The population of a city increases at the rate of $3 \%$ per year. If the population is $p$ at time $t$,then the equation of $p$ in terms of $t$ is . . . . . . .

If the slope of the tangent on a curve at any point $(x, y)$ is equal to $\frac{y^2-x^2}{2xy}$,then the equation of the normal at the point $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ is

The assets of a person 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 ₹ $10$ lakhs and due to loss they reduce to ₹ $10000$ after $3$ years,then the number of years required for the person to be bankrupt will be

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half the quantity of ice melts in $20 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $40 \text{ minutes}$ the amount of ice left is $Kx_0$,then $K=$

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k (k > 0)$ is the constant of proportionality. The time during which an original mass of $1.5 \text{ gm}$ will disintegrate to a mass of $0.5 \text{ gm}$ is ...

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to