NIMCET 2019 — Mathematics PYQ

CONCEPT:
MEAN: The meaning of mean is to evaluate the average. It is defined as the average of the given numbers.
Mean = Sum of Observations/Total number of observations
Mean Deviation: For n observation x₁, x₂, x₃ . . . . . . xₙ, the mean deviation about their mean $\overline{x}$ is given by:
$M.D(\overline{x})=\frac{\sum |x_i-\overline{x}|}{n}$
CALCULATIONS:

Given series of numbers are 1,1+d,1+2d,...,1+100d

The given numbers are in arithmetic progression with first term a=1, number of terms n=101 and last term l=1+100d

∴ Sum of the numbers is $\frac{n}{2}(a+1)=\frac{101}{2}(1+1+100d)$

$\Rightarrow Mean=\frac{101(1+50d)}{101}=1+50d$

Mean deviation is $\frac{\sum |x_i-\bar{x}|}{n}$

$=\frac{(1+50d-1)+(1+50d-1-d)+\dots +(1+100d-1-50d)}{n}$

$=\frac{50d+49d+\dots +0+d+\dots +50d}{101}=\frac{2d(1+2+\dots +50)}{101}$

$\Rightarrow \frac{2d(50)(51)}{101\times 2}$ (As sum of first n natural numbers is given by $\frac{n(n+1)}{2}$)

$=\frac{d(50)(51)}{101}$

Given that mean deviation is 255.

∴ $255=\frac{d(50)(51)}{101}$

$\Rightarrow d=\frac{101\times 255}{50\times 51}$

$\Rightarrow d=10.1$