Question

Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.

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 particular solution of the differential equation $\left(\tan^{-1}y-x\right)d y=(1+y^{2})\,d x,$ given that when x = 0, y = 0.

Solve the differential equation : ${\frac{d y}{d x}}-3y\cot x=\sin2x\operatorname{given}y=2,{\mathrm{~when~}}x={\frac{\pi}{2}}.$

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

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

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.