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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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