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Question

Consider the following system of equations:
x+2y3z=a
2x+6y11z=b
x2y+7z=c,
Where a,b and c are real constants. Then the system of equations:

A
has a unique solution when
5a=2b+c
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B
has infinite number of solutions when 5a=2b+c
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C
has no solution for all a,b and c
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D
has a unique solution for all a,b and c
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Solution

The correct option is B has infinite number of solutions when 5a=2b+c
Δ=∣ ∣1232611127∣ ∣
=202(25)3(10)
=2050+30=0

Δ1=∣ ∣a23b611c27∣ ∣
=20a2(7b+11c)3(2b6c)
=20a14b22c+6b+18c
=20a8b4c
=4(5a2bc)

Δ2=∣ ∣1a32b111c7∣ ∣
=7b+11ca(25)3(2cb)
=7b+11c25a6c+3b
=25a+10b+5c
=5(5a2bc)

Δ3=∣ ∣12a26b12c∣ ∣
=6c+2b2(2cb)10a
=10a+4b+2c
=2(5a2bc)

for infinite solution
Δ=Δ1=Δ2=Δ3=0
5a=2b+c

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