The solution of the differential equation $\frac{dy}{dx} = \frac{a + bx}{c + dy}$ represents a family of circles centered at the origin if...

Let $f$ be a differentiable function defined on $\left[0, \frac{\pi}{2}\right]$ such that $f(x) > 0$ and $f(x)+\int \limits_0^x f(t) \sqrt{1-\left(\log _e f(t)\right)^2} d t=e, \forall x \in\left[0, \frac{\pi}{2}\right]$. Then $\left(6 \log _{ e } f \left(\frac{\pi}{6}\right)\right)^2$ is equal to $.............$

$A$ piece of steel is heated to $100^{\circ}C$ and allowed to cool in a room. Which graph correctly represents the cooling process?

If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 3y = 0$, then the values of $m$ are

The orthogonal trajectories of the family of curves $x^{2/3} + y^{2/3} = a^{2/3}$,where $a$ is an arbitrary constant,are given by:

The bacteria increases at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $4$ hours,then the number of bacteria in $12$ hours will be: (in $N$)