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 $n$ 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 :

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