Determinants Question 1
Question 1 - 24 January - Shift 2
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
(1) $(\frac{72}{5}, \frac{21}{5})$
(2) $(\frac{-72}{5}, \frac{-21}{5})$
(3) $(\frac{72}{5}, \frac{-21}{5})$
(4) $(\frac{-72}{5}, \frac{21}{5})$
Show Answer
Answer: (3)
Solution:
Formula: Properties of determinants, System of equations with 3 variables, consistency of solutions
$x+2 y+3 z=3$
$4 x+3 y-4 z=4$
$8 x+4 y-\lambda z=9+\mu$
(i) $\times 4$-(ii) $\Rightarrow 5 y+16 z=8$
(ii) $\times 2$ - (iii) $\Rightarrow 2 y+(\lambda-8) z=-1-\mu$
(iv) $\times 2-$ (iii) $\times 5 \Rightarrow(32-5(\lambda-8)) z=16-5(-1-\mu)$
For infinite solutions $\Rightarrow 72-5 \lambda=0 \Rightarrow \lambda=\frac{72}{5}$
$21+5 \mu=0 \Rightarrow \mu=\frac{-21}{5}$
$\Rightarrow(\lambda, \mu) \equiv(\frac{72}{5}, \frac{-21}{5})$
