Question

A bag contains (2n + 1) coins. It is known that (n – 1) of these coins have a head on both sides, whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x² – 3xy²) dx = (y³ – 3x²y) dy, where C is a parameter.

Using properties of determinants, prove that : $\begin{vmatrix}2y & y-z-x & 2y \\ 2z & 2z & z-x-y \\ x-y-z & 2x & 2x\end{vmatrix}$ = $(x+y+z)^{3}.$

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

Using properties of determinants, prove that $\begin{vmatrix}x+λ & 2x & 2x \\ 2x & x+λ & 2x \\ 2x & 2x & x+λ\end{vmatrix}$ = $(5x+{\lambda})({\lambda}-x)^{2}.$

If the function f : R → R be given by f(x) = x² + 2 and g : R → R be given by g(x) = ${\frac{x}{x-1}},x\neq1,$ find fog and gof and hence find fog(2) and gof(– 3).

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 ?