A) \[=-{{x}^{2}}\]
B) \[={{x}^{2}}\]
C) All of these
D) None of these
Correct Answer: A
Solution :
Sol. Sum of \[{{\operatorname{x}}^{2}}- {{y}^{2}}-1, \,{{y}^{2}}-{{x}^{2}}-1 and 1 -{{x}^{2}}-{{y}^{2}}\] \[=\,\,{{x}^{2}}-{{y}^{2}}-1+{{y}^{2}}-{{x}^{2}}-1+1-{{x}^{2}}-{{y}^{2}}\] On combining the like terms, \[=\,\,\,{{x}^{2}}-{{x}^{2}}-{{x}^{2}}-{{y}^{2}}+{{y}^{2}}-{{y}^{2}}-1-1+1\] \[=\,\,-{{x}^{2}}-{{y}^{2}}-1\] Now, subtract \[-(1 + {{y}^{2}})\,\,from\,\,-{{x}^{2}}- {{y}^{2}}-1\] \[=-{{x}^{2}}-{{y}^{2}}-1-\left[ -(1+{{y}^{2}}) \right]\] \[=\,\,\,-{{x}^{2}}-{{y}^{2}}-1+1+{{y}^{2}}\] \[=\,\,-{{x}^{2}}-{{y}^{2}}+{{y}^{2}}-1+1\] \[=-{{x}^{2}}\]
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