A) \[x(x-p)(x-q)\]
B) \[(x-p)(x-q)(x+p+q)\]
C) \[(p-q)(x-q)(x-p)\]
D) \[pq(x-p)(x-q)\]
Correct Answer: B
Solution :
Given, \[\left| \begin{matrix} x & p & q \\ p & x & q \\ p & q & x \\ \end{matrix} \right|\] \[=x[{{x}^{2}}-{{q}^{2}}]-p[px-pq]+q[pq-px]\] \[={{x}^{3}}-x{{q}^{2}}-{{p}^{2}}x+{{p}^{2}}q+p{{q}^{2}}-pqx\] \[=x(x-q)(x+q)-{{p}^{2}}(x-q)-pq(x-q)\] \[=(x-q)[{{x}^{2}}+xq-{{p}^{2}}-pq]\] \[=(x-q)(x-p)(x+p+q)\]
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