Monthly Archives: July 2016

Could a one-sided limit not exist ?

Here is basic concept of limit :

Bus Conductor’s son from Khar slum makes it to IIT-Bombay

Never Say Die!

(Reproduced from DNA Newspaper, Mumbai, Tuesday 12.07.2016, page 2, print edition for the purpose of motivating my own students and readers of my blogs)

(Author: Bilal Khan; E-mail: correspondent@dnaindia.net)

“Beating the odds”

They say, “if you can dream it, then you achieve it”, and how true this has turned out to be for this 18-year old Arbaz Shaikh, son of a bus conductor from a Khar (a suburb in Mumbai) slum, has got selected for IIT-B’s Bachelor of Technology course.

Arbaz stays with his parents and younger sister, in a 10 x 12 square feet room. His father Rajvalli (43) has been working as a BEST conductor for five years. “I was a technician before I joined the Undertaking. I did not have a shop; I used to go door to door to repair TV’s and other appliances. Those were tough days — trying to earn a livelihood enough to feed the family and pay the rent,” he said.

Arbaz has been good in his studies right from childhood. He secured 95.64 % and 85.85 % in standards X and XII, respectively. In JEE Advanced, he secured an AIR (All India Rank) 2262 in the second attempt.

The teenager, who used to escape to the college library to study, said, “I stay in a slum area, where it’s always noisy. Neither did I have a room to myself for studying. The library was my refuge.”

He added that he never took stress over studies and always managed to find the right balance.

This year was Arbaz’s second attempt to get admission to IIT-B. “My rank was not good enough last year. My father motivated me to try once more,” he said.

When asked why he pushed his son to give it another go, Rajvalli said Arbaz had been taking scholarship exams and other tests apart from those conducted in school and college. “He had got scholarship for Std XI and XII too without tuition classes. Because of that I was sure that he would get through in the second try,” added Rajvalli.

Arbaz’s father has big dreams for his son. “I want my children to be well-educated so that they can provide a good education and a good life to their children.”

Arbaz is trying to get a scholarship for his IIT-B course too. At the moment, he has got assistance assurance from an NGO, the Association of Muslim Professionals.

(With thanks to Bilal Khan and DNA newspaper),

(Nothing is impossible, thus spake the wise, famous French warrior Napoleon Bonaparte!)

You are requested to share more motivational stories about pursuit of excellence in Math, IITJEE, RMO etc. here in my blog 🙂

-Nalin Pithwa

Tower of Brahma

One example of very large numbers is found in the story of Brahma’s Tower supposed to be set in a temple in the holy city of Varanasi. The tower consists of 64 discs of gold of decreasing sizes with hole in the middle of each disc. To begin with, there was a tower built by placing discs in descending order of size, one on top of another around a peg. There are two other similar pegs which were empty to begin with. The priests have to transfer the tower from one peg to another making sure that each time the disc on top is transferred on to another disc (around another peg) larger in size, until the whole tower is moved to a neighbouring peg. Estimate the time taken to do this job, assuming that one move takes one second. A great deal hinges on your solution; for once the job is completed it will be the end of the universe, till Brahma creates another!!

(Ref: Fun and Fundamentals of Mathematics by Jayant V Narlikar and Mangala Narlikar).

More later,

Nalin Pithwa

Cyclic Numbers

Consider the number 2387. We can construct new numbers from the same digits by moving them cyclically:

3872, 8723, and 7238.

in general, we can construct n cyclic numbers from a number containing n digits:

$N_{1}=a_{1}a_{2}a_{3} \ldots a_{n-1}a_{n}$

$N_{2}=a_{2}a_{3} \ldots a_{n-1}a_{n}a_{1}$

$\vdots$

$N_{3}=a_{3}a_{4}a_{5}\ldots a_{n-1}a_{n}a_{1}a_{2}$

$N_{n}=a_{n}a_{1}a_{2}a_{3}\ldots a_{n-1}$.

What is the sum of all the cyclic numbers of such a set $\{N_{1}, N_{2}, \ldots, N_{n} \}$? it is easy to compute this from the above description. When we add all the numbers above, we find that in each vertical column, the sum simply is

$S = a_{1}+ a_{2} + a_{3} + \ldots + a_{n-1} + a_{n}$

so that if we factor it out, the remaining factor will be $111\ldots11$, with the number repeated n times. In our example, chosen above, we should get

$2387+3872+8723+7238=(2+3+8+7) \times 1111 = 20 \times 1111 = 22220$.

You can verify that this is indeed true.

There are several puzzles related to cyclic numbers. You are most welcome to share with us.

-Nalin Pithwa

Some problems for IITJEE Foundation mathematics

Problem 1.

A cistern can be filled by two pipes in 33$\frac{1}{3}$; if the larger pipe takes 15 minutes less than the smaller to fill the cistern, find in what time it will be filled by each pipe singly.

Problem 2.

By rowing half the distance and walking the other half, a man can travel 24 km. on a river in 5 hours with the stream, and in 7 hours against the stream. If there were no current, the journey would take 5$\frac{2}{3}$ hours; find the rate of his walking, and rowing and the rate of the stream.

Problem 3.

Factorize: $2a^{2}x^{2}-2(3b-4c)(b-c)y^{2}+abxy$

Problem 4:

Find the fourth root of $81x^{4}-216x^{3}y+216x^{2}y^{2}-96x^{3}y+16y^{4}$

Problem 5:

Find the sixth root of

$(x^{3}-\frac{1}{x^{3}})^{2}-6(x-\frac{1}{x})(x^{3}-\frac{1}{x^{3}})+9(x-\frac{1}{x})$

More interesting stuff on IITJEE foundation maths later,

Nalin Pithwa

Kids of pushy parents ‘face higher risk of depression’: NUS study

The greatest gift we can give our children is the gift of enthusiam. (Charles Schwab). I came across this on Professor Terence Tao’s blog.

Recently there are two articles on “Tiger Moms” and “Kiasu (translated as “overly afraid of losing”) Parents” in Singapore. Interesting to read.

Parents in Singapore are indeed at a dilemma, overly pushing their child will lead to negative consequences (as mentioned in the articles), but not pushing their child may lead to falling behind academically.

This quote sums it up:

A housewife, who wanted to be known only as Mrs Lim, 43, said: “In Singapore, the pressure to do well starts early. Parents have no choice but to set high expectations of their kids’ performance.

“But I will be more mindful of the way I speak to my kids, so that they won’t feel bad about making mistakes in their work.”

The solution, ideally, is for children to be self-motivated rather than being pushed by parents. Check out some motivational educational books here.

The worst consequence of pushing children…

View original post 192 more words