Matrices and Determinants 4 Question 3
3. If the system of linear equations
$x+y+z=5$
$x+2 y+2 z=6$
$x+3 y+\lambda z=\mu,(\lambda, \mu \in R), \quad$ has infinitely many solutions, then the value of $\lambda+\mu$ is
(2019 Main, 10 April I)
(a) 7
(b) 12
(c) 10
(d) 9
Show Answer
Answer:
Correct Answer: 3. (c)
Solution:
- Given system of linear equations
$ \begin{gathered} x+y+z=5 \\ x+2 y+2 z=6 \\ x+3 y+\lambda z=\mu \end{gathered} $
$(\lambda, \mu \in R)$
The above given system has infinitely many solutions, then the plane represented by these equations intersect each other at a line, means $(x+3 y+\lambda z-\mu)$
$=p(x+y+z-5)+q(x+2 y+2 z-6)$
$=(p+q) x+(p+2 q) y+(p+2 q) z-(5 p+6 q)$
On comparing, we get
$ \begin{array}{lc} & p+q=1, p+2 q=3, p+2 q=\lambda \\ \text { and } & 5 p+6 q=\mu \\ \text { So, } & (p, q)=(-1,2) \\ \Rightarrow & \lambda=3 \text { and } \mu=7 \\ \Rightarrow & \lambda+\mu=3+7=10 \end{array} $
