Question

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

Solve the differential equation $x\;\frac{d y}{d x}\;+y=$ x cos x + sin x, given that y = 1 when $x={\frac{\pi}{2}}$

Solve the following differential equation. ${\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}}+x y{\frac{d y}{d x}}\,=0$

Find the equations of tangents to the curve 3x² – y² = 8, which passes through the point $\left({\frac{4}{3}},\,0\right).$

Consider the experiment of tossing a coin. If the coin shows head, toss is done again, but if it shows tail, then throw a die. Find the conditional probability of the events that ‘the die shows a number greater than 4’, given that ‘there is atleast one tail’.

Find the value of p for which the curves x² = 9p(9 – y) and x² = p(y + 1) cut each other at right angles.

Show that the function f : R {x ∈ R – 1 < x < 1} defined by f(x) =${\frac{x}{1+\left|\,x\right|}},$ x ∈ R is one-one and onto function. Hence find f⁻¹(x).