Let equation of line is $y=m x+c$

$x$	$0$	$1$	$2$	$3$	$4$	$R -\{0,1,2,3,4\}$
$P ( x )$	$C$	$m + c$	$2 m + c$	$3 m + c$	$4 m+c$	$0$

$\sum_{ x =0}^4( mx + c )=1 \Rightarrow 10 m +5 c =1 \Rightarrow 2 m + c =\frac{1}{5}$   $. . . (1)$

$\text { mean }=\sum x _{ i } P _{ i }=\sum_{ i =0}^4\left( mx _{ i }+ c \right) x _{ i }=30 m +10 c =\frac{5}{2}$

$\therefore 3 m + c =\frac{1}{4} \ldots(2)$

$\text { from (1) and (2) m= } \frac{1}{20}, c =\frac{1}{10}$

$\sum P _{ i } x _{ i }^2=\sum_{ i =0}^4\left( mx _{ i }+ c \right) x _1^2$

$=\sum_{ i =0}^4\left( mx _{ i }^3+ cx _{ i }^2\right) \Rightarrow 100 m +30 c (\text { Now putting } m \text { and } c )$

$\Rightarrow \Sigma P _{ i }^2=5+3=8$

$\text { Variance }=\Sigma P _{ i } x _{ i }^2-\left(\Sigma P _{ i } x _{ i }\right)^2=8-\left(\frac{5}{2}\right)^2=\frac{7}{4}$

$\therefore 24 \alpha=42$