Question

Find the equation of the normal at the point (am², am³) for the curve ay² = x³.

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

If the function f : R → R is given by f(x) = $\frac{x+3}{3}$ and g : R → R is given by g(x) = 2x – 3, then find (i) fog (ii) gof Is f⁻¹ = g?

A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

Find the general solution of the following differential equation : $y-x{\frac{d y}{d x}}=a\left(y^{2}+{\frac{d y}{d x}}\right).$

Solve the differential equation $(x^{2}\ -\ y x^{2})d y$ + $(y^{2}+x^{2}y^{2})d x=0,$ given that y = 1 when x = 1.

Solve the following differential equation : $\left(1+e^{\frac{x}{y}}\right)d x+e^{\frac{x}{y}}\left(1-{\frac{x}{y}}\right)d y=0.$