TIFR 2014 Problem 20 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front […]
Let's discuss a problem from TIFR 2014 Problem 18. Question: What is the last digit of (97^{2013})? Discussion: (97 \equiv -3 (\mod 10 ) ) (97^2 \equiv (-3)^2 \equiv -1 (\mod 10 ) ) (97^3 \equiv (-1)\times (-3) \equiv 3 (\mod 10 ) ) (97^4 \equiv (3)\times (-3) \equiv 1 (\mod 10 ) ). Now, (2013=4\times […]
A ring of radius (R) carries a linear charge density ($\lambda$). It is rotating with angular speed ($\omega$). What is the magnetic field at the centre? Discussion: Linear charge density $$ \lambda=\frac{Q}{2\pi R} $$ When the ring is rotated about the axis, the motion of the electrons in a circular orbit is equivalent to a […]
A rigid triangular molecule consists of three non-collinear atoms joined by rigid rods. The constant pressure molar specific heat (C_p) of an ideal gas consisting of such molecules is (a) (6R) (b) (5R) (c) (4R) (d) (3R) Degrees of freedom are the number of independent parameters that define its configuration If (N) be the number […]
A cylinder contains (16g) of (O_2). The work done when the gas is compressed to (75\%) of the original volume at constant temperature of (27^\circ) is ________. Discussion: Given mass of (O_2), m=(16g) Number of moles of (O_2), $$n=\frac{m}{M}$$ where (M)=molecular weight of (O_2)=(32g) $$n=\frac{16}{32}=\frac{1}{2}$$ Temperature (T=27^\circ=300K) If (V_1) be the original volume and (V_2) […]
Question: How many subgroups does the group (\mathbb{Z\times Z}) have? A. 1 B. 2 C. 3 D. infinitely many. Discussion: How many subgroups does (\mathbb{Z}) have? Well, for every (m\ge 0) we have a subgroup (m\mathbb{Z}). And these are all distinct. So there are infinitely many subgroups of (\mathbb{Z}). So how many subgroups does (\mathbb{Z}\times {\bar{0}}) […]
TIFR 2014 Problem 15 Solution has been written for TIFR entrance preparation series. A problem on maps for discrete space from mathematical analysis.
Cheenta Opportunity is an initiative for the benefit of Cheenta Olympiad candidates. We dig up opportunities and resources available all around the world for our students. Here we are exploring UCLA Full Scholarship Program - MUMS. University of California, Los Angles is one of the leading universities in the world. Its mathematics department is held in […]
TIFR 2014 Problem 14 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects. The image is a front […]
Try this Problem based on Doppler Effect where we find the tone of the whistle and speed of the Train. First, do it yourself and then read the solution. The Problem: Doppler Effect A train passes through a station with constant speed. A stationary observer at the station platform measures the tone of the train […]