Let $f : N \rightarrow R$ be a function such that $f(x+y)=2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which

$\sum \limits_{k=1}^{10} f(\alpha+k)=\frac{512}{3}\left(2^{20}-1\right)$ holds, is

Let $f(x)$ and $g(x)$ be two functions given by $f\left( x \right) = \frac{{2\sin \pi x}}{x}$ and $g\left( x \right) = f\left( {1 - x} \right) + f\left( x \right).$ If $g\left( x \right) = kf(\frac{x}{2})f\left( {\frac{{1 - x}}{2}} \right)$,then the value of $k$ is

If function $f : R \to S, f(x) = (\sin x -\sqrt 3 \cos x+1)$ is onto, then $S$ is equal to

The domain of $f(x) = [\sin x] \cos \left( {\frac{\pi }{{[x - 1]}}} \right)$ is (where $[.]$ denotes $G.I.F.$)

If the domain and range of $f(x){ = ^{9 - x}}{C_{x - 1}}$ contains $m$ and $n$ elements respectively, then 

Let $f(\theta)$ is distance of the line $( \sqrt {\sin \theta } )x + (  \sqrt {\cos  \theta })y +1 = 0$ from origin. Then the range of $f(\theta)$ is -