Question

If the system of equations
$ x+2 y+3 z=3$
$ 4 x+3 y-4 z=4 $
$ 8 x+4 y-\lambda z=9+\mu$
has infinitely many solutions, then the ordered pair $(\lambda, \mu)$ is equal to :

• A. $\left(\frac{72}{5},-\frac{21}{5}\right)$
• B. $\left(\frac{72}{5}, \frac{21}{5}\right)$
• C. $\left(-\frac{72}{5} \cdot-\frac{21}{5}\right)$
• D. $\left(-\frac{72}{5}, \frac{21}{5}\right)$

Answer 1

Answer: (A)

$ x+2 y+3 z=3 ......$(i)
$ 4 x+3 y-4 z=4.......$(ii)
$ 8 x+4 y-\lambda z=9+\mu ......$(iii)
$ \text { (i) } \times 4 \text { - (ii) } \Rightarrow 5 y +16 z =8......$(iv)
(ii) $\times 2-$ (iii) $\Rightarrow 2 y+(\lambda-8) z=-1-\mu ......$(v)
(iv) $\times 2-$ (iii) $\times 5 \Rightarrow(32-5(\lambda-8)) z=16-5(-1-\mu)$
For infinite solutions $\Rightarrow 72-5 \lambda=0 \Rightarrow \lambda=\frac{72}{5}$
$21+5 \mu=0 \Rightarrow \mu=\frac{-21}{5}$
$\Rightarrow(\lambda, \mu) \equiv\left(\frac{72}{5}, \frac{-21}{5}\right)$