JEE Main 2023 (Online) 24th January Evening ShiftQuestion 1
If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to
Determinant
Solution
Infinitely many solutions need $R_3=aR_1+bR_2$ on the left and right: $a+4b=8$, $2a+3b=4$ give $a=-\frac85$, $b=\frac{12}5$. Then $-\lambda=3a-4b=-\frac{72}5$, so $\lambda=\frac{72}5$, and $9+\mu=3a+4b=\frac{24}5$, so $\mu=-\frac{21}5$.