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Question

The system of linear equations
x+y+z=2
2x+3y+2z=5
2x+3y+(a2āˆ’1)z=a+1

A
is inconsistent when |a|=3
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B
has infinitely many solutions for a=4
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C
is incosistent when a=4
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D
has a unique solution for |a|=3
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Solution

The correct option is A is inconsistent when |a|=3
For the given system of equation, the augmented matrix can be written as
[A:B]=1112232523a21a+1

Row operation by R2R22R1 and R3R32R1 we will get,

[A:B]=1112010101a23a3

Row operation by R3R3R2

[A:B]=1112010100a23a4

Now if a=±3, then rank of matrix A=2 and rank of the augmented matrix [A:B]=3

rank of Arank [A:B] and the system of linear equation becomes incosistent.
Hence, for |a|=3 the system of linear equation is inconsistent.

Alternate solution :

If we consider all the three equations representing a plane and if we substitute a21=2, i.e., a=±3 the last two planes become parallel in nature.
So the given system of linear equations does not have a solution for a=±3 and will become inconsistent.

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