| If \[{{x}^{3}}+{{y}^{3}}=9\]and \[x+y=3,\] then the value of \[{{x}^{4}}+{{y}^{4}}\]is |
A) 81
B) 32
C) 27
D) 17
Correct Answer: D
Solution :
| Given, \[(d)x+y=3\] |
| On cubing both sides, |
| \[{{(x+y)}^{3}}=27\] |
| \[{{x}^{3}}+{{y}^{3}}+3xy\,\,(x+y)=27\] |
| \[\Rightarrow \] \[9+3xy\,\,(3)=27\] |
| \[\Rightarrow \] \[9xy=18\]\[\Rightarrow \]\[xy=2\] |
| Now, \[(x+y)=3\] |
| On squaring both sides, |
| \[{{x}^{2}}+{{y}^{2}}+2xy=9\] |
| \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}+4=9\] |
| \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}=5\] |
| Again, squaring both sides, |
| \[{{x}^{2}}+{{y}^{2}}+2{{x}^{2}}{{y}^{2}}=25\] |
| \[{{x}^{4}}+{{y}^{4}}=25-2\,\,{{(xy)}^{2}}=25-2\,\,{{(2)}^{2}}\] |
| \[=25-8=17\] |
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