EXAMPLE 43 Solve for x: a+x a-x a-x...
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EXAMPLE 43 Solve for x: a+x a-x a-x a+x a-x a -x a -x| a-x= 0. a+x|

11th - 12th Class
Maths
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sounow Let the given determinant be ( Delta ). Then, [ begin{array}{l} Delta=left|begin{array}{lll} a+x & a-x & a-x a-x & a+x & a-x a-x & a-x & a+x end{array}right| =left|begin{array}{lll} 3 a-x & a-x & a-x 3 a-x & a+x & a-x 3 a-x & a-x & a+x end{array}right| quadleft[C_{1} rightarrowleft(C_{1}+C_{2}+C_{3}right)right] end{array} ] ( =(3 a-x) cdotleft|begin{array}{ccc}1 & a-x & a-x 1 & a+x & a-x 1 & a-x & a+xend{array}right| quadleft[operatorname{taking}(3 a-x) text { common from } C_{1}right] ) ( =(3 a-x) cdotleft|begin{array}{ccc}1 & a-x & a-x 0 & 2 x & 0 0 & 0 & 2 xend{array}right|left[R_{2} rightarrowleft(R_{2}-R_{1}right) text { and } R_{3} rightarrowleft(R_{3}-R_{1}right)right] ) ( left.=(3 a-x) cdot 1 cdotleft|begin{array}{cc}2 x & 0 0 & 2 xend{array}right| text { [expanding by } C_{1}right] ) [ =4(3 a-x) x^{2} ] ( therefore quad Delta=0 Leftrightarrow 4(3 a-x) x^{2}=0 ) ( Leftrightarrow x=0 quad ) or ( quad x=3 a ) Hence, solution set ( ={0,3 a} )
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