Try this true/false problem from TIFR 2013 Problem 12 based on Limit. Question: TIFR 2013 problem 12 True/False? $$ \lim_{x\to 0} \frac{sin(x^2)}{x^2}sin(\frac{1}{x}) = 1$$ Discussion: As \(x\) goes to zero, \(x^2\) also goes to zero, and consequently, since $$ \lim_{y\to 0} \frac{sin(y)}{y}=1 $$ we have $$ \lim_{x\to 0} \frac{sin(x^2)}{x^2}= 1$$. If the statement were true, then […]
Try this beautiful problem, useful for Physics Olympiad, based on Button on a Rotating Platform. The Problem: A small button placed on a horizontal rotating platform with diameter (0.320m) will revolve with the plate when it is brought up to a speed of (40rev/min), provided the button is no more than (0.150m) from the axis. […]
Try this beautiful problem, useful for the Physics Olympiad, based on Minimum Speed of a Rotating Pail of Water. The Problem: Minimum Speed of a Rotating Pail of Water You tie a cord to a pail of water and you swing the pail in a vertical circle of radius (0.6m). What minimum speed must you […]
TIFR 2013 Problem 11 Solution has been written for TIFR entrance preparation series. A problem on convergence of cauchy sequence from mathematical analysis.
Try this beautiful problem, useful for Physics Olympiad, based on Tossing a Coin in a Moving Train. The Problem: Tossing a Coin in a Moving Train A person is sitting in a moving train and is facing the engine. He tosses up a coin that falls behind him. He concludes that the train is moving. […]
Try this problem from TIFR-2013 problem 1 based on the Countable group-Uncountable Subgroup. Question: TIFR-2013 problem 1 True/False? Every countable group \(G\) has only countable number of distinct subgroups. Hint: How many subgroups does \(\mathbb{Q}\) have? Discussion: A very natural way to go from countably infinite set to uncountable set is to take its power […]
Try this problem, useful for Physics Olympiad based on Vertical Motion. The Problem: Vertical Motion A boy is standing on top of a tower of height 85m and throws a ball in the vertically upward direction at a certain speed. If 5.25 secs later he hears the ball hitting the ground, then the speed with […]
Try this problem useful for Physics Olympiad based on Relative motion. The Problem: Relative motion Problem A 40kg boy is standing on a plank of mass 160Kg. The plank originally at rest, is free to slide on a smooth frozen lake. The boy walks along the plank at a constant speed of 1.5m/s relative to […]
Let's solve a beautiful and tricky integral problem. The Problem: Let $$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$ J$$=$$ \int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$. Find the value of (J-I). Solution: $$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$J$$= $$\int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$ Let (e^x)=(z) $$ J-I$$=$$\int\frac{e^x(e^{2x-1})}{e^{4x}+e^{2x}+1}dx$$=$$\int\frac{z^2-1}{z^4+z^2+1}dz $$ $$ =\frac{1}{2}ln\frac{(e^x+e^-x-1)}{(e^x+e^-x+1)}+c$$ (where c is a constant of integration)
A canoe has a velocity of 0.40m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50m/s east relative to the earth. Find the velocity (magnitude and direction ) of the canoe relative to the river. Discussion: We apply the relative velocity relation. The relative velocities are (\vec{v_{CE}}), the canoe […]