Let us learn MAKS

The Stirling's approximation formula: n ! ∼ 2 π n ( n e ) n ,   where the sign ~ means that the two quantities are asymptotic: their ratio tends to 1 as n tends to infinity.   This formula can be used to solve limits of sequences involving factorial.   Practice problems: 1. The following problem is asked in CSIR NET 2020(TN-PY) Find the limit: $\lim_{n \to \infty} \frac{((n+1)(n+2)\cdots(n+n))^{\frac{1}{n}}}{n}$  Note: Here we use ${n}^\frac{1}{n} \to 1.$ (Prove it also.) 2. $\lim_{n \to \infty}( \frac{(n!)}{(kn)^n})^{\frac{1}{n}}$ where $k \neq 0$      

Here we list some of the descriptive type questions asked in various competitive examinations on higher mathematics. We hope these questions will be helpful for the aspirants preparing for upcoming TN Arts and Science college recruitment exam.   (The list given is not a final one. Visit the page frequently, as the questions will be updated often.) You can type more questions(missed here) in comment box. They also will be updated.    (Updated on 10 Nov 2022) Algebra: 1. Prove that every group of order $p^2$ is abelian, where $p$ is a prime number. 2. P. T. any group of order 65 must be cyclic.  3. S. T. group of order 30 has a subgroup of order 15. Is it necessarily normal? ******************** Linear Algebra: 1.  If T is an nxn matrix that commutes with all nxn matrices, then prove that T is a scalar matrix. 2. Obtain the characteristic polynomial and minimal polynomial for a given matrix. 3. Let A and B be two 3 × 3 complex matrices. Show that A and B are similar if and only if χ