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

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