If $x=e^{y+e^{y+e^{y+\ldots}}}$,then $\frac{d y}{d x}=$

$A$ man running round a race-course notes that the sum of the distances of two flag-posts from him is always $10 \ m$ and the distance between the flag-posts is $8 \ m$. The area of the path he encloses in square metres is (in $\pi$)

The solution of $x \frac{dy}{dx} = y + x e^{y/x}$ with $y(1) = 0$ is

Which one of the following pairs of terms/names mean one and the same thing?

For any integer $n \geq 1$, the number of positive divisors of $n$ is denoted by $d(n)$. Then, for a prime $P$, $d(d(d(P^7)))$ is equal to

Two beams of light having intensities $I$ and $4I$ interfere to produce a fringe pattern on a screen. The phase difference between the beams is $\pi / 2$ at point $A$ and $\pi$ at point $B$. Then the difference between the resultant intensities at $A$ and $B$ is (in $I$)