If S is set of distinct values of 'b' for which following system of linear equations&n

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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

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