A) \[5-\sqrt{6}\]only
B) \[5+\sqrt{6}\]only
C) \[5-\sqrt{6}\]or \[5+\sqrt{6}\]
D) Neither\[5-\sqrt{6}\]nor \[5+\sqrt{6}\]
Correct Answer: C
Solution :
We know that in triangle larger the side larger the angle. Since angles \[\angle A,\,\angle B\] and \[\angle C\] are in AP. Hence \[\angle B={{60}^{o}}\] \[\cos B=\frac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac}\Rightarrow \frac{1}{2}=\frac{100+{{a}^{2}}-81}{20a}\] \[\Rightarrow \] \[{{a}^{2}}+19=10a\Rightarrow {{a}^{2}}-10a+19=0\] \[\] \[a=\frac{10\pm \sqrt{100-76}}{2}\Rightarrow a+c\sqrt{2}=5\pm \sqrt{6}\].
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