Question

Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.

Find the general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0.

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}\;=1+x^{2}+y^{2}+x^{2}y^{2},$ given that y = 1, when x = 0.

Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..

Evaluate : $\large\textstyle\int\!{\frac{x+2}{\sqrt{x^{2}+5x+6}}}d x~.$

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 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.