If for some $x \in \mathbb{R}^{+} \cup\{0\}$, the frequency distribution of the marks obtained by 20…

If for some $x \in \mathbb{R}^{+} \cup\{0\}$, the frequency distribution of the marks obtained by 20 students in a test is,
then the mean of the marks is
  1. $3.0$
  2. $2.8$
  3. $2.5$
  4. $3.2$

Solution

$\begin{aligned} & \text {Here, } \Sigma \mathrm{f}_{\mathrm{i}}=(x+1)^2+2 x-5+x^2-3 x+x \\ & =2 x^2+2 x-4 \\ & \sum \mathrm{f}_{\mathrm{i}} x_{\mathrm{i}}=2(x+1)^2+3(2 x-5)+5\left(x^2-3 x\right)+7 x \\ & =7 x^2+2 x-13 \end{aligned}$ $\begin{aligned} & \mathrm{N}=20 \\ & \Rightarrow \sum \mathrm{f}_{\mathrm{i}}=20 \\ & \Rightarrow 2 x^2+2 x-4=20 \\ & \Rightarrow x=-4,3 \\ & \Rightarrow x=3 \end{aligned}$ $\ldots\left[\because x \in \mathrm{R}^{+} \cup\{0\}\right]$
Now mean $\begin{aligned} (\bar{x}) & =\frac{\sum \mathrm{f}_{\mathrm{i}} x_{\mathrm{i}}}{\mathrm{~N}} \\ & =\frac{7(3)^2+2(3)-13}{20} \end{aligned}$ $=2.8$

Asked in: MHT CET 2024 (11 May Shift 2)

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