The function defined by $f(x) = \begin{cases} |x - 3|, & x \ge 1 \\ \frac{1}{4}x^2 - \frac{3}{2}x + \frac{13}{4}, & x < 1 \end{cases}$ is

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi / 4$ meets the curve again at the point $Q$,then $|PQ|$ is equal to

Let $f(x) = \begin{cases} 3-x & \text{if } x < -3 \\ 6 & \text{if } -3 \leq x \leq 3 \\ 3+x & \text{if } x > 3 \end{cases}$. Let $\alpha$ be the number of points of discontinuity of $f$ and $\beta$ be the number of points where $f$ is not differentiable. Then $\alpha+\beta=$

Let $f(x)$ be a differentiable function, $f^{\prime}(x) > f(x)$ and $f(0) = 0$. Then

Match the items given in List-$I$ with those of the items of List-$II$:

List-$I$	List-$II$
$a$. If $y=|x|+|x-2|$, then at $x=2, \frac{dy}{dx}=$	$i$. $2$
$b$. If $f(x)=|\cos 2x|$, then $f^{\prime}(\frac{\pi}{4}+)=$	$ii$. $0$
$c$. If $f(x)=\sin(\pi[x])$, where $[\cdot]$ denotes the greatest integer function, then $f^{\prime}(1-)=$	$iii$. $-2$
$d$. If $f(x)=\log|x-1|, x \neq 1$, then $f^{\prime}(\frac{1}{2})=$	$iv$. Does not exist

If $y=(x+1)(x^2+1)(x^4+1)(x^8+1)$,then $\lim _{x \rightarrow-1} \frac{dy}{dx}=$