JEE Main 2026 — 24 January, Evening Shift — Question 14

$\Sigma \mathrm{y}_{\mathrm{i}}=\mathrm{a} \Sigma \mathrm{x}_{\mathrm{i}}+\Sigma \mathrm{b}$

$=\mathrm{a} \times(1+2+\ldots . .+19)+19 \mathrm{~b}$

$\frac{\Sigma \mathrm{y}_{\mathrm{i}}}{19}=\frac{\mathrm{a} \times 19 \times 20}{2 \times 19}+\mathrm{b}$

$$\begin{gathered} 30=10 \mathrm{a}+\mathrm{b} \end{gathered}$$

Variance of $\mathrm{X}=\frac{\Sigma \mathrm{x}_{\mathrm{i}}^{2}}{19}-\left(\frac{\Sigma \mathrm{x}_{\mathrm{i}}}{19}\right)^{2}$

$=\frac{19 \times 20 \times 39}{19 \times 6}-(10)^{2}=30$

Variance of $\mathrm{Y}=\mathrm{a}^{2}$ (variance of X )

$750=\mathrm{a}^{2} \times 30$ $\mathrm{a}^{2}=25 \Rightarrow \mathrm{a}= \pm 5$ if $\mathrm{a}=+5 \Rightarrow \mathrm{~b}=30-50=-20 \quad \ldots$ from (i)

if $\mathrm{a}=-5 \Rightarrow \mathrm{~b}=30+50=80 \quad \ldots$ from (i)

sum of values of $b=80-20=60$ .