Category Archives: IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha)

Theory of Equations: III: IITJEE maths: algebra

January 9, 2019 – 4:24 am

may overlap a bit with previous lecture(s)…

 

By Nalin Pithwa | Also posted in algebra, IITJEE Mains, ISI Kolkatta Entrance Exam, KVPY | Tagged , | Comments (0)

Applications of Derivatives: IITJEE Maths tutorial problem set: III

October 23, 2018 – 9:43 pm

Slightly difficult questions, I hope, but will certainly re-inforce core concepts:

  1. Prove that the segment of the tangent to the curve y=c/x which is contained between the co-ordinate axes, is bisected at the point of tangency.
  2. Find all tangents to the curve y=\cos{(x+y)} for -\pi \leq x \leq \pi that are parallel to the line x+2y=0.
  3. Prove that the curves y=f(x), where f(x)>0 and y=f(x)\sin(x), where f(x) is a differentiable function, have common tangents at common points.
  4. Find the condition that the lines x\cos{\alpha} + y \sin{\alpha}=p may touch the curve (\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1.
  5. If p_{1} and p_{2} are lengths of the perpendiculars from origin on the tangent and normal to the curve x^{2/3} + y^{2/3}=a^{2/3} respectively, prove that 4p_{1}^{2}+p_{2}^{2}=a^{2}.
  6. Show that the curve x=1-3t^{2}, y=t-3t^{3} is symmetrical about x-axis and has no real points for x>1. If the tangent at the point t is inclined at an angle \psi to OX, prove that 3t = \tan{\psi} + \sec{\psi}. If the tangent at P(-2,2) meets the curve again at Q, prove that the tangents at P and Q are at right angles.
  7. A tangent at a point P_{1} other than (0,0) on the curve y=x^{3} meets the curve again at P_{2}. The tangent at P_{2} meets the curve at P_{3} and so on. Show that the abscissae of P_{1}, P_{2}, \ldots, P_{n} form a GP. Also, find the ratio of area \frac{\Delta P_{1}P_{2}P_{3}}{area \hspace{0.1in} P_{2}P_{3}P_{4}}.
  8. Show that the square roots of two successive natural numbers greater than N^{2} differ by less than \frac{1}{2N}.
  9. Show that the derivative of the function f(x) = x \sin {(\frac{\pi}{x})}, when x>0, and f(x)=0 when x=0 vanishes on an infinite set of points of the interval (0,1).
  10. Prove that \frac{x}{(1+x)} < \log {(1+x)} < x for x>0.

More later, cheers,

Nalin Pithwa.

By Nalin Pithwa | Comments (0)

Primality testing: recap

September 11, 2018 – 4:13 am

via Primality testing: recap

By Nalin Pithwa | Comments (0)

How to solve equations: Dr. Vicky Neale: useful for Pre-RMO or even RMO training

April 25, 2018 – 12:51 pm

Dr. Neale simply beautifully nudges, gently encourages mathematics olympiad students to learn to think further on their own…

By Nalin Pithwa | Also posted in IITJEE Foundation mathematics, Pre-RMO, RMO, RMO Number Theory | Tagged , | Comments (0)

Happy Pie with Pi…

March 14, 2018 – 12:05 pm

via Everything you wanted to know about pi but were afraid to ask

By Nalin Pithwa | Comments (0)

Paul Erdos, Mathematics, Russia and USA:

February 2, 2018 – 5:42 am

It is true …universally, including India…

Hats off to the “Intellect of the Wise Mathematicians”, late, adorable professor of mathematics, Paul Erdos.

— humble tribute …from Nalin Pithwa.

Paul Erdos, most profound quote ever made by any mathematician

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Birch and Swinnerton Dyer Conjecture from the inimitable Manjul Bhargava

January 25, 2018 – 2:45 pm

via Birch and Swinnerton-Dyer Conjecture

By Nalin Pithwa | Comments (0)

How to deal with failure

November 24, 2017 – 1:17 am

via How to deal with failure?

By Nalin Pithwa | Comments (0)

Some light moments with an IITJEE foundation math student

July 20, 2017 – 8:35 pm

I have a bright kid working towards his IITJEE Foundation math. Some days back I had suggested to him to solve some hard word problems based on simultaneous equations from a very old classic text. He started on his own, almost with some guidance from me. Until he attacked very well — those “age” kind of problems. Father’s age vs. son’s age, etc. But, in this case, he suddenly sprang to his feet: He almost yelled, “Sir! Please check my solution! I am getting the answer as husband’s age is 48 and wife’s age is 23!” Actually, I too was a bit shocked; but, I checked his calculations in detail; they were mathematically correct. It suddenly flashed in my head:

“Do you know Vedant? Mathematics is not human! It has no emotions, no feelings, whatsoever! I told him the following quote of Bertrand Russell: ” Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty, cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.””

And, of course, equations//numbers don’t lie! 🙂 🙂 🙂

Nalin Pithwa.



				

				

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Maths in a minute! Thanks to PlusMath!

				
June 23, 2017 – 2:02 pm

				


https://plus.maths.org/content/maths-in-a-minute?nl=0


				

				

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