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Question

If S is the set of distinct values of 'b' for which the following system of linear equation
x + y + z = 1
x + zy + z = 1
ax + by + z = 0
has no solution, then S is

A
an empty set
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B
an infinite set
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C
a finite set containing two or more elements
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D
a singleton
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Solution

The correct option is D a singleton
These three equations will have no solution if derterminant of these equations is 0.

Δ=∣ ∣1111a1ab1∣ ∣=0

ab+a1+b2a=0

(a1)2=0

a=1

If we put, a=1,

Eqaution (2) and (1), will become same.

Now, if we put b=1 in third equation, then it will become parallel to first two equations.

So, for this system to have no solutions, b should be 1.

Thus, S is a singleton set containing one element 1.

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