Let $f : [0,1] \to R$ be such that $f(xy) = f(x)f(y)$ for all $x, y \in [0,1],$ and $f(0) \ne 0.$ If $y = y(x)$ satisfies the differential equation $\frac{dy}{dx} = f(x)$ with $y(0) = 1,$ then $y\left( \frac{1}{4} \right) + y\left( \frac{3}{4} \right)$ is equal to

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

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half of the initial quantity of ice melts in $15 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $30 \text{ minutes}$ the amount of ice left is $k x_0$,then the value of $k$ is:

At any point $(x, y)$ of a curve,the slope of the tangent is twice the slope of the line segment joining the point of contact to the point $(-4, -3)$. Find the equation of the curve given that it passes through $(-2, 1)$.

Suppose a continuous function $f:(0, \infty) \rightarrow R$ satisfies $f(x)=2 \int_0^x t f(t) d t+1, \forall x \geq 0$. Then,$f(1)$ equals

If $y = e^{-mx}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0$, then the values of $m$ are