Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.

Consider the statements:
$$P:$$ There exists some $$x \in R$$ such that $$f\left( x \right) + 2x = 2\left( {1 + {x^2}} \right)$$
$$Q:\,\,$$ There exists some $$x \in R$$ such that $$2\,f\left( x \right) + 1 = 2x\left( {1 + x} \right)$$
Then

Let $$f\left( x \right) = {\left( {1 - x} \right)^2}\,\,{\sin ^2}\,\,x + {x^2}$$ for all $$x \in IR$$ and let
$$g\left( x \right) = \int\limits_1^x {\left( {{{2\left( {t - 1} \right)} \over {t + 1}} - In\,t} \right)f\left( t \right)dt} $$ for all $$x \in \left( {1,\,\infty } \right)$$.

Which of the following is true?

If $$f\left( x \right) = \int_0^x {{e^{{t^2}}}} \left( {t - 2} \right)\left( {t - 3} \right)dt$$ for all $$x \in \left( {0,\infty } \right),$$ then

Let $$PQR$$ be a triangle of area $$\Delta $$ with $$a=2$$, $$b = {7 \over 2}$$ and $$c = {5 \over 2}$$; where $$a, b,$$ and $$c$$ are the lengths of the sides of the triangle opposite to the angles at $$P.Q$$ and $$R$$ respectively. Then $${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$$ equals.