Question

If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.

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

Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?

Solve the differential equation ${\frac{d y}{d x}}\ +\ y\ \cot\ x$= 2 cos x, given that y = 0 when $x={\frac{\pi}{3}}\,.$

Evaluate : $\large\textstyle\int\!{\frac{\ d x}{x(x^{5}+3)}}\,.$

Evaluate : ($\large\textstyle\int(3x-2){\sqrt{x^{2}+x+1}}\,d x~.$

The amount of pollution content added in air in a city due to x-diesel vehicles is given by p(x) = 0·005x³ + 0·02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added.