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

### Applications of Derivatives: IITJEE Maths tutorial problem set: III

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.

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

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

### Paul Erdos, Mathematics, Russia and USA:

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

### Some light moments with an IITJEE foundation math student

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.

### Petronet Kashmir Super 30: Nine crack IITJEE 2017 exams

(This news is a bit delayed here on this blog, nevertheless, inspirational.)

Reference: DNA, Mumbai print edition, June 15 2017:

Srinagar: Petronet LNG Ltd. (PLL), which started its CSR voyage from western India and traversing South and East, have now reached Northern state of Jammu and Kashmir.

Considering the lack of facilities/faculty in J and K to impart coaching for Engineering Entrance Examination to facilitate admissions to IITs/NITs/other institutions of repute, Petronet LNG Ltd. sponsored 40 underprivileged students in 2016-17 to fulfill their dream of higher engineering education (35 boys and 5 girls) under Petronet Kashmir Super 30 programme in association with Indian Army, CSRL. This 11 months’ residential programme was attended by 40 students (kargil 7, Pulwama 5, Bandipura 6, Baramulla 4, Anantnag 7, Ganderbal 4, Kulgam 1, Tangmarg 2, Ramban 1, Shopian 1, Sopora 1, Nagrota 1).

Despite hardships in Kashmir and educational institutions being closed last year, the project continued with its mission. The declaration of IITJEE results on April 27 2017 saw selection of 28 students (including 2 girls) who were coached under Petronet Kashmir Super 30 cracking the IITJEE mains exams. Six other students will join Regional Engineering Colleges.

Shri Dharmendra Pradhan, Hon’ble Minister of State (I/C) Petroleum and Natural Gas and Mr. Chowdhary Zulfkar Ali ji, Hon’ble Minister for Dept. of Food, Civil Supplies and Consumer Affairs and Information Dept., J and K, interacted with students of Petronet Kashmir Super 30 at Delhi and felicitated them for their hard work and achievement.

Shri Pradhan expressed happiness that students from underprivileged backgrounds from places like Kupwara, Pulwama, Anantnag, Shopian etc. have shown their determination and got selected in prestigious institutes. He also announced on behalf of Petronet LNG Ltd. that 100 students will be sponsored from Kashmir for engineering entrance next year.

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🙂 🙂 🙂 Congratulations to Petronet (Kashmir) LNG Team including their faculties and the students from Nalin Pithwa !