If $f(x) = \frac{{{{\cos }^2}x + {{\sin }^4}x}}{{{{\sin }^2}x + {{\cos }^4}x}}$ for $x \in R$, then $f(2002) = $

Consider the function $f (x) = x^3 - 8x^2 + 20x -13$
Number of positive integers $x$ for which $f (x)$ is a prime number, is

Let $\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, - \,1),$ where the function $f$ satisfies $f(x + y) = f(x) f(y)$ for all natural numbers $x, y$ and $f(1) = 2.$ Then the natural number $‘ a '$ is

If the domain of the function $f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)$ is $[\alpha, \beta) \cup(\gamma, \delta]$, then $|3 \alpha+10(\beta+\gamma)+21 \delta|$ is equal to $.......$.

Consider a function $f:\left[0, \frac{\pi}{2}\right]$ $ \rightarrow$ $R$ given by $f(x)=\sin x$ and $g:\left[0, \frac{\pi}{2}\right] $ $\rightarrow$ $R$ given by $g(x)=\cos x .$ Show that $f$ and $g$ are one-one, but $f\,+\,g$ is not one-one.

For a real number $x,\;[x]$ denotes the integral part of $x$. The value of $\left[ {\frac{1}{2}} \right] + \left[ {\frac{1}{2} + \frac{1}{{100}}} \right] + \left[ {\frac{1}{2} + \frac{2}{{100}}} \right] + .... + \left[ {\frac{1}{2} + \frac{{99}}{{100}}} \right]$ is