A) 0
B) 12 xyz
C) 24 xyz
D) 36 xyz
Correct Answer: C
Solution :
| [c] \[{{(x+y+z)}^{3}}-{{(y+z+x)}^{3}}-\,{{[{{(z+x+y)}^{3}}+(-z+x)+y)}^{3}}]\]Rearranging, \[{{(y+z+x)}^{3}}-{{(y+z-x)}^{3}}-\,{{[{{(z+x+y)}^{3}}+(-z+x)+y)}^{3}}]\] Let \[(x+y+z)=a,\]\[(z+x-y)=c,\] \[(y+z+-x)=b,\]\[(-\,z+x+y)=d\] Then, \[{{a}^{3}}-{{b}^{3}}-[{{c}^{3}}+{{d}^{3}}]\] \[=\underbrace{(a-b)({{a}^{2}}+ab+{{b}^{2}})}_{1st}-\underbrace{[(c+d)({{c}^{2}}+cd+{{d}^{2}})]}_{2nd}\]Solving 1st, \[(x+y+z-y-z+x)[(y+z+{{x}^{2}})+\,(y+z+x)\] \[(y+z-x)+{{(y+z-x)}^{2}}]\] \[=2x\,[{{(y+z+x-y-z+x)}^{2}}+2\,(y+z+x)\] \[(y+z-x)+{{(y+z)}^{2}}-{{x}^{2}}]\] [adding and subtracting\[2(y+z+x)(y+z-x)]\] \[=2x\,[{{x}^{2}}+2\{{{(y+z)}^{2}}-{{x}^{2}}\}+{{(y+z)}^{2}}-{{x}^{2}}]\] \[=2x\,[{{x}^{2}}+3{{(y+z)}^{2}}-3{{x}^{2}}]\] \[=2x\,[3{{(y+z)}^{2}}-2{{x}^{2}}]\] \[=2x\,[3{{y}^{2}}+3{{z}^{2}}+6yz-2{{x}^{2}}]\] Solving 2nd, \[(z+x-y-z+x+y){{\{z+x-y)}^{2}}-(z+x-y)\] \[(-z+x+y)+{{(-z+x+y)}^{2}}\}\] \[=2x\,{{[z+x-y-z+x+y)}^{2}}-2\,(z+x-y)\] \[(-z+x+y)-(z+x+-y)(-z+x+y)]\] [adding and subtracting\[2\,(z+x+-y)(-z+x+y)]\] \[=2x\,[{{(2x)}^{2}}-3(z+x-y)(-z+x+y)]\] \[=2x\,[4{{x}^{2}}-3\,(-{{z}^{2}}+zx+yz+xz+{{x}^{2}}+xy\] \[+yz-yx-{{y}^{2}})]\] \[=2x\,(4{{x}^{2}}+3{{z}^{2}}+6yz+3{{y}^{2}}-2{{x}^{2}})\] On combing both, sides, we get \[6x{{y}^{2}}+6x{{z}^{2}}+12xyz-4{{x}^{3}}\] \[-\,\,6x{{y}^{2}}-6x{{z}^{2}}+12xyz\] \[=24xyz\] |
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