Question

Sand is pouring from the pipe at the rate of 12 cm³/s. The falling sand forms a cone on a ground in such a way that the height of cone is always one-sixth of radius of the base. How fast is the height of sand cone increasing when the height is 4 cm ?

Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.

Solve the following differential equation : $\mathrm{cosec}\,x\,\mathrm{log}\,y\,\frac{d y}{d x}+x^{2}y^{2}=0\,.$

Find the particular solution of the differential equation satisfying the given condition ${\frac{d y}{d x}}=y\tan x,$ given that y = 1, when x = 0.

Prove the following : $\begin{vmatrix}a² & bc & ac+c² \\ a²+ab & b² & ac \\ ab & b²+bc & c²\end{vmatrix}$ = $4a^{2}b^{2}c^{2}.$

Find the equation of tangent to the curve 4x² + 9y² = 36 at the point (3 cos θ, 2 sin ).

Find the particular solution of the differential equation $\frac{d y}{d x}\ \ +\ 2y\ \tan\ x\ =\ \sin\ x,$ given that y = 0 when $x={\frac{\pi}{3}}\,.$