## Monthly Archives: August 2019

### From Passive to Active Learning: India Today: Jamshed Bharucha: Aug 19 2019

(By Jamshed Bharucha; Vice Chancellor, SRM Amravati University)

https://www.indiatoday.in/magazine/education/story/20190819-from-passive-to-active-learning-1578655-2019-08-09

### Some practical uses of maths

1. MPEG 4, audio/video/speech recognition/speaker identification/face recognition/HDTV and mp3 —- all these use logarithms and trigonometry. The special jargon is — Fourier Series, and Fourier Transforms.
2. In finance, compound interest is calculated by using a power function; the inverse problem of finding the duration of deposit is calculated using logarithms.
3. All (digital) phones are touch phones and they use DTMF (Dual Tone Multi Frequency) standard — implemented using sines and cosines.
4. Quadratic equations are used to design/model/develop certain kind of electronic amplifiers.
5. Probability theory is used in computer networks, routing of telephone calls, and also in Wall Street — stock market !!
6. There are ways to compute the numerical value of the irrational number $\pi$ up to a million digits and these ways are used to test the efficiency and efficacy of supercomputers.
7. Quadratic equations are used to study projectile motion (or to put it playfully, suppose we throw a pebble at a certain angle from horizontal ground, (angle less than 90 degrees (which would mean vertically up)) — the projectile is subject only to the force of gravity of the earth — the path or curve or trajectory of the projectile is a parabole, which is characterized by a quadratic equation. This can be easily proved using laws of straight line motion in two dimensional using resolution of vectors.

More later,

Nalin Pithwa.

### Elementary algebra: fractions: IITJEE foundation maths

Expertise in dealing with algebraic fractions is necessary especially for integral calculus, which is of course, a hardcore area of IITJEE mains or advanced maths.

Below is a problem set dealing with fractions; the motivation is to develop super-speed and super-fine accuracy:

A) Find the value of the following: (the answer should be in as simple terms as possible, which means, complete factorization will be required):

1) $\frac{1}{6a^{2}+54} + \frac{1}{3a-9} - \frac{a}{3a^{2}-27}$

2) $\frac{1}{6a-18} - \frac{1}{6a+18} -\frac{1}{a^{2}+9} + \frac{18}{a^{4}+81}$

3) $\frac{1}{8-8x} - \frac{1}{8+8x} + \frac{x}{4+4x^{2}} - \frac{x}{2+2x^{4}}$

4) $\frac{x+1}{2x^{3}-4x^{2}} + \frac{x-1}{2x^{3}+4x^{2}} - \frac{1}{x^{2}-4}$1

5) $\frac{1}{3x^{2}-4xy+y^{2}} + \frac{1}{x^{2}-4xy+3y^{2}} -\frac{3}{3x^{2}-10xy+3y^{2}}$

6) $\frac{1}{x-1} + \frac{2}{x+1} - \frac{3x-2}{x^{2}-1} - \frac{1}{(x+1)^{2}}$

7) $\frac{108-52x}{x(3-x)^{2}} - \frac{4}{3-x} - \frac{12}{x} + (\frac{1+x}{3-x})^{2}$

8) $\frac{(a+b)^{2}}{(x-a)(x+a+b)} - \frac{a+2b+x}{2(x-a)} + \frac{(a+b)x}{x^{2}+bx-a^{2}-ab} + \frac{1}{2}$

9) $\frac{3(x^{2}+x-2)}{x^{2}-x-2} -\frac{3(x^{2}-x-2)}{x^{2}+x-2} - \frac{8x}{x^{2}-4}$

More later,

Nalin Pithwa