A) 5
B) 18
C) 12
D) 36
Correct Answer: B
Solution :
We have, \[{{(\alpha +\beta +\gamma )}^{2}}={{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}+2(\alpha \beta +\beta \gamma +\gamma \alpha )\] \[\Rightarrow \] \[4=6+2(\alpha \beta +\beta \gamma +\gamma \alpha )\] \[\Rightarrow \] \[\beta \gamma +\gamma \alpha +\alpha \beta =-1\] ... (i) Also, \[{{\alpha }^{3}}+{{\beta }^{3}}+{{\gamma }^{3}}-3\alpha \beta \gamma \] \[=(\alpha +\beta +\gamma )({{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}-\alpha \beta -\beta \gamma -\gamma \alpha )\] \[\Rightarrow \] \[8-3\alpha \beta \gamma =2(6+1)\] \[\Rightarrow \] \[3\alpha \beta \gamma =8-14=-6\] or \[\alpha \beta \gamma =-2\] ... (ii) Now, \[{{({{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}})}^{2}}={{\alpha }^{4}}+{{\beta }^{4}}+{{\gamma }^{4}}\] \[+2({{\alpha }^{2}}{{\beta }^{2}}+{{\beta }^{2}}{{\gamma }^{2}}+{{\gamma }^{2}}{{\alpha }^{2}})\] \[=({{\alpha }^{4}}+{{\beta }^{4}}+{{\gamma }^{4}})+2[{{(\alpha \beta +\beta \gamma +\gamma \alpha )}^{2}}\] \[-2\alpha \beta \gamma (\alpha +\beta +\gamma )]\] \[\Rightarrow \] \[({{\alpha }^{4}}+{{\beta }^{4}}+{{\gamma }^{4}})=36\] \[-2[{{(-1)}^{2}}-2(-2)(2)]=18\]
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