| If \[x=5,\]\[y=6\]and \[z=-\,\,11,\]then the value of \[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}\] is |
A) \[-\,\,890\]
B) \[-\,\,970\]
C) \[-\,\,870\]
D) \[-\,\,990\]
Correct Answer: D
Solution :
| Given, \[x=5,\]\[y=6\]and \[z=-\,\,11\] |
| \[\therefore \] \[x+y+z=5+6-11\] |
| \[\Rightarrow \] \[x+y+z=0\] (i) |
| Now, \[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz\] |
| \[=(x+y+z)({{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-xz)\] |
| \[\Rightarrow \]\[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=0\] [from Eq. (i)] |
| \[\Rightarrow \]\[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}=3xyz\] |
| \[\Rightarrow \]\[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}=3\,\,(5)(6)(-11)\] |
| \[\therefore \]\[{{x}^{3}}+{{y}^{3}}+{{z}^{3}}=-\,\,990\] |
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