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Solve this problem - If , then is equal to : - Matrices and Determinants - JEE Main

If \begin{vmatrix} a-b-c & 2a & 2a\\ 2b & b-c-a & 2b\\ 2c & 2c & c-a-b \end{vmatrix}= (a+b+c) (x+a+b+c)^{2}, x\neq 0 \: and\:

a+b+c\neq 0 , then x is equal to :

  • Option 1)

    -2(a+b+c)

  • Option 2)

    abc

  • Option 3)

    -(a+b+c)

  • Option 4)

    2(a+b+c)

Answers (1)
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A admin

Value of determinants of order 3 -

-

\begin{vmatrix} a-b-c &2a & 2a\\ 2b & b-c-a &2b \\ 2c & 2c & c-a-b \end{vmatrix}

 R_{1}\rightarrow R_{1}+R_{2}+R_{3}

=\begin{vmatrix} a+b+c &a+b+c & a+b+c\\ 2b & b-c-a &2b \\ 2c & 2c & c-a-b \end{vmatrix}

C_{2}\rightarrow C_{2}-C_{1}               C_{3}\rightarrow C_{3}-C_{1}

=(a+b+c)\begin{vmatrix} 1 &0 & 0\\ 2b & -(a+b+c) &0 \\ 2c & 2c & c-a-b \end{vmatrix}

=(a+b+c)\cdot (a+b+c)^{2}

=> x=0 \:\: or\: \: -2(a+b+c)

 

 

 

 


Option 1)

-2(a+b+c)

Option 2)

abc

Option 3)

-(a+b+c)

Option 4)

2(a+b+c)

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