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 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 population of towns $A$ and $B$ increases at a rate proportional to their population present at that time. At the end of the year $1984$,the population of both the towns was $20,000$. At the end of the year $1989$,the population of town $A$ was $25,000$ and that of town $B$ was $28,000$. The difference of populations of towns $A$ and $B$ at the end of $1994$ was

The rate at which the population of a city increases varies as the population present. Within the period of $30$ years,the population grew from $20$ lakhs to $40$ lakhs. Then,the population after a further period of $15$ years will be (Take $\sqrt{2} = 1.41$) (in $lakhs$)

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by the differential equation $\frac{dV(t)}{dt} = -k(T - t)$,where $k > 0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is:

$A$ radioactive substance has a half-life of $h \ days$. Its initial decay rate is given by (Note that at $t = 0, M = m_0$):