The slope at any point of a curve $y=f(x)$ is given by $\frac{dy}{dx}=3x^2$ and it passes through $(-1,1)$. The equation of the curve is

The population $p$ of the city at time $t$ is given by $\frac{dp}{dt} = \frac{p}{2} - 100$. If the initial population at $t = 0$ is $100$,then $p$ is:

The money invested in a company is compounded continuously. If ₹ $400$ invested today becomes ₹ $800$ in $6$ years,then at the end of $30$ years,it will become (in ₹)

If it is mentioned that for a curve passing through $(3,4)$,the slope of the curve at any point is the reciprocal of twice the ordinate of that point,then that curve is a

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 $\%$)

If the tangent at a point $P(x, y)$ of a curve is perpendicular to the line that joins the origin with the point $P$,then the curve is