QJEEM20Mabec2: JEE Main 2020 Mathematics - Class 11 JEEM20Mabec2 Solution: Let X = {x N : 1 x 17} and Y = {ax + b: x X and a, b R, a > 0}. …

Mean of X = ${{\sum\limits_{x = 1}^{17} x } \over {17}}$ = ${{17 \times 18} \over {17 \times 2}}$ = 9

Mean of Y = ${{\sum\limits_{x = 1}^{17} {\left( {ax + b} \right)} } \over {17}}$ = 17

$\Rightarrow$ $a{{\sum\limits_{x = 1}^{17} x } \over {17}} + b$ = 17

$\Rightarrow$ 9a + b = 17 ....(1)

Given Var(Y) = 216

$\Rightarrow$ ${{\sum\limits_{x = 1}^{17} {{{\left( {ax + b} \right)}^2}} } \over {17}} - {\left( {17} \right)^2}$ = 216

$\Rightarrow$ ${\sum\limits_{x = 1}^{17} {{{\left( {ax + b} \right)}^2}} }$ = 8585

$\Rightarrow$ (a + b)² + (2a + b)² +....+ (17a + b)² = 8585

$\Rightarrow$ 105a² + b² + 18ab = 505 ....(2)

From equation (1) & (2)

a = 3 & b = -10

$\therefore$ a + b = –7