Try this problem based on convergence from TIFR 2013, Problem 15. Question: TIFR 2013 problem 15 True/False? Let \(x_1\in(0,1)\). For \(n>1\) define \(x_{n+1}=x_n-x_n^{n+1} \) Then \(lim_{n\to \infty} x_n \) exists. Hint: A bounded monotone sequence is convergent. Discussion: By induction, we can prove that the sequence is between 0 and 1 always. Suppose, \(x_n\in(0,1)\). Since […]
TIFR 2013 Problem 14 Solution has been written for TIFR entrance preparation series. A problem on uniform continuity from mathematical analysis.
Try this true/false problem from TIFR 2013 Problem 13 based on composite of Riemann integrable functions. Question: TIFR 2013 problem 13 True/False? Let \(f:[a,b]\to[c,d]\) and \(g:[c,d]\to\mathbb{R}\) be Riemann integrable functions. Then the composite \(gof\) is also Riemann integrable. Hint: Can the Thomae function(which is known to be integrable) be composed with some function to give […]
Try this problem, useful for Physics Olympiad, based on Mass over a Smooth Pulley. The Problem: One end of a string is attached to a rigid wall at point O, passes over a smooth pulley and carries a hanger S of mass M at its other end. Another object P of mass M is suspended […]
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 […]