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  • Need explanation for: If S is the set of distinct values of ‘b’ for which the following system of linear equationsx+y+z=1x+ay+z=1ax+by+z=0has no solution, then S is :

 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 :

 

  • Option 1)

     an infinite set

     

  • Option 2)

     a finite set containing two or more elements

     

  • Option 3)

    a singleton

     

  • Option 4)

     an empty set

 

Answers (2)

best_answer

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 -a2 =0

=> a2- 2a +1 =0

=> (a - 1)2 = 0

=> a = 1  

It is independent of b

 


Option 1)

 an infinite set

 

Incorrect

Option 2)

 a finite set containing two or more elements

 

Incorrect

Option 3)

a singleton

 

Correct

Option 4)

 an empty set

Incorrect

Posted by

prateek

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