If system of linear equations 2 x+y-z=3 x-y-z=α 3 x+3 y+β z=3 has infinitely many solution, then ?

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3 and 4 .Determinants and Matrices

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If the system of linear equations

$2 x+y-z=3$

$x-y-z=\alpha$

$3 x+3 y+\beta z=3$

has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to .... .

A

$1$

B

$2$

C

$3$

D

$5$

(JEE MAIN-2021)

Solution

$2 \times(\mathrm{i})-(\mathrm{ii})-(\mathrm{iii}) \text { gives : }$

$-(1+\beta) \mathrm{z}=3-\alpha$

For infinitely many solution

$\beta+1=0=3-\alpha \Rightarrow(\alpha, \beta)=(3,-1)$

Hence, $\alpha+\beta-\alpha \beta=5$

Standard 12

Mathematics

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