If $f: R \rightarrow R$ defined by $f(x) = \begin{cases} a^2 \cos ^2 x+b^2 \sin ^2 x, & x \leq 0 \\ e^{ax+b}, & x>0 \end{cases}$ is a continuous function, then:

$f(x) = \begin{cases} \frac{x-4}{|x-4|} + a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|} + b, & x > 4 \end{cases}$
If $f(x)$ given above is continuous at $x=4$,then find the values of '$a$' and '$b$'.

If $f(x) = |x|$,then $f(x)$ is

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x) = \begin{cases} b^2 \sin \left(\frac{\pi}{2} \left[\frac{\pi}{2}(\cos x + \sin x) \cos x\right]\right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$ is continuous at $x = 0$, then $a^2 + b^2$ is equal to

If the function $f(x) = \begin{cases} \frac{\sqrt{1+px}-\sqrt{1-px}}{x}, & \text{if } -1 \leq x < 0 \\ \frac{2x+1}{x-2}, & \text{if } 0 \leq x \leq 1 \end{cases}$ is continuous in the interval $[-1, 1]$,then $p = $

If $f: R \rightarrow R$ defined as $f(x) = \frac{x^3+2x^2+x+2}{x^2+x-2}$ (when $x \neq -2$) is continuous at $x = -2$, then $f(-2)$ is equal to