Question: Let (C \subset \mathbb{ZxZ} ) be the set of integer pairs ((a,b)) for which the three complex roots (r_1,r_2,r_3) of the polynomial (p(x)=x^3-2x^2+ax-b) satisfy (r_1^3+r_2^3+r_3^3=0). Then the cardinality of (C) is A. (\infty) B. 0 C. 1 D. (1<|C|<\infty) Discussion: We have (r_1+r_2+r_3=-(-2)=2) (r_1r_2+r_2r_3+r_3r_1=a) and (r_1r_2r_3=-(-b)=b) Also, of the top of our head, we […]
Let's discuss a problem based on light through prisms from Kishore Vaigyanik Protsahan Yojana, KVPY 2010. Try it and then check your solution. The Problem: Light through Prisms White light is split into a spectrum by a prism and it is seen on a screen. If we put another identical inverted prism behind in contact, […]
Let's discuss a problem based on the velocity of efflux at the bottom of a tank. Try the problem yourself and read the solution here. The Problem: A large tank is filled with water. The total pressure at the bottom is (3.0atm). If a small hole is punched at the bottom, what is the velocity […]
TIFR 2014 Problem 9 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 […]
TIFR 2014 Problem 8 Solution has been written for TIFR entrance preparation series. A problem on function from real analysis.
Let's discuss a problem and know how to find the power consumption of electric heater. Try the problem and read the solution here. The Problem: An electric heater coonsists of a nichrome coil and under (220V) consuming (1KW) power. Part of its coil burned out and it was reconnected after cutting off the burnt portion. […]
TIFR 2014 Problem 7 Solution has been written for TIFR entrance preparation series. A problem on point wise limit from real analysis.
TIFR 2014 Problem 6 Solution has been written for TIFR entrance preparation series. A problem on Cauchy-Schwartz inequality for an inner product space.
Let's discuss a problem based on Removed Charge from Kishore Vaigyanik Protsahan Yojana, KVPY, 2010. First, try the problem yourself, then read the solution here. The Problem: (12) positive charges of magnitude (q) are placed on a circle of radius (R) in a manner that they are equally spaced. A charge (Q) is placed at […]