In a bank,the principal increases continuously at the rate of $r \%$ per year. Find the value of $r$ if $Rs \, 100$ doubles itself in $10$ years $\left(\log _{e} 2=0.6931\right)$. (in $\%$)

The rate of increase of the population of a country is proportional to the number present. If the population doubles in $50$ years,then the time taken by it to become four times itself is (in $years$)

$A$ spherical mothball has an initial radius of $3 \text{ cm}$. Due to evaporation, the radius of the ball reduces to $1 \text{ cm}$ in $4 \text{ months}$. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface area?

If the half-life of a substance is $5$ years,then the total amount of the substance left after $15$ years,when the initial amount is $64$ gms,is: (in $gm$)

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 solution of the equation $\frac{x^2 d^2y}{dx^2} = \ln x$,given that at $x = 1$,$y = 0$ and $\frac{dy}{dx} = -1$,is: