11. Tip - Don't join the most popular coaching in town

I beg, DO NOT do this blunder. Well, it's under condition that that the coaching has more than 20-25 people in a batch / class. If not, you may join.

I know if the coaching people read this, they'd be behind me. But it's a fact. More often than not they would attempt to coach more than 100 students in a class. What if you have any doubts? I bet, you won't get an answer in the class. What more... if you get your basics wrong, they will always remain unanswered. Complex questions may be, sometimes, answered.

Join some correspondence test series, like of FIIT-JEE, Brilliant Tutorials etc. A lot more have come since I joined in 1996. The national scale test series give you two benefits:

• You get an idea of the examination room. Your heartbeat will run fast in first paper and soon get normal. It will run normally when you sit in real exams.
• You know, right from first test, where you stand nationally. If not precisely, at least approximately.

Oh yes... if you need to join coaching, join a small center where the batch will not be big, the teacher will be enthusiatic to talk and work and your queries will be duly attended.

10. Tip - Don't listen to what your mother says!

No... I am not talking about taking food but about her worrying about your board exams or JEE screening. Believe it or not, more often than not, you will be pressurized by your mother or grandmother or some other relative to qualify JEE. If you pay more attention towards JEE, you will get scolding on not studying for your board exams. It happened to me also.

Luckily, for me, my father was with me. He said to me just one thing, which I still remember and probably will never forget. He said, "If you qualify in JEE and pass boards with even 33% marks, nobody will complain. If you don't beat JEE, your board marksheet if everything." I always kept it in my mind.

I still remember it was January of 1998 and my board exams were to be in March / April, 1998. I had some idea of English course but a very little idea of Hindi. I took the board exams with no more than 15 days of exclusive study of these subjects and passed with handsome marks.

But who looks at them. All I have to say now is - "I am an IIT Kanpur Graduate / B.Tech.". That's it.

So, don't listen to anybody on this topic. But also don't overlook them completely. After all, you'll be giving your internal examinations.

9. Trick - Yet another multiplication trick

Ok, so far we've seen multiplication of numbers near powers of 10 and a generic multiplication. In this we'd see multiplication of numbers near a multiple of 10. For example, multiplication of 37 and 39 or 42, both being near to 40 = 10*4. The key here is 4.

Here's the full calculation:

   37  -3
   42  +2
 ----------
  39  | 6
 ----------

  => 39*4 | 6
  => 156 | 6
  => 1554

Superb! Hmm... note what I did in the second step 39 * 4. I multiplied by 4 because to get 40, I had multiplied 10 by 4. And since 10 has one zero, the value on the right side consists of only one digit.

Here's another way to do the same. But I will take another example: 47 * 49. Both are near 50 = 100 / 2. We will take 100 as base and the key is 2. Point to note here is that we are dividing 100 by 2 to get 50. Watch now...

   47  -3
   49  -1
 ----------
   46 | 03
 ----------
 => 46/2 | 03
 => 2303

What more do you ask for than converting multiplication to simple additions and at max a division by 2!!!

8. Trick - Squaring numbers near powers of 10

In this article, we shall learn about squaring numbers that are near to powers of 10 like 10, 100, 1000 etc.

The sutra for this is "Yavdunam Jaavdunikritya Varga Cha Yojayet" meaning whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square of that deficiency.

To understand the sutra, let's take an example. We would square 96. One way is to do it by squaring 104. Even shorter is this method.

96 is closer to 100 (we need to take power of 10). 100 - 96 = 4.. We further reduce 4 from 96 to get 92. Square of 4 is 16. So, the answer is 9216. Since 100 has two zeroes, the square must be of two significant digits.

Let's take a few more examples:

982 = (98 - 2)|(22) = 96|04 = 9604.
1022 = (102 + 2)|(22) = 104|04 = 10404

Now, we would examine is algebraically. Let the number be 100x - y.

(100x - y)2
   = 10000x2 - 2*100xy + y2
   = 100(100x - 2y) + y2
   = 100(100x - y - y) + y2

Now, (100x - y) is our original number from which we again subtract the dificiency y.

What if y² extends to 3 digits? We add the carry over to the front. Here's an example:

892 = (100 - 11)2
    = (89 - 11)|112
    = 78|121
    = (78 + 1)|21
    = 7921

As again, if you have any doubts or queries mail me at the email provided in the contact info.

