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Can someone explain - Matrices and Determinants - JEE Main-2

Let S be the set of all real values of ‘a’ for which the following system of linear equations

ax+2y+5z=1

   2x+y+3z=1

            3y+7z=1

is consistent. Then the set S is :

  • Option 1)

    an empty set

  • Option 2)
    equal to R
  • Option 3)
    equal to R − {1}
  • Option 4)

    equal to  {1}

 
Answers (2)
148 Views
N neha

As we learnt 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} a & 2 & 5\\ 2 & 1 & 3\\ 0 & 3 & 7 \end{vmatrix}\neq 0

a\left ( -2 \right )-2\left ( 14 \right )+5\left ( 6 \right )\neq 0

a\neq 1


Option 1)

an empty set

This option is incorrect.

Option 2)

equal to R

This option is incorrect.

Option 3)

equal to R − {1}

This option is correct.

Option 4)

equal to  {1}

This option is incorrect.

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