QJEEM23Ma5173: JEE Main 2023 Mathematics - Class 11 JEEM23Ma5173 Solution: Let the mean and standard deviation of marks of class A of 100…

Question

Verified Solution • 2025–26 Session moderate

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and

\alpha(&gt;

0 ), and the mean and standard deviation of marks of class

B

of

n

students be respectively 55 and 30

-\alpha

. If the mean and variance of the marks of the combined class of

100+\mathrm{n}

studants are respectively 50 and 350 , then the sum of variances of classes

A

and

B

is :

Accepted Answer

\begin{array}{cll}\quad \mathbf{A} &amp; \mathbf{B} &amp; \mathbf{A}+\mathbf{B} \\\overline{\mathrm{x}}_{1}=40 &amp; \overline{\mathrm{x}}_{2}=55 &amp; \overline{\mathrm{x}}=50 \\\sigma_{1}=\alpha &amp; \sigma_{2}=30-\alpha &amp; \sigma^{2}=350 \\\mathrm{n}_{1}=100 &amp; \mathrm{n}_{2}=\mathrm{n} &amp; 100+\mathrm{n} \\\\\overline{\mathrm{x}}=\frac{100 \times 40+55 \mathrm{n}}{100+\mathrm{n}} &amp;\end{array}

\begin{aligned}&amp; 5000+50 \mathrm{n}=4000+55 \mathrm{n} \\\\&amp; 1000=5 \mathrm{n} \\\\&amp; \mathrm{n}=200 \\\\&amp; \sigma_{1}^{2}=\frac{\sum \mathrm{x}_{\mathrm{i}}^{2}}{100}-40^{2} \\\\&amp; \sigma_{2}^{2}=\frac{\sum \mathrm{x}_{\mathrm{j}}^{2}}{100}-55^{2} \\\\&amp; 350=\sigma^{2}=\frac{\sum \mathrm{x}_{\mathrm{i}}{ }^{2}+\sum \mathrm{x}_{\mathrm{j}}^{2}}{300}-(\overline{\mathrm{x}})^{2} \\\\&amp; 350=\frac{\left(1600+\alpha^{2}\right) \times 100+\left[(30-\alpha)^{2}+3025\right] \times 200}{300}-(50)^{2} \\\\&amp; 2850 \times 3=\alpha^{2}+2(30-\alpha)^{2}+1600+6050 \\\\&amp; 8550=\alpha^{2}+2(30-\alpha)^{2}+7650 \\\\&amp; \alpha^{2}+2(30-\alpha)^{2}=900 \\\\&amp; \alpha^{2}-40 \alpha+300=0 \\\\&amp; \alpha=10,30 \\\\&amp; \sigma_{1}^{2}+\sigma_{2}^{2}=10^{2}+20^{2}=500\end{aligned}

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