Question

Assertion :Consider the system of equation $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$. If the system has infinite number of solutions, then $\mu =10$.. Reason: The determinant $\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0$ for $\mu =10$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given system of equations $AX=B$
where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}6\\ 10\\ \mu \end{array}\right]$
For infinitely many solution,
$D=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0$
$\Rightarrow 2(\lambda -3)-(\lambda -3)=0$
$\Rightarrow \lambda =3$
Also, $D_1=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& 3\end{array}\right|=0$
$\Rightarrow \mu =10$
$D_2=\left|\begin{array}{ccc}1& 6& 1\\ 1& 10& 3\\ 1& \mu & 3\end{array}\right|=0$
$\Rightarrow \mu =10$
$D_3=\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0$
$\Rightarrow \mu =10$
Hence, the assertion is true that the system has infinitely many solutions then $\mu =10$
Reason is also true but it is not the correct explanation for assertion .