Question

Sol. [Key Concept: Mean gives $\alpha + \beta = 16$ and variance gives $\alpha^{2} + \beta^{2} = 136$, so $\alpha = 6,\beta = 10$; the roots $3\alpha + 2,2\beta + 1$ build the quadratic.]

$$\frac{2 + 4 + \alpha + \beta + 8 + 12 + 14}{7} = 8$$

$$\begin{array}{r} \alpha + \beta = 56 - 40 = 16 \Rightarrow \alpha + \beta = 16\#(i) \end{array}$$

Variance $= \frac{\sum_{}^{}\ x_{i}^{2}}{n} - \left( \frac{\sum_{}^{}\ x_{i}}{n} \right)^{2}$

$${16 = \frac{2^{2} + 4^{2} + \alpha^{2} + \beta^{2} + 8^{2} + 12^{2} + 14^{2}}{7} - 64 }{560 = \alpha^{2} + \beta^{2} + 424}$$

$$\begin{array}{r} \Rightarrow \alpha^{2} + \beta^{2} = 136\#(ii) \end{array}$$

From (i) & (ii) $\alpha = 6\&\beta = 10$

$$x_{1} = 3\alpha + 2 = 20;x_{2} = 2\beta + 1 = 21 $$Q.E. : $x^{2} - 41x + 420 = 0$