The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

$$1 + {}^n{C_1}\cos \theta + {}^n{C_2}\cos 2\theta + ... + {}^n{C_n}\cos n\theta $$ equals

Let $$Q = \left[ {\matrix{ {\cos {\pi \over 4}} & { - \sin {\pi \over 4}} \cr {\sin {\pi \over 4}} & {\cos {\pi \over 4}} \cr } } \right]$$ and $$x = \left[ {\matrix{ {{1 \over {\sqrt 2 }}} \cr {{1 \over {\sqrt 2 }}} \cr } } \right]$$, then Q³x is equal to

If the function f : R $$ \to $$ R is defined by f(x) = (x² + 1)³⁵, $$\forall $$ x$$ \in $$R, then f is