The values of , for which vmatrix 1 (3)/(2) +(3)/(2) 1 (1)/(3) +(1)/(3) 2 +3 3 +1 0 vmatri…

Question

The values of $\alpha $, for which $\begin{vmatrix} 1 \quad  \frac{3}{2} \quad  \alpha +\frac{3}{2} \ 1 \quad  \frac{1}{3} \quad  \alpha +\frac{1}{3} \ 2\alpha +3 \quad  3\alpha +1 \quad  0 \end{vmatrix}=0$, lie in the interval

Accepted Answer

$\begin{vmatrix} 1 \quad  \frac{3}{2} \quad  \alpha +\frac{3}{2} \ 1 \quad  \frac{1}{3} \quad  \alpha +\frac{1}{3} \ 2\alpha +3 \quad  3\alpha +1 \quad  0 \end{vmatrix}=0$

$\begin{gathered} \Rightarrow (2\alpha +3){\frac{7\alpha }{6}}-(3\alpha +1){\frac{-7}{6}}=0 \ \Rightarrow (2\alpha +3)\cdot \frac{7\alpha }{6}+(3\alpha +1)\cdot \frac{7}{6}=0 \ \Rightarrow 2{\alpha }^{2}+3\alpha +3\alpha +1=0 \ \Rightarrow 2{\alpha }^{2}+6\alpha +1=0 \ \Rightarrow \alpha =\frac{-3+\sqrt{7}}{2},\frac{-3-\sqrt{7}}{2} \end{gathered}$

Hence option (C) is correct.

Solutions · 1