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3 and 4 .Determinants and Matrices
hard
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 :
A
a singleton
B
an empty set
C
an infinate set
D
a finate set contatining two or more elements
(JEE MAIN-2017)
Solution
$D = \left| \begin{array}{l}
1\,\,\,1\,\,\,\,1\\
1\,\,\,a\,\,\,1\\
1\,\,\,b\,\,\,1
\end{array} \right|\, = 0$
$ \Rightarrow 1\left[ {a – b} \right]\, – 1\left[ {1 – a} \right] + 1\left[ {b – {a^2}} \right] = 0 \Rightarrow {\left( {a – 1} \right)^2} = 0$
$ \Rightarrow a = 1$
For $ \Rightarrow a = 1$, First two equation are identical ie. $x+y+z=1$
To have no solution with $x+by+z=0$
$b=1$
So $b = \left\{ 1 \right\} \Rightarrow $ It is singleton set.
Standard 12
Mathematics
