Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and <img alt="\alpha(> 0 )," src="https://entrancecorner.oncodecogs.com/gif.latex?%5Calpha%28%3E%200%20

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $\alpha(> 0 ),$ and the mean and standard deviation of marks of class B of $\mathrm{n}$ students be respectively 55 and 30 $\mathrm{-\alpha}$ . If the mean and variance of the marks of the combined class of $\mathrm{100+\mathrm{n}}$ studants are respectively 50 and 350 , then the sum of variances of classes $\mathrm{A \: and \: B}$ is :

Option: 1
$650$

Option: 2
$450$

Option: 3
$900$

Option: 4
$500$

Answers (1)

$\begin{aligned} & \mathrm{m}_{\mathrm{A}}=40 \mathrm{~S} \cdot \mathrm{d}_{\mathrm{A}}=\alpha>0 \quad \mathrm{n}_{\mathrm{A}}=100 \\ \end{aligned}$

$\begin{aligned} & \mathrm{~m}_{\mathrm{B}}=55 \mathrm{~S} \cdot \mathrm{d}_{\mathrm{B}}=30-\alpha \mathrm{n}_{\mathrm{B}}=\mathrm{n} \end{aligned}$

$$$ \mathrm{m}_{\mathrm{AVB}}=50 \quad \text { Variance }_{\mathrm{AVB}}=350 \mathrm{n}_{\mathrm{AVB}}=100+\mathrm{n}$

$\mathrm{A=\left\{x_1, \ldots \ldots ., x_{100}\right\} B=\left\{y_1 \ldots ., y_n\right\} }$

$\begin{aligned} & \sum \mathrm{x}_{\mathrm{i}}=4000 \\ & \sum \mathrm{y}_{\mathrm{i}}=55 \mathrm{n} \end{aligned}$

$\sum\left(\mathrm{x}_{\mathrm{i}}+\mathrm{y}_{\mathrm{i}}\right)=50(100+\mathrm{n})$

$4000+55 \mathrm{n}=5000+50 \mathrm{n}$

Using formula of standard deviation

$\alpha^2=\frac{\sum \mathrm{x}_{\mathrm{i}}^2}{100}-(40)^2 \quad(30-\alpha)^2 \frac{\sum \mathrm{y}_{\mathrm{i}}^2}{200}-(55)^2$
$\sum \mathrm{x}_{\mathrm{i}}^2=100\left(1000+\alpha^2\right)$
$\sum \mathrm{y}_{\mathrm{i}}^2=200\left((55)^2+(30-\alpha)^2\right)$
$350=\frac{\sum\left(\mathrm{x}_{\mathrm{i}}^2+\mathrm{y}_{\mathrm{i}}^2\right)}{300}-(50)^2$
$\sum \mathrm{x}_{\mathrm{i}}^2+\sum \mathrm{y}_{\mathrm{i}}^2=\left((50)^2+350\right) 300$
$160000+100 \alpha^2+200(55)^2+200(30-\alpha)^2$
$(50)^2 300+350 \times 300$
$1600+\alpha^2+6050+2(30-\alpha)^2=7500+1050$
$\alpha^2+1800-120 \alpha+2 \alpha^2-900=0$
$3 \alpha^2-120 \alpha+900=0$
$\alpha^2-40 \alpha+300=0$
$(\alpha-10)(\alpha-30)=0$
$\alpha=10 \text { or } \alpha=30$
$\text { if } \alpha=10 \text { VarA }=100 \& \text { VarB }=400$