A) \[7\]
B) \[17\]
C) \[27\]
D) \[37\]
Correct Answer: C
Solution :
[c] \[\frac{{{\alpha }^{15}}+{{\beta }^{15}}+{{\alpha }^{11}}+{{\beta }^{11}}}{{{\alpha }^{13}}+{{\beta }^{13}}}\] \[=\frac{{{\alpha }^{15}}+{{\beta }^{15}}+{{\alpha }^{2}}{{\beta }^{2}}({{\alpha }^{11}}+{{\beta }^{11}})}{{{\alpha }^{13}}+{{\beta }^{13}}}\,\,\,\,\,\,\,(\alpha \beta =-1)\] \[=\frac{{{\alpha }^{15}}+{{\alpha }^{13}}{{\beta }^{2}}+{{\beta }^{15}}+{{\alpha }^{2}}{{\beta }^{13}}}{{{\alpha }^{13}}+{{\beta }^{13}}}\] \[=\frac{({{\alpha }^{13}}+{{\beta }^{13}})({{\alpha }^{2}}+{{\beta }^{2}})}{{{\alpha }^{13}}+{{\beta }^{13}}}\] \[={{\alpha }^{2}}+{{\beta }^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta =27\]
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