QJEEM24Maae21: JEE Main 2024 Mathematics - Class 11 JEEM24Maae21 Solution: Let the mean and the variance of 6 observations be and ,…

$\begin{aligned}& a, b, 68,44,48,60 \\& \text { Mean }=55 \quad a>b \\& \text { Variance }=194 \quad a+3 b \\& \frac{a+b+68+44+48+60}{6}=55 \\& \Rightarrow 220+a+b=330 \\& \therefore a+b=110 \ldots . .(1)\end{aligned}$

Also,

$\begin{aligned}& \sum \frac{\left(x_i-\bar{x}\right)^2}{n}=194 \\& \Rightarrow(a-55)^2+(b-55)^2+(68-55)^2+(44-55)^2 \\& +(48-55)^2+(60-55)^2=194 \times 6 \\& \Rightarrow(a-55)^2+(b-55)^2+169+121+49+25=1164 \\& \Rightarrow(a-55)^2+(b-55)^2=1164-364=800 \\& a^2+3025-110 a+b^2+3025-110 b=800 \\& \Rightarrow a^2+b^2=800-6050+12100 \\& a^2+b^2=6850 \ldots \ldots . .(2)\end{aligned}$

Solve (1) & (2);

$\begin{aligned}& a=75, b=35 \\& \therefore a+3 b=75+3(35)=75+105=180\end{aligned}$