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 […]
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 […]
Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Area of the triangle from AMC-8, 2000, Problem-25. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on mixture from AMC-8, 2002.. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry based on Trapezium from PRMO , 2018. You may use sequential hints to solve the problem.
Try this beautiful problem from Probability from AMC-8, 2016 Problem 21. You may use sequential hints to solve the problem.
Try this beautiful problem from Pattern from AMC-8(2002) problem no 23.You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on Quadratic equation from PRMO 8, 2018. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2012 based on Arrangement. You may use sequential hints.
Try this beautiful problem from AMC-8, 2002, (Problem-20) based on area of Trapezoid.You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on Number theory fro AMC-8(2002) problem no 25.You may use sequential hints to solve the problem.