Question

If the system of equations
$x+2y+3z=3$
$4x+3y-4z=4$
$8x+4y-\lambda z=9+\mu$
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)

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