If |{1, 3/2, α + 3/2}, {1, 1/3, α + 1/3}, {2α+3, 3α + 1, 0}| = 0 then α lies in the interval

Question

If \(\begin{vmatrix}1&\frac 32& \alpha + \frac 32\\1&\frac 13&\alpha + \frac 13\\2\alpha +3 &3\alpha + 1 & 0 \end{vmatrix} = 0\) then α lies in the interval

(1) (0, 3)

(2) (-3, 0)

(3) (-2, 1)

(4) (-2, 0)

Answer 1

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

\(C_3 \to C_3 - (αC_1 + C_2)\)

\(\begin{vmatrix}1&\frac 32&0\\1&\frac 13&0\\2\alpha +3 &3\alpha + 1 & -(2\alpha^2 + 6\alpha + 1) \end{vmatrix} = 0\)

⇒ \((2\alpha^2 + 6\alpha + 1) = 0\)

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