Question

Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II. List I	List II
(A)	$\lambda=8, \mu \neq 15$
(B)	$\lambda \neq 8, \mu \in R$
(C)	$\lambda=8, \mu=15$

• A. A $\to$ 1, B $\to$ 2, C $\to$ 3.
• B. A $\to$ 2, B $\to$ 3, C $\to$ 1.
• C. A $\to$ 3, B $\to$ 1, C $\to$ 2.
• D. A $\to$ 3, B $\to$ 2, C $\to$ 1.

Answer 1

Answer: (B)

A $\to$ 2, B $\to$ 3, C $\to$ 1.

Answer 2

Given system of linear equations is
$x+2 y+3 z=6$
$x+3 y+5 z=9$
and $2 x+5 y+\lambda z=\mu$
Now, according to Cramer's rule,
$\Delta=\begin{vmatrix}1 & 2 & 3 \\\ 1 & 3 & 5 \\\ 2 & 5 & \lambda\end{vmatrix}$
$=1(3 \lambda-25)-2(\lambda-10)+3(5-6)$
$=\lambda-8$
$\Delta_{1}=\begin{vmatrix}6 & 2 & 3 \\\ 9 & 3 & 5 \\\ \mu & 5 & \lambda\end{vmatrix}$
$=6(3 \lambda-25)-(29 \lambda-5 \mu) +3(45-3 \mu)=\mu-15$
$\Delta_{2}=\begin{vmatrix}1 & 6 & 3 \\\ 1 & 9 & 5 \\\ 2 & \mu & \lambda\end{vmatrix}$
$=1(9 \lambda-5 \mu)-6(\lambda-10)+3(\mu-18)$
$= 3 \lambda-2 \mu+6$
and
$\Delta_{3}=\begin{vmatrix}1 & 2 & 6 \\\ 1 & 3 & 9 \\\ 2 & 5 & \mu\end{vmatrix}$
$=1(3 \mu-45-2(\mu-18)+6(5-6)$
$=\mu-15$
Now, if $\lambda=8$ and $\mu \neq 15$, then system of linear equations has no solution.
If $\lambda \neq 8$ and $\mu \in R$, then system of linear equations has unique solution.
And, if $\lambda=8$ and $\mu=15$, then system of linear equations has infinite number of solutions, because $\Delta_{2}=3 \lambda-2 \mu+6$ is also be zero.