7. Trick - Division by numbers ending in 9 (Part II)

In the previous article, we learnt about division by numbers ending in 9. It's ok if you need exact value, like what I calculated for 1/19. But for JEE, approximate value of 1/19 = 0.05263 may do.

How do you plan to do it? Divide by 19 everytime? No... you don't need to learn tables of 19. How about dividing by 2 or 5 or such small numbers!

We will make use of previous sutra - by one more than the previous one.

Let's start with 1/19 as example, followed be 1/7.

one more than the previous one for 19 is 2. So, we would divide all the way by 2.

1 divided by 2 gives 0 as quotient and 1 remainder.

1/2 = 0.10

Now, we divide ₁0 = 10 by 2 to get 5 as quotient and 0 as remainder.

0.10 => 0.005

Now, we divide ₀5 = 5 by 2 to get 2 as quotient and 1 as remainder.

0.10 => 0.005 => 0.0512

Similarly, the full calculations yields:

0.0512 => 0.05206
      => 0.052603
      => 0.0526311
      => 0.05263115

... and so on.

The advantage of this method is that you can stop whenever you feel you are ok with the precision of the result. For example, you can stop at 0.053 as an approximation or at 0.05263.

We shall carry the direct calculation for 1/7 = 7/49 where divisor is 4 + 1 = 5. We will start by dividing 7 by 5.

7/49 => 0.21
     => 0.114
     => 0.1442
     => 0.14228

... and so on!

Kewl... ain't. Note that all these tricks that I have been telling you till now are what I learnt in "Vedic Mathematics".

6. Trick - Division by numbers ending in 9 (Part I)

The sutra (formula) that empowers me to divide by numbers ending in 9 is again "Ekadhikena Purvena" meaning by one more than the previous one

The division can be done in two ways - by division and by multiplication .

I will take up by multiplication in this article. I will illustrate by taking two examples, 1/19 and 1/7 = 7/49

The numerator in 1/19 is 1. So, we start with 1.

Digit previous to 9 is 1 and one greater than 1 is 2. We will deal with multiplications by 2 only.

Multiply 1 by 2 to get 2

Now, we have 21.

Multiply 2 by 2 to get 4

Now, we have 421.

Multiply 4 by 2 to get 8

Now, we have 8421.

Multiply 8 by 2 to get 6 as base-number and 1 as carry over.

Now, we have ₁68421.

Multiply base-number 6 by 2 to get 2 as base-number and 1 as carry over. Add the previous carry over 1 to get 3 as base number.

Now, we have ₁368421.

Carry on till you have a total of (19-1)/2 = 9 digits in place. And they would be 947368421. We call is Lower Half. To calculate the Upper Half, take 9's complement.

  999999999
 -947368421
------------
  052631578
------------

The result, therefore, is:

 0.052631578947368421

Enjoy! Now the example of 1/7. It will contain a total of 7 - 1 = 6 digits that will recur.

1/7 = 7/49. The number previous to 9 is 4 and one next to 4 is 5.

First, write 7. Multiply by 5, yielding 5 as base-number and 3 as carry overy - ₃57. The full chain is as below:

7 => 357 => 2857
  => 42857 => 142857

Therefore,

 1/7 = 0.142857

5. Trick - More Multiplication

This trick is an extension of the previous trick. It empowers you to multiply two numbers that can be expressed as z|y and z|(10-y).

The result of multiplication of the two numbers is z(z+1)|y(10-y)

To illustrate by an example,

37 * 33 = 3(3+1)|7*3 = 1221
141 * 149 = 14(15)|1*9 = 210|09 = 21009

Hmm... what do you say? Isn't maths really a fun!

4. Trick - Squaring numbers ending in 5

The sutra (formula) that empowers me to do so at ease is "Ekadhikena Purevena", meaning by one more than the previous one.

To calculate square of any number ending in 5, say z5 is z(z+1)|25, ie, multiply z by z+1 and attach 25 next to it.

For example, 75² is:

752 = 7(7+1)|25 = 5625

Simple and powerful, isn't it? Now, let us see what goes behind it. It's simple algebra.

(10x + 5)2 = 100x2 + 100x + 25
  = 100x(x+1) + 25
  = x(x+1)|25

3. Trick - Multiplication

Today we will learn about multiplication... once again, the same trick but with a difference.

The trick that I will tell you now will give you more power. After this, you will never need to know any table more than 5x5. I call this extension as "10's complement".

10's complement of any digit x is 10 - x. In two numbers that you are multiplying, take 10's complement of any digit greater than 5 and increase the digit next to it (higher significant value) by one. Reverse this process while calculating final value. The digits whose complements are taken shall be called "barred".

Calculation with barred digits will be considered as being done with negative numbers. For example, bar-4 * 4 = bar-16 and bar-8 + 3 = bar-5.

Now, I shall demonstrate it with an example:

    478*129 = 522*131

    522
   x131
 --------
   53342
   11
 --------

  = 62342 = 61662

2. Tip - Multiple Choice Questions

JEE (Joint Entrance Examination) consists of two phases: Screening, consisting of objective questions and Mains, consisting of subjective questions. Screening plays a major role in determining your entry - not only you need to be accurate, you also need to be fast.

In this article, let us see how can we avoid actual calculations and reduce time to arrive at the correct solution.

More often than not, you will not be required to completely solve the problem. You would be able to reject some answers by mere inspection. For example, 234*567 would be greater than 10,000. Now, 25*55 = 1375 (quick, mental calculation by Vedic Maths trick). So, the answer must be somewhere close to 137500 but less than it. How? Guess. Failed? Mail me!

Ok... that may be an artificial example. Let's take a real example - from 2004 screening.

Three distinct numbers are selected from first 100 natural
numbers. The probability that all the three numbers are
divisible by 2 and 3 is:

(a) 4/25            (b) 4/35
(c) 4/55            (d) 4/1155

According to requirement, the numbers must be divisible by 6. There are 16 numbers divisible 6. So, the probability is:

16C3 / 100C3

  16*15*14       16*15
----------- = ----------- (98/14 = 7)
 100*99*98     100*99* 7

       16
 < ---------- (1/99 < 1/96)
    100*6*7

       4         4
 < -------- < -------
    25*6*7     100*7

The only number that is less than 4/700 is 4/1155, hence the answer.

These calculations given above are very trivial, mere approximations. You never did an actual calculation. No calculation was complex. Well, except for 98/14 = 7, if you consider it.

1. Trick - Multiplication

Today we will learn about probably one of the first "Sutras" (meaning: formulas, Hindi) called Urdhva-Tiryak Sutra

In English, it means "Vertically and Crosswise". The demonstration below shows how one can shorten the calculation done:

Conventional Method	Vedic Method

How is it done? What's the trick? Well... nothing too great. It can be easily observed if you are very strong in Algebra. Ok.. you'll get it when I explain.

The Vedic calculation is done as told by the sutra: Vertically and crosswise. First, take unit digits of the two numbers, 2 and 6 in this case. Multiply. Units digit of the result, 2, is written in first row while tens digit, 1, is written in second row. The first row is result row. The second is the carry-over.

Now, take two digits at a time - 32 and 46 - multiply crosswise and vertically, ie, 3 by 6 and 2 by 8. Add. The result is 26. Add the carry-over. Units digit - 7 - is written in the result row. 2 is carry over.

Now, take three digits - 232 and 246. Multiple 2 by 6, 3 by 4 and 2 by 2. Add. Add the carryover - 2 - to it. You get 30.

Carry on till you are done will all digits - increasing the number of digits into consideration by one. One you reach the limit, start removing digits from the last.

For example, after you are done with 28232 and 53246, apply the same algorithm with 2823 and 5324. Notice that I've removed the units digits - 2 and 6. Now start removing one by one and complete.

Here is the algebraic explanation of the calculation:

          ax2 + bx + c
          dx2 + ex + f
   ----------------------------
      adx4 + (ae + bd)x3 + 
       (af + be + cd)x2 +
           (bf + ce)x + cf
   ----------------------------

Does it catch your eyes now? Still not... mail me at the email provided in contact info section.

Free-JEE Yahoo Group

Courtsey Rambo, an IITM alumnus, a Yahoo group has been created towards Free-JEE education. Click here to learn more about it.

Introduction

I am creating this blog with an intention to provide tips and tricks for all Indian Institute of Technology aspirants.

Given below is the list of all IITs (in alphabetical order):

• IIT Bombay
• IIT Delhi
• IIT Guwahati
• IIT Kanpur
• IIT Kharagpur
• IIT Madras
• IIT Roorkee

Below is the list of the website of their alumni networks:

• IIT Bombay
• IIT Delhi
• IIT Guwahati
• IIT Kanpur
• IIT Kharagpur
• IIT Madras
• IIT Roorkee