If P = 7 + 4√3 and PQ = 1, then what is the value of $$\frac{1}{{{P^2}}} + \frac{1}{{{Q^2}}}?$$

Solution (By Examveda Team)

$$\eqalign{ & P = 7 + 4\sqrt 3 \cr & PQ = 1,\,Q = \frac{1}{P} \cr & Q = \frac{1}{{7 + 4\sqrt 3 }} = 7 - 4\sqrt 3 \cr & P + Q \cr & = 7 + 4\sqrt 3 + 7 - 4\sqrt 3 \cr & = 14 \cr & \Rightarrow \frac{1}{{{P^2}}} + \frac{1}{{{Q^2}}} \cr & = \frac{{{P^2} + {Q^2}}}{{{{\left( {PQ} \right)}^2}}} \cr & = \frac{{{{\left( {P + Q} \right)}^2} - 2PQ}}{{{{\left( {PQ} \right)}^2}}} \cr & = {\left( {14} \right)^2} - 2 \cr & = 196 - 2 \cr & = 194 \cr} $$