Question

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\cot x=2x+x^{2}\cot x,$ x ≠ 0. Given that y = 0, when $x={\frac{\pi}{2}}\ .$

Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.

Show that f : N → N, given by f(x) = |``x+1, if x is odd x- 1, if x is even is both one-one and onto.``|

If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

Find : $\large\textstyle\int$$\frac{x^{4}+1}{x\left(x^{2}+1\right)^{2}}d x$.

Find : $\large\textstyle\int\!{\large\frac{x^{2}+x+1}{(x^{2}+1)\left(x+2\right)}}\,d x.$

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls ? Given that : (i) the youngest is a girl. (ii) atleast one is a girl.