Let $\mathrm{A}, \mathrm{B}$ and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16 x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle A B C$ be a conic $C_0$. Then three times the length of latus rectum of $\mathrm{C}_0$ is $\_\_\_\_$

Let $f$ be a twice differentiable function such that

$$ f(x)=\int_0^x \tan (t-x) d t-\int_0^x f(t) \tan t d t, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $$

Then $f^{\prime \prime}\left(\frac{\pi}{6}\right)+12 f^{\prime}\left(-\frac{\pi}{6}\right)+f\left(\frac{\pi}{6}\right)$ is equal to _______.

$$ \text { Match the LIST-I with LIST-II } $$

Choose the correct answer from the options given below:

Two cars $A$ and $B$ are moving in the same direction along a straight line with speeds $100 \mathrm{~km} / \mathrm{h}$ and $80 \mathrm{~km} / \mathrm{h}$, respectively such that car $A$ is moving ahead of car $B$. A person in car $B$ throws a stone with a speed $v$ so that it hits the car $A$ with a speed of $5 \mathrm{~m} / \mathrm{s}$. The value of $v$ is $\_\_\_\_$ $\mathrm{km} / \mathrm{h}$.