At any point $(x, y)$ on a curve,if the length of the subnormal is $(x - 1)$ and the curve passes through $(1, 2)$,then the curve is a conic. $A$ vertex of the curve is:

The population of a town increases at a rate proportional to the population at that time. If the population increases from $40,000$ to $80,000$ in $40$ years,then the population in another $40$ years will be (in $,000$)

Let $f:[0,2] \rightarrow R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)=\int_0^{x^2} f(\sqrt{t}) dt$ for $x \in [0,2]$. If $F'(x)=f'(x)$ for all $x \in (0,2)$,then $F(2)$ equals

Let $f$ be a non-negative function defined in $[0, \pi / 2]$,$f^{\prime}$ exists and is continuous for all $x$,and $\int_0^x \sqrt{1-\left(f^{\prime}(t)\right)^2} dt = \int_0^x f(t) dt$ with $f(0) = 0$. Then

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