Question

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).

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

If C = 0·003x³ + 0·02x² + 6x + 250 gives the amount of carbon pollution in air in an area on the entry of x number of vehicles, then find the marginal carbon pollution in the air, when 3 vehicles have entered in the area.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.

Let A = {1, 2, 3, . ...., 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d =b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].

Find the particular solution of differential equation : ${\frac{d y}{d x}}=-{\frac{x+y\cos x}{1+\sin x}}$ given that y = 1 when x = 0.

Find the solution of the differential equation $(x d y-y d x)\,y\,\mathrm{sin}{\left({\frac{y}{x}}\right)}=(y d x+x d y)x\cos\biggl({\frac{y}{x}}\biggr).$