The population of a city increases at a rate proportional to the population at that time. If the population of the city increases from $20$ lakhs to $40$ lakhs in $30$ years,then after another $15$ years,the population is:

The area bounded by a curve,the $x$-axis,and the ordinate of some point $(x, y)$ of the curve is equal to the length of the corresponding arc of the curve from a fixed point. If the curve passes through the point $P(0, 1)$,then the equation of this curve is:

If it is mentioned that for a curve passing through $(3,4)$,the slope of the curve at any point is the reciprocal of twice the ordinate of that point,then that curve is a

The assets of a person are reduced in his business such that the rate of reduction is proportional to the square root of the existing assets. If the assets were initially ₹ $10,00,000$ and due to loss they reduce to ₹ $10,000$ after $3$ years,then the number of years required for the person to go bankrupt will be

The bacteria increase at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $8$ hours,then the number of bacteria in $24$ hours will be (in $N$)

$A$ normal is drawn at a point $P(x, y)$ of a curve $y=f(x)$. The normal meets the $X$-axis at $Q$. If the length $l(PQ) = k$,where $k$ is a constant,find the equation of the curve passing through $(0, k)$.