Try this problem from TIFR 2013 problem 21 based on no fixed homeomorphism. Question: TIFR 2013 problem 21 True/False? Every homeomorphism of the 2-sphere to itself has a fixed point. Hint: \(z= -z\) implies \(z=0\) Discussion: 2-sphere means \( S^2=\left \{(x,y,z)\in\mathbb{R}^3 | x^2+y^2+z^2=1 \right \} \). i.e, \( S^2=\left \{v\in\mathbb{R}^3 | ||v||=1 \right \} \). \(||.||\) […]
Try this Problem, useful for Physics Olympiad, based on the period of a planet. The Problem: Suppose that the gravitational force varies inversely as the \(n^{th}\) power of the distance. Then, the period of a planet in circular orbit of radius \(R\) around the sun will be proportional to (A) \(R^{\frac{n+1}{2}}\) (B)\(R^{\frac{n-1}{2}}\) (C) \(R^n\) (D) […]
Try this problem 20 from TIFR 2013 based on Sequence Limit - Fixed Point. This problem is an application of the Banach Fixed Point Theorem. Question: TIFR 2013 problem 20 True/False? Consider the function \(f(x)=ax+b\) with \(a,b\in\mathbb{R}\). Then the iteration \(x_{n+1}=f(x_n)\); \(n\ge0\) for a given \(x_0\) converges to \(\frac{b}{1-a}\) whenever \(0<a<1\). Hint: Banach Fixed Point […]
Try this problem, useful for Physics Olympiad based on Collision Between Two Balls. The Problem: A ball A moving with a velocity 5m/s collides elastically with another identical ball at rest such that the velocity of the A makes an angle of \(30^\circ\) with the line joining the centres of the balls. Then, (A) speed […]
This problem from TIFR 2013, Problem 19 discusses the example of a non-uniformly continuous function. Question: TIFR 2013 problem 19 True/False? Every differentiable function $f:(0,1) \to [0,1]$ is uniformly continuous. Hint: $\sin(1/x)$ is not uniformly continuous. However, range is not $[0,1]$. But its a simple matter of scaling. Discussion: Let $f(x)=\sin(\frac{1}{x})$ for all $x\in(0,1)$. For […]
Try this problem, useful for Physics Olympiad based on Constructing Parallel Plate Capacitor. The Problem: Constructing Parallel Plate Capacitor Suppose you are to construct a parallel plate capacitor of (1\mu F) by using paper sheets of thickness (0.05mm) as dielectric and a number of circular parallel metal foils connected alternately. If the dielectric constant of […]
Try this problem, useful for Physics Olympiad, based on Capacitance of a Spherical Capacitor. The Problem: A spherical capacitor has inner radius (a) and outer radius (b). It is filled with an inhomogeneous dielectric with permittivity $$ \epsilon=\epsilon_0K/r^2$$ for (a<r<b). The outer sphere is grounded and a charge is placed on the inner sphere. Find […]
Try this problem from TIFR 2013 problem 18 based on limit. Question: TIFR 2013 problem 18 True/False? Let \(P(x)= 1+x+ \frac{x^2}{2!} +... \frac{x^n}{n!} \) where n is a large positive integer. Then \(\lim_{x\to\infty} \frac{e^x}{P(x)} = 1 \) Hint: n really doesn't matter! Discussion: As \(x\) approaches 'infinity', both \(e^x\) and \(P(x)\) tends to 'infinity' (i.e, […]
TIFR 2013 Problem 17 Solution has been written for TIFR entrance preparation series. A problem on convergence of improper integral from real analysis.
TIFR 2013 Problem 16 Solution has been written for TIFR entrance preparation series. A problem on bounded sequence from real analysis.