$\overline{ x }=\frac{\sum \limits_{ i =11}^{41} i }{31}=\frac{11+41}{2}=26 \quad(31 \text { elements) }$

$\overline{ y }=\frac{\sum \limits_{ j =61}^{91} j }{31}=\frac{61+91}{2}=76 \quad \text { (31 elements) }$

$\text { Combined mean, }$

$\mu =\frac{31 \times 26+31 \times 76}{31+31}$

$=\frac{26+76}{2}=51$

$\sigma^2=\frac{1}{62} \times\left(\sum_{i=1}^{31}\left(x_i-\mu\right)^2+\sum_{i=1}^{31}\left(y_i-\mu\right)^2\right)=705$

Since, $x _{ i } \in X$ are in $A.P.$ with $31$ elements and common difference $1$,same is $y _{ i } \in y$, when written 

in increasing order.

$\therefore \sum \limits_{i=1}^{31}\left(x_i-\mu\right)^2=\sum \limits_{i=1}^{31}\left(y_i-\mu\right)^2$

$=10^2+11^2+\ldots . .+40^2$

$=\frac{40 \times 41 \times 81}{6}-\frac{9 \times 10 \times 19}{6}=21855$

$\therefore \left|\bar{x}+\bar{y}-\sigma^2\right|=|26+76-705|=603$