EXAMPLE 40 Show that b+c cta a+b la...
Question

# EXAMPLE 40 Show that b+c cta a+b la b q+r r + p p + 9 = 2 p 9 y +z z+x x+y x y

11th - 12th Class
Maths
Solution
109
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soumon We have [ begin{aligned} text { LHS } &=left|begin{array}{lll} b+c & c+a & a+b q+r & r+p & p+q y+z & z+x & x+y end{array}right| &=2left|begin{array}{lll} a+b+c & c+a & a+b p+q+r & r+p & p+q x+y+z & z+x & x+y end{array}right| end{aligned} ] [applying ( C_{1} rightarrowleft(C_{1}+C_{2}+C_{3}right) ) and taking out 2 common from [ =2left|begin{array}{ccc} a+b+c & -b & -c p+q+r & -q & -r x+y+z & -y & -z end{array}right| quadleft[C_{2} rightarrowleft(C_{2}-C_{1}right), C_{3} rightarrowleft(C_{3}-C_{1}right)right] ] ( =2(-1)(-1) cdotleft|begin{array}{lll}a+b+c & b & c p+q+r & q & r x+y+z & y & zend{array}right| ) [taking out (-1) common from each one of ( C_{2} ) and ( C_{3} ) [ left.=2left|begin{array}{ccc} a & b & c p & q & r x & y & z end{array}right|=text { RHS } quad text { [applying } C_{1} rightarrow C_{1}-left(C_{2}+C_{3}right)right] ] Hence, ( L H S=R H S )