SAMPLE 23 Prove that (b + c)2 b? 2 ...
Question

# SAMPLE 23 Prove that (b + c)2 b? 2 22 (c + a) 2 a? b2 = 2abela + b +03 (a + b)2 ICBSE 2010 100

11th - 12th Class
Maths
Solution
57
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Let the given determinant be ( Delta ). Then ICBSE ( 2010,{ }^{~} 10 mathrm{C} ) ( Delta=left|begin{array}{ccc}(b+c)^{2} & a^{2} & a^{2} b^{2} & (c+a)^{2} & b^{2} c^{2} & c^{2} & (a+b)^{2}end{array}right| ) ( =left|begin{array}{ccc}(b+c)^{2}-a^{2} & 0 & a^{2} 0 & (c+a)^{2}-b^{2} & b^{2} c^{2}-(a+b)^{2} & c^{2}-(a+b)^{2} & (a+b)^{2}end{array}right| ) ( left[C_{1} rightarrow C_{1}-C_{3} text { and } C_{2} rightarrow C_{2}-C_{3}right] ) ( =left|begin{array}{ccc}(a+b+c)(b+c-a) & 0 & a^{2} 0 & (a+b+c)(c+a-b) & b^{2} (a+b+c)(c-a-b) & (a+b+c)(c-a-b) & (a+b)^{2}end{array}right| ) ( =(a+b+c)^{2} cdotleft|begin{array}{ccc}(b+c-a) & 0 & a^{2} 0 & c+a-b & b^{2} c-a-b & c-a-b & (a+b)^{2}end{array}right| ) ( left[operatorname{taking}(a+b+c) text { common from } C_{1} text { and } C_{2} text { both }right] ) ( =(a+b+c)^{2} cdotleft|begin{array}{ccc}(b+c-a) & 0 & a^{2} 0 & c+a-b & b^{2} -2 b & -2 a & 2 a bend{array}right| quadleft[R_{3} rightarrow R_{3}-left(R_{1}+R_{2}right)right] ) ( =(a+b+c)^{2}left[(b+c-a)left{(c+a-b) cdot 2 a b+2 a b^{2}right}+a^{2}{0+2 b(c+a-b) mid]right. ) ( left.=(a+b+c)^{2}[(b+c-a) cdot 2 a b(c+a-b+b)}+2 a^{2} b(c+a-b)right] ) ( =2 a b(a+b+c)^{2}{(b+c-a)(c+a)+a(c+a-b)} ) ( =2 a b(a+b+c)^{2} cdotleft{b c+a b+c^{2}+a c-a c-a^{2}+a c+a^{2}-a bright} ) ( =2 a b(a+b+c)^{2}left{b c+c^{2}+a cright}=2 a b c(a+b+c)^{3} ) Hence, ( Delta=2 a b c(a+b+c)^{3} )