With what minimum velocity should a block of mass $m$ be projected from the left end $A$ towards end $B$ such that it reaches the other end $B$ of a conveyor belt of length $L$ moving with a constant velocity $v$ in the opposite direction? The friction coefficient between the block and the belt is $\mu$.

$\int_{-1}^1 \frac{\cosh x}{1+e^{2 x}} d x$ is equal to :

If the lines $x^2+2xy-35y^2-4x+44y-12=0$ and $5x+\lambda y-8=0$ are concurrent,then the value of $\lambda$ is

Match the following (Take the relative strength of the strongest fundamental forces in nature as $1$):
| List-$I$ (Fundamental forces in nature) | List-$II$ (Relative strength) |
| :--- | :--- |
| $(A)$ Strong nuclear force | $(e)$ $10^{-2}$ |
| $(B)$ Weak nuclear force | $(f)$ $1$ |
| $(C)$ Electromagnetic force | $(g)$ $10^{10}$ |
| $(D)$ Gravitational force | $(h)$ $10^{-13}$ |
| | $(i)$ $10^{-39}$ |
The correct match is:

$\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}$ is equal to

Let $M$ be the foot of the perpendicular from a point $P$ on the parabola $y^2=8(x-3)$ onto its directrix and let $S$ be the focus of the parabola. If $\triangle SPM$ is an equilateral triangle,then $P$ is equal to