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Mathematics Question on Determinants

The solution set of the equation ∣1420 1−25 12x5x2∣=0\begin{vmatrix}1&4&20\\\ 1&-2&5\\\ 1&2x&5x^{2}\end{vmatrix} = 0 is

A

{0, 1}

B

{1, 2}

C

{1, 5}

D

{2, - 1}

Answer

{2, - 1}

Explanation

Solution

Given ∣1420 1−25 12x5x2∣=0\begin{vmatrix}1&4&20\\\ 1&-2&5\\\ 1&2x&5x^{2}\end{vmatrix} = 0 Operate, R2→R2−R1R3→R3−R1R_{2} \to R_{2} - R_{1} R_{3} \to R_{3} - R_{1} ∣1420 0−6−15 02x−45x2−20∣=0\begin{vmatrix}1&4&20\\\ 0&-6&-15\\\ 0&2x-4&5x^{2}-20\end{vmatrix}=0 ⇒1[−30x2+120+30x−60]=0\Rightarrow 1\left[-30x^{2} + 120 + 30x -60\right] =0 ⇒30x2−30x−60=0\Rightarrow 30x^{2} -30x -60 =0 ⇒x2−x−2=0 \Rightarrow x^{2} -x -2 =0 ⇒(x−2)(x+1)=0 \Rightarrow \left(x -2\right)\left(x+1\right)=0 ⇒x=−1,2\Rightarrow x=-1,2 Thus, solution set is {2, - 1}.