Question

If the mean and variance of eight numbers $3,$ $7,$ $9,$ $12,$ $13,$ $20,$ $x,$ $y$ are $10$ and $25$ respectively, then which of the following is false $(x>y)$

• A. $x^2+y^2=148$
• B. $xy=54$
• C. $x-y=2\sqrt{10}$
• D. $x,y$ are rational numbers

Answer 1

The correct option is D $x,y$ are rational numbers

Given, $\frac{64+x+y}{8}=10$
$\Rightarrow x+y=16\dots \left(1\right)$
$\frac{\sum x_i^2}{n}-{\left(\overline{x}\right)}^2=25$
$\Rightarrow x^2+y^2=148\dots \left(2\right)$
$\Rightarrow (x+y{)}^2-2xy=148$
$\Rightarrow xy=54\dots \left(3\right)$
Using equations $\left(1\right),\left(2\right)$ and $\left(3\right)$, we get
$x=8+\sqrt{10}$ and $y=8-\sqrt{10}$ $(\because x>y)$
$\Rightarrow x-y=2\sqrt{10}$
Clearly $x,y$ are irrational numbers