In my previous avatar as a research analyst, there was one nugget that the department head gave us that has stuck around with me for a while. As a time-management exercise, he asked all of us to list down a few things that we were doing currently that we would stop doing. The funda is simple – in a world with finite time, you can do something new only if you stop doing something else. CAT Preparation can also be structured in similar fashion. In this blog, I am going to give an outline of the rules and theorems that you are better off ignoring.
I am going to limit this to-don’t list of rules and theorems to academic-sounding stuff, otherwise we will be rehashing the same Amazon to Zomato list that we all know we should avoid.
Euler was a genius – so stay away from him
Euler introduced the concept of a function, devised the notation for trigonometric functions, the letter e for natural logarithmic base and the symbol i for imaginary numbers. Euler’s totient function is a wonderful idea. Having a sense of the number of natural numbers less than N and coprime to it is beautiful. If we want to find the last two digits of 9981 it probably comes of use.
But our exam is not the mathematical Olympiad and it has not tested any of Eulerian stuff for decades. The downside to knowing and learning obscure stuff is two-fold
- You can end up wasting time in rabbit-holes and
- Look for avenues to use and showcase your knowledge
I have seen students use Euler’s totient function correctly for remainder questions. But all of the relevant remainders questions for CAT can be solved without using Euler’s function. The temptation to use Euler’s totient function is so high than far simpler methods get frequently overlooked.
Studying with Fermat is good, but do not go overboard
Pierre de Fermat came up with lots of glorious things, the most commonly used of these is perhaps Fermat’s little theorem. Now, Fermat’s little theorem is a neat idea that is like a poor man’s Euler