If the mean and the variance of $6,4, a, 8, b, 12,10$, 13 are 9 and 9.25 respectively, then $a+b+a b$ is…

If the mean and the variance of $6,4, a, 8, b, 12,10$, 13 are 9 and 9.25 respectively, then $a+b+a b$ is equal to :
  1. $105$
  2. $103$
  3. $100$
  4. $106$

Solution

$\begin{aligned} & \because \text { mean }=9 \\ & \therefore 53+\mathrm{a}+\mathrm{b}=72 \\ & \Rightarrow \mathrm{a}+\mathrm{b}=19 \\ & \because \sigma^2=\frac{37}{4} \text { and }(\overline{\mathrm{X}})^2+\sigma^2=\frac{\sum \mathrm{x}_1^2}{\mathrm{~N}} \\ & \Rightarrow 81+\frac{37}{4}=\frac{529+\mathrm{a}^2+\mathrm{b}^2}{8} \\ & \Rightarrow 648+74=529+\mathrm{a}^2+\mathrm{b}^2 \\ & \Rightarrow \mathrm{a}^2+\mathrm{b}^2=193 \\ & \because \mathrm{a}+\mathrm{b}=19 \Rightarrow \mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ab}=361 \\ & \Rightarrow 2 \mathrm{ab}=168 \\ & \Rightarrow \mathrm{ab}=84 \\ & \therefore \mathrm{a}+\mathrm{b}+\mathrm{ab}=103\end{aligned}$ *

Asked in: JEE Main 2025 (02 Apr Shift 2)

Practice more Statistics questions on Aicharya