The reaction of compound $$P$$ with $$C{H_3}MgBr$$ (excess) in $${\left( {{C_2}{H_5}} \right)_2}O$$ followed by addition of $${H_2}O$$ gives $$Q.$$ The compound $$Q$$ on treatment with $${H_2}S{O_4}$$ at $${0^ \circ }C$$ gives $$R.$$ The reaction of $$R$$ with $$C{H_3}COCl$$ in the presence of anhydrous $$AlC{l_3}$$ in $$C{H_2}C{l_2}$$ followed by treatment with $${H_2}O$$ producess compound $$S.$$ [$$Et$$ in compound $$P$$ is ethyl group]

The reactions, $$Q$$ to $$R$$ and $$R$$ to $$S,$$ are

If f : R $$ \to $$ R is a twice differentiable function such that f"(x) > 0 for all x$$ \in $$R, and $$f\left( {{1 \over 2}} \right) = {1 \over 2}$$, f(1) = 1, then

If y = y(x) satisfies the differential equation

$${8\sqrt x \left( {\sqrt {9 + \sqrt x } } \right)dy = {{\left( {\sqrt {4 + \sqrt {9 + \sqrt x } } } \right)}^{ - 1}}}$$

dx, x > 0 and y(0) = $$\sqrt 7 $$, then y(256) =

How many 3 $$ \times $$ 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of M^TM is 5?