Pythagoras is famous. Even those who do not like mathematics, have heard of Pythagoras' Theorem (regarding right angled triangles). He lived about 2500 years ago and did about 2500 wonderful things (well may be a little less) but all that is not the subject matter of this note. We want to talk about the shattered […]
At the college (and university), on each topic, there are simply too many good books to choose from. It is easy to get overwhelmed. Then there are the tests. For semester exams, there are simply no substitute for class notes. But what about the entrance tests like GRE Math Subject Test or TIFR, M.Math and […]
Compute I = $latex (\int_e^{e^4}\sqrt{\log(x)}dx)$ if it is given that $latex (\int _1^2 e^{t^2} dt = \alpha )$ I = $latex ([x \sqrt{\log(x)}]_e^{e^4} - \int_e^{e^4} x \frac{1}{2 \sqrt{log(x)}} \frac {1}{x} dx )$ = $latex ([e^4 \sqrt {\log_e e^4} - e \sqrt {\log _e e}] - \frac{1}{2} \int_e^{e^4}\frac{1}{\sqrt{log(x)}} dx )$ = $latex (2 e^4 - e […]
P148. Show that there is no real constant c > 0 such that $latex (\cos\sqrt{x+c}=\cos\sqrt{x})$ for all real numbers $latex (x\ge 0)$.Solution: If the given equation holds for some constant c>0 then, f(x) = $latex (\cos\sqrt{x}-\cos\sqrt{x+c}=0)$ for all $latex (x\ge 0)$$latex (\Rightarrow 2\sin\frac{\sqrt{x+c}+\sqrt{x}}{2}\sin\frac{\sqrt{x+c}-\sqrt{x}}{2}=0)$Putting x=0, we note$latex (\Rightarrow\sin^2\frac{\sqrt{c}}{2}=0)$As $latex (c\not=0)$$latex (\sqrt{c}=2n\pi)$$latex (\Rightarrow c=4n^2\pi^2)$We put n=1 and […]
P164. Show that the area of the bounded region enclosed between the curves $latex (y^3=x^2)$ and $latex (y=2-x^2)$, is $latex (2\frac{2}{15})$. Solution: Note that $latex (y=x^{\frac{2}{3}})$ is an even function (green line). P165. Find the area of the region in the xy plane, bounded by the graphs of $latex (y=x^2)$, x+y = 2 and $latex […]
That is a good start. And a demanding one. All good starts are demanded by birthright. They ask you to do more in the subsequent days. This article is mainly targeted at class 12 pass-outs who are targeting I.S.I. 2012 (or those 12th graders who are able to devote some serious time to mathematics). Target […]
ABCD is a quadrilateral and P, and Q are mid-points of CD, and AB respectively. Let AP, DQ meet at X, and BP, CQ meet at Y . Prove that area of ADX + area of BCY = area of quadrilateral PXQY The number of ways in which three non-negative integers \( n_1, n_2, n_3 […]
I want to discuss the 'idea behind the famous Chinese Remainder Theorem. Let us leave the jargon and start our exploration by a problem. Find a number that leaves the remainder 1, when divided by 5, 7 and 13. Clearly, such a number can be found by trial. A stupid method is to check out […]
Indian Statistical Institute (I.S.I.), Chennai Mathematical Institute (C.M.I.) and Institute of Mathematics and Application (I.M.A.) can be regarded as three Indian institutions that provided world class mathematics course at undergraduate level. The B.Stat Course at I.S.I. is also world famous. The courses at C.M.I. and I.M.A. have computer science as second major. Each of these […]