Question

If $x\ne 0,\,\left| \begin{matrix} x+1 & 2x+1 & 3x+1 \\\ 2x & 4x+3 & 6x+3 \\\ 4x+4 & 6x+4 & 8x+4 \\\ \end{matrix} \right|=0,$ then $2x+1$ is equal to

• A. $x$
• B. $0$
• C. $2x$
• D. $3x$

Answer 1

Answer: (B)

$0$

Answer 2

Given, $\left| \begin{matrix} x+1 & 2x+1 & 3x+1 \\\ 2x & 4x+3 & 6x+3 \\\ 4x+4 & 6x+4 & 8x+4 \\\ \end{matrix} \right|=0$
$\Rightarrow$ $2\left| \begin{matrix} 0 & x & 2x \\\ 2x & 4x+3 & 6x+3 \\\ 2x+2 & 3x+2 & 4x+2 \\\ \end{matrix} \right|=0$
Applying $({{R}_{1}}\to 2{{R}_{1}}-{{R}_{3}})$
$\Rightarrow$ $2\left| \begin{matrix} 0 & x & 0 \\\ 2x & 4x+3 & -2x-3 \\\ 2x+2 & 3x+2 & 4x+2 \\\ \end{matrix} \right|=0$ Applying $({{C}_{3}}\to {{C}_{3}}-2{{C}_{2}})$
$\Rightarrow$ $-4\left| \begin{matrix} 0 & x & 0 \\\ x & 4x+3 & 2x+3 \\\ x+1 & 3x+2 & 2x+2 \\\ \end{matrix} \right|=0$
$\Rightarrow$ $-4x[2{{x}^{2}}+2x-(2x+3)(x+1)]=0$
$\Rightarrow$ $-4x[2{{x}^{2}}+2x-(2{{x}^{2}}+5x+3)]=0$
$\Rightarrow$ $4x[3x+3]=0$
$\Rightarrow$ $x+1=0$ $(\because \,\,x\ne 0\,\,given)$