Question

Investigate the values of $\lambda$ and $\mu$ for the system $x+2y+3z=6,x+3y+5z=9$,
$2x+5y+\lambda z=\mu$ and match the values in List - I with the items in List - II.

List I		List II
(A)	$\lambda =8,\mu \ne 15$	1.	Infinitely many solutions
(B)	$\lambda \ne 8,\mu \in R$	2.	No solution
(C)	$\lambda =8,\mu =15$	3.	Unique solution

• 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)

Given system of linear equations is
$x+2y+3z=6$
$x+3y+5z=9$
and $2x+5y+\lambda z=\mu$
Now, according to Cramer's rule,
$\Delta =\left|\begin{array}{ccc}1& 2& 3\\ 1& 3& 5\\ 2& 5& \lambda \end{array}\right|$
$=1(3\lambda -25)-2(\lambda -10)+3(5-6)$
$=\lambda -8$
${\Delta }_1=\left|\begin{array}{ccc}6& 2& 3\\ 9& 3& 5\\ \mu & 5& \lambda \end{array}\right|$
$=6(3\lambda -25)-(29\lambda -5\mu )+3(45-3\mu )=\mu -15$
${\Delta }_2=\left|\begin{array}{ccc}1& 6& 3\\ 1& 9& 5\\ 2& \mu & \lambda \end{array}\right|$
$=1(9\lambda -5\mu )-6(\lambda -10)+3(\mu -18)$
$=3\lambda -2\mu +6$
and
${\Delta }_3=\left|\begin{array}{ccc}1& 2& 6\\ 1& 3& 9\\ 2& 5& \mu \end{array}\right|$
$=1(3\mu -45-2(\mu -18)+6(5-6)$
$=\mu -15$
Now, if $\lambda =8$ and $\mu \ne 15$, then system of linear equations has no solution.
If $\lambda \ne 8$ 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.