Question

The only integral root of the equation
$\left|\begin{array}{ccc}2-y& 2& 3\\ 2& 5-y& 6\\ 3& 4& 10-y\end{array}\right|=0$ is

• A. y = 3
• B. y = 2
• C. y = 1
• D. None of these

Answer 1

Answer: (C)

We have, $\left|\begin{array}{ccc}2-y& 2& 3\\ 2& 5-y& 6\\ 3& 4& 10-y\end{array}\right|=0$
$\Rightarrow \left(2-r\right)\left[\left(5-y\right)\left(10-y\right)-24\right]-2\left[2\left(10-y\right)-18\right]$
$+3\left[8-3\left(5-y\right)\right]=0$
$\Rightarrow \left(2-y\right)\left[50-15y+y^2-24\right]-2\left[20-2y-18\right]$
$+3\left[8-15+3y\right]=0$
$\Rightarrow \left(2-y\right)\left[y^2-15y+26\right]$
$-2\left[2-2y\right]+3\left[3y-7\right]=0$
$\Rightarrow 2y^2-30y+52-y^3+15y^2-26y-4+4y$
$+9y-21=0$
$\Rightarrow -y^3+17y^2-43y+27=0$
$\Rightarrow y^3-17y^2+43y-27=0$
$\Rightarrow \left(y-1\right)\left(y^2-16y+27\right)=0$
$\Rightarrow y-1=0$ or $y^2-16y+27=0$
$\Rightarrow y=1$