Let $D$ be the domain of a twice differentiable function $f$. For all $x \in D, f^{\prime \prime}(x)+f(x)=0$ and $f(x)=\int g(x) \, dx + \text{constant}$. If $h(x)={f(x)}^2+{g(x)}^2$ and $h(0)=5$,then $h(2015)-h(2014)$ is equal to

Let $f(x)$ be a non-negative differentiable function on $[0, \infty)$ such that $f(0)=0$ and $f^{\prime}(x) \leq 2 f(x)$ for all $x>0$. Then,on $[0, \infty)$:

$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

Let $f(x) = \begin{cases} e^{x-1}; x < 0 \\ x^2-5x+6; x \ge 0 \end{cases}$ and $g(x) = f(|x|) + |f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha + \beta$ is equal to ————

If $y = \frac{\tan x \cos^{-1} x}{\sqrt{1-x^2}}$, then the value of $\frac{dy}{dx}$ when $x = 0$ is:

In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column $I$ with the properties in Column $II$.

Column $I$	Column $II$
$(A)$ $f(x) = x|x|$	$(p)$ continuous in $(-1, 1)$
$(B)$ $f(x) = \sqrt{|x|}$	$(q)$ differentiable in $(-1, 1)$
$(C)$ $f(x) = x + [x]$	$(r)$ strictly increasing in $(-1, 1)$
$(D)$ $f(x) = |x - 1| + |x + 1|$	$(s)$ not differentiable at least at one point in $(-1, 1)$