A group of individuals contains ten people with blood group O, five with group A and five with group B. Give a formula for the probability that a random sample of size six will contain two people with each blood group.

An athlete conceals two performance enhancing tablets in a bottle containing eight vitamin tablets that are similar in appearance. If three tablets are selected at random for testing by drugs surveillance officials, what is the probability the cheating will be detected? You may assume that analysis is not error prone, i.e. if a performance enhancing... Read More

Locate the global maximum point on the curve $y=xe^{-x^2}$.

Find the inflexion point on $y=e^{-x^2}$.

Prove that $1 + x < e^x < \frac{1}{1-x}$ for all $x\in(0,1)$.

Prove that $e^a (b-a) < e^b – e^a < e^b(b-a)$ for all $a<b$ in $\mathbb{R}$.

Consider the non-zero real-valued function $exp$ on the reals such that $exp(a+b)=exp(a)exp(b)$ and $exp'(a)=exp(a)$ for all real values of $a$ and $b$. Prove that this function is strictly increasing.

Where does the following converge $\int_{1}^{\infty} e^{-\alpha x} dx$?

For what values of $\alpha$ will $\int_1^\infty \! x^{-\alpha} \, \mathrm{d}x$ converge?

Show that $f(x) = x^4 – 3x^2 + 1$ has two positive and two negative roots.