JEE MAIN - Mathematics (2017 (Offline) - No. 4)
If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :
an empty set
an infinite set
a finite set containing two or more elements
a singleton
Explanation
$$\left| {\matrix{
1 & 1 & 1 \cr
1 & a & 1 \cr
a & b & 1 \cr
} } \right| = 0$$
$$ \Rightarrow $$ 1 [a – b] – 1 [1 – a] + 1 [b – a2] = 0
$$ \Rightarrow $$ (a - 1)2 = 0
$$ \Rightarrow $$ a = 1
For a = 1, the equations become
x + y + z = 1
x + y + z = 1
x + by + z = 0
These equations give no solution for b = 1
$$ \Rightarrow $$ S is singleton set.
$$ \Rightarrow $$ 1 [a – b] – 1 [1 – a] + 1 [b – a2] = 0
$$ \Rightarrow $$ (a - 1)2 = 0
$$ \Rightarrow $$ a = 1
For a = 1, the equations become
x + y + z = 1
x + y + z = 1
x + by + z = 0
These equations give no solution for b = 1
$$ \Rightarrow $$ S is singleton set.
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