Showing posts with label CMI. Show all posts
Showing posts with label CMI. Show all posts

Saturday, June 6, 2015

Problems : Limits

Problems for Indian Statistical Institute, Chennai Mathematical Institute, JEE Main and Advanced. $$$$ \[1.Evaluate: \lim_{x \to \infty} \frac{20+2\sqrt{x}+3\sqrt[3]{x}}{2+\sqrt{4x-3}+\sqrt[3]{8x-4}}\] \[2.Evaluate: \lim_{x \to \infty} \big( x \sqrt{x^2+a^2}-\sqrt{x^4+a^4}\big)\] \[3.Evaluate: \lim_{x \to \infty} x^3 \big\{ \sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2}x \big \}\] \[4.Evaluate: \lim_{x \to \infty} \sqrt{\frac{x-\cos^2 x}{x+\sin x}}\] \[5.Evaluate: \lim_{x \to \infty} [2\log(3x)-\log(x^2+1) ]\] 6. Let \( R_n =2+\sqrt{2+\sqrt{2+\dots+\sqrt{2}}}\) (n square roots signs). Then evaluate \(\lim_{n \to \infty} R_n \) $$$$ 7. If \(a_n = \bigg( 1+\frac{1}{n^2}\bigg)\bigg( 1+\frac{2^2}{n^2}\bigg)^2 \bigg( 1+\frac{3^2}{n^2}\bigg)^3 \dots \bigg( 1+\frac{n^2}{n^2}\bigg)^n \), then evaluate \[ \lim_{n \to \infty} a_n^{-\frac{1}{n^2}} \] $$$$ \[8.Evaluate \lim_{x \to \infty} \sqrt{x^2+x}-\sqrt{x^2+1}\] \[9. \lim_{x \to \frac{\pi}{2}} (\sin x)^{\tan x}\] \[10. \lim_{x \to 0} \frac{\cos x -1}{\sin^2 x}\] For PDF click here

Saturday, May 30, 2015

Matrices & Determinants

Mathematics Olympiad ~ Vinod Sing, Kolkata $Problem$ #1 $$ $$ If $A$ and $B$ are different matrices satisfying \( A^3 = B^3 \) and \(A^2B = B^2A\), find \(det(A^2+B^2)\) $$ $$ Since $A$ and $B$ are different matrices \( A-B \neq O \), Now \((A^2+B^2)(A-B) = A^3-A^2B+B^2A-B^3\) $$ $$ =$O$ since \(A^3 = B^3\) and \(A^2B = B^2A\) $$ $$ This shows that \((A^2+B^2)\) has a zero divisor, so it is not invertible hence \(det(A^2+B^2) = 0\)

Sunday, May 19, 2013

Integer Solution


Snippet of the problem is given below. To download follow the link http://sdrv.ms/17ORpFU or http://kolkatamaths.yolasite.com/problems-for-isi.php and download the file Integer Solution.pdf.
This type of problem is appropriate for ISI or CMI entrance test for B.Math or B.Stat, rather than a straight forward integer solution is has a restriction. 




Wednesday, September 5, 2012

Pigeon Hole Principle and Divisibility

Here is a good problem Pigeon Hole Principle and Divisibility of integers. These type of problems are important for  Olympiads, Indian Statistical Institute (ISI) and  Chennai Mathematical Institute (CMI) .
Get the pdf file here. Please leave your comment.


Saturday, August 20, 2011

Indian Statistical Institute

I have started a new page based on Indian Statistical Institute exam for B.Stat and B. Math. Most of the problems are previous years paper of ISI, for these type of question it is difficult to get solutions I hope this section will be helpful to students. Problems for ISI and CMI
google.com, pub-6701104685381436, DIRECT, f08c47fec0942fa0