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3 and 4 .Determinants and Matrices
hard
For the system of linear equations
$2 x+4 y+2 a z=b$
$x+2 y+3 z=4$
$2 x-5 y+2 z=8$
which of the following is NOT correct?
A
It has infinitely many solutions if $a=3, b=6$
B
It has unique solution if $a=b=6$
C
It has unique solution if $a=b=8$
D
It has infinitely many solution if $a=3, b=8$
(JEE MAIN-2023)
Solution
$\Delta=\left|\begin{array}{ccc}2 & 4 & 2 a \\ 1 & 2 & 3 \\ 2 & -5 & 2\end{array}\right|=18(3-a)$
$\Delta_x=\left|\begin{array}{ccc}b & 4 & 2 a \\ 4 & 2 & 3 \\ 8 & -5 & 2\end{array}\right|=(64+19 b-72 a)$
For unique solution $\Delta=0$
$\Rightarrow a \neq 3$ and $b \in R$
For Infinitely many solution ;
$\Delta=\Delta_x=\Delta_y=\Delta_z=0$
$\Rightarrow a =3 \quad \because \Delta=0$
$\text { and } b =8 \quad \because \Delta_x=0$
Standard 12
Mathematics
