ be 10 and 25 respectively, then $x \cdot y$ is equal to

Professor PC Thomas Classes · Answer

(54) : Mean $(\bar{x})$\ $=\frac{3+7+9+12+13+20+x+y}{8}=10$ $$\begin{equation*}\Rightarrow \quad x+y=16 	ag{i} \end{equation*}$$ Variance, $(\sigma)^{2}=\frac{\sum\left(x_{i}ight)^{2}}{8}-(\bar{x})^{2}=25$\ $9+49+81+144+169$\ $\Rightarrow \frac{+400+x^{2}+y^{2}}{8}-100=25$ $$\begin{equation*}\Rightarrow \quad x^{2}+y^{2}=148 	ag{ii} \end{equation*}$$ From (i), $(x+y)^{2}=(16)^{2}$\ $\Rightarrow \quad x^{2}+y^{2}+2 x y=256$\ $\Rightarrow \quad x y=54$\ (Using (ii))\