Let equation of line is $$\mathrm{y=mx+c}$$

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

$$\begin{aligned} & \text { mean }=\sum x_i P_i=\sum_{i=0}^4\left(m x_i+c\right) \cdot x_i=30 m+10 c=\frac{5}{2} \\ & \therefore 3 m+c=\frac{1}{4} \ldots(2) \\ & \text { from (1) and (2) } \mathrm{m}=\frac{1}{20}, \mathrm{c}=\frac{1}{10} \\ & \Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^2=\sum_{\mathrm{i}=0}^4\left(\mathrm{mx}_{\mathrm{i}}+\mathrm{c}\right) \mathrm{x}_1^2 \\ & =\sum_{\mathrm{i}=0}^4\left(\mathrm{mx}_{\mathrm{i}}^3+c \mathrm{x}_{\mathrm{i}}^2\right) \Rightarrow 100 \mathrm{~m}+30 \mathrm{c}(\text { Now putting } \mathrm{m} \text { and } \mathrm{c}) \\ & \Rightarrow \Sigma \mathrm{P}_{\mathrm{i}}^2=5+3=8 \\ & \text { Variance }=\Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^2-\left(\Sigma \mathrm{P}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}\right)^2=8-\left(\frac{5}{2}\right)^2=\frac{7}{4} \\ & \therefore 24 \alpha=42 \end{aligned}$$