JEE Advanced 2024 — Paper 1 — Question 13

Let $P\left(x_{i}\right)=m x_{i}+c, i=0,1,2,3,4$

$\sum_{i=0}^{4} P\left(x_{i}\right)=1 \Rightarrow 2 m+c=\frac{1}{5}$

$\sum_{i=0}^{4} x_{i} P\left(x_{i}\right)=\frac{5}{2} \Rightarrow 3 m+c=\frac{1}{4} \Rightarrow m=\frac{1}{20}, c=\frac{1}{10}$

$\Rightarrow \alpha=\sum_{i=0}^{4} x_{i}^{2} P\left(x_{i}\right)-\left(\sum_{i=0}^{4} x_{i} P\left(x_{i}\right)\right)^{2}=8-\frac{25}{4}=\frac{7}{4} \Rightarrow 24 \alpha=42$