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 JEE MAINS Mathematics 2017

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As we leant in

Cramer's rule for solving system of linear equations -

When $\Delta =0$  and $\Delta _{1}=\Delta _{2}=\Delta _{3}=0$ ,

then  the system of equations has infinite solutions.

- wherein

$a_{1}x+b_{1}y+c_{1}z=d_{1}$

$a_{2}x+b_{2}y+c_{2}z=d_{2}$

$a_{3}x+b_{3}y+c_{3}z=d_{3}$

and

$\Delta =\begin{vmatrix} a_{1} &b_{1} &c_{1} \\ a_{2} & b_{2} &c_{2} \\ a_{3}&b _{3} & c_{3} \end{vmatrix}$

$\Delta _{1},\Delta _{2},\Delta _{3}$ are obtained by replacing column 1,2,3 of $\Delta$ by $\left ( d_{1},d_{2},d_{3} \right )$  column

$\begin{vmatrix} 1 & 1&1\\ 1 & a & 1\\ a&b & 1 \end{vmatrix} = 0$

= 1 (a -b) - 1(1 -a) + 1 (b - a2) = 0

=a - b - 1 + a + b - a2 = 0

=> 2a - 1 -a=0

=> a2- 2a +1 =0

=> (a - 1)= 0

=> a = 1

It is independent of b

Hence ,option 1 is correct

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