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 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

Radium decomposes at a rate proportional to the amount present. If half the original amount disappears in $1600$ years,then the percentage loss in $100$ years is (Given $\log 2 = 0.6931$ and $e^{-0.0433} = 0.9576$) (in $\%$)

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 population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt} = 0.5p(t) - 450$. If $p(0) = 850$,then the time at which the population becomes zero is:

The equation of a curve passing through $\left( 2, \frac{7}{2} \right)$ and having gradient $1 - \frac{1}{x^2}$ at $(x, y)$ is: