3 and 4 .Determinants and Matrices
medium

Consider the following system of equations : $x+2 y-3 z=a$ ; $2 x+6 y-11 z=b$ ; $x-2 y+7 z=c$    where $a , b$ and $c$ are real constants. Then the system of equations :

A

has a unique solution when $5 a =2 b + c$

B

has infinite number of solutions when $5 a =2 b + c$

C

has no solution for all $a , b$ and $c$

D

has a unique solution for all $a , b$ and $c$

(JEE MAIN-2021)

Solution

$P_{1}: x+2 y-3 z=a$

$P_{2}: 2 x+6 y-11 z=b$

$P_{3}: x-2 y+7 z=c$

Clearly

$5 P _{1}=2 P _{2}+ P _{3} \quad$ if $5 a =2 b + c$

$\Rightarrow$ All the planes sharing a line of intersection

$\Rightarrow$ infinite solutions

Standard 12
Mathematics

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