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3 and 4 .Determinants and Matrices
hard
જો સમીકરણ સંહતિ $x+y+z=6 \,; \,2 x+5 y+\alpha z=\beta \,; \, x+2 y+3 z=14$ એ અનંત ઉકેલ ધરાવે છે તો $\alpha+\beta$ ની કિમંત મેળવો.
A
$8$
B
$36$
C
$44$
D
$48$
(JEE MAIN-2022)
Solution
$x+y+z=6$
$2 x+5 y+\alpha z=\beta \quad(1)$
$x+2 y+3 z=14$
$x+y=6-z$
$x+2 y=14-3 z$
On solving
$x=z-2 \Rightarrow y=8-2 z$ in $(2)$
$2(z-2)+5(8-2 z)+\alpha z=\beta$
$(\alpha-8) z=\beta-36$ For having infinite solutions
$\alpha-8=0 \quad \& \quad \beta-36=0$
$\alpha=8, \beta=36$
Standard 12
Mathematics