If for some x ∈ R , the frequency distribution of the marks...

Total students $= 20$

$⇒ {\left(x + 1\right)}^{2} + \left(2x - 5\right) + \left({x}^{2} - 3x\right) + x = 20$

$⇒ {x}^{2} + 2x + 1 + 2x - 5 + {x}^{2} - 3x + x = 20$

$⇒ 2{x}^{2} + 2x - 24 = 0 ⇒ {x}^{2} + x - 12 = 0$

$⇒ x = - 4, 3\left(∵ x > 0\right)$

$∴ x = 3$

Average $= \dfrac{{\left(x + 1\right)}^{2} \times 2 + \left(2x - 5\right)3 + 5\left({x}^{2} - 3x\right) + 7 \times x}{20}$

$= \dfrac{32 + 3 + 0 + 21}{20} = 2.8$

Hence, required mean is $2 . 8$