The values of α, for which |{1, 3/2, α + 3/2}, {1, 1/3, α + 1/3}, {2α + 3, 3α + 1, 0}| = 0, lie in the interval

Question

The values of α, for which \(\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0\) lie in the interval

(1) \((-2,1)\)

(2) \((-3,0)\)

(3) \(\left(-\frac{3}{2}, \frac{3}{2}\right)\)

(4) \((0,3)\)

Answer 1

Correct option is (2) \((-3,0)\)

\(\left|\begin{array}{ccc} 1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0 \end{array}\right|=0\)

\(\Rightarrow(2 \alpha+3)\left\{\frac{7 \alpha}{6}\right\}-(3 \alpha+1)\left\{\frac{-7}{6}\right\}=0\)

\(\Rightarrow(2 \alpha+3) \cdot \frac{7 \alpha}{6}+(3 \alpha+1) \cdot \frac{7}{6}=0\)

\(\Rightarrow 2 \alpha^{2}+3 \alpha+3 \alpha+1=0\)

\( \Rightarrow 2 \alpha^{2}+6 \alpha+1=0\)

\(\Rightarrow \alpha=\frac{-3+\sqrt{7}}{2}, \frac{-3-\sqrt{7}}{2}\)