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3 and 4 .Determinants and Matrices
medium
જો સુરેખ સમીકરણોની સંહતિ $x+y+2 z=6$, $2 x+3 y+a z=a+1$, $-x-3 y+b z=2 b$ જ્યાં $a, b \in R$, ને અસંખ્ય ઉકેલો હોય, તો $7 a+3 b=$ ______
A$9$
B$12$
C$16$
D$22$
(JEE MAIN-2025)
Solution
$\Delta=\left|\begin{array}{ccc}1 & 1 & 2 \\ 2 & 3 & a \\ -1 & -3 & b\end{array}\right|=0$
$\Rightarrow 2 a+b-6=0$
$\Delta_1=\left|\begin{array}{ccc}1 & 1 & 6 \\ 2 & 3 & a+1 \\ -1 & -3 & 2 b\end{array}\right|=0$
$\Rightarrow a+b-8=0$
$\text { Solving }(1)+(2)$
$a=-2, b=10$
$\Rightarrow 7 a+3 b=16$
$\Rightarrow 2 a+b-6=0$
$\Delta_1=\left|\begin{array}{ccc}1 & 1 & 6 \\ 2 & 3 & a+1 \\ -1 & -3 & 2 b\end{array}\right|=0$
$\Rightarrow a+b-8=0$
$\text { Solving }(1)+(2)$
$a=-2, b=10$
$\Rightarrow 7 a+3 b=16$
Standard 12
Mathematics
