Number of values of $m \in N$ for which $y = e^{mx}$ is a solution of the differential equation $D^3y - 3D^2y - 4Dy + 12y = 0$ is

$A$ spherical balloon expands at a rate proportional to its surface area. Initially, its radius is $2 \text{ cm}$ and $5 \text{ minutes}$ later it increases to $7 \text{ cm}$. What will be the surface area of the spherical balloon after $12 \text{ minutes}$ (in $\text{ cm}^2$)?

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

$A$ body at an unknown temperature is placed in a room which is held at a constant temperature of $30^{\circ} F$. If after $10 \text{ minutes}$ the temperature of the body is $0^{\circ} F$ and after $20 \text{ minutes}$ the temperature of the body is $15^{\circ} F$,then the expression for the temperature of the body at any time $t$ is

The curve passing through $(3, 0)$ and satisfying the differential equation $(9 - x^2)(\frac{dy}{dx})^2 = 9 - y^2$ represents:

The solution of the differential equation $\frac{dy}{dx} + \frac{x}{y} \cdot \frac{x^2+y^2-1}{2(x^2+y^2)+1} = 0$ is