Question

Consider the system of equations$\begin{array}{rcl}x+y+z& =& 1,\\ 2x+3y+2z& =& 1,\\ 2x+3y+(a^2-1)z& =& a+1\end{array}$ then

• A. System has a unique solution for$\left|a\right|=\sqrt{3}$
• B. System is inconsistence for$\left|a\right|=\sqrt{3}$
• C. System is inconsistence for$a=4$
• D. System is inconsistence for$a=3$

Answer 1

The correct option is B

System is inconsistence for$\left|a\right|=\sqrt{3}$

Step 1: Use Cramer's rule

We have given a system of equations

$\begin{array}{rcl}x+y+z& =& 1,\\ 2x+3y+2z& =& 1,\\ 2x+3y+(a^2-1)z& =& a+1\end{array}$

by Cramer's rule,

$\begin{array}{rcl}D& =& \left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 2\\ 2& 3& a^2-1\end{array}\right|\\ D& =& 1\left[3\right(a^2-1)-6]-1\left[2\right(a^2-1)-4]+1[6-6]\\ D& =& (a^2-1)-6+4+0\\ D& =& a^2-3\end{array}$

Step 2: Checking the system of equations consistence or inconsistence.

If$D=0$, System of equations has inconsistence solution.

If$D\ne 0$, System of equations has a unique solution.

$\begin{array}{rcl}\therefore a^2-3& =& 0\\ a^2& =& 3\\ a& =& \pm \sqrt{3}\\ \left|a\right|& =& \sqrt{3}\end{array}$

Hence, option$\left(B\right)$ is correct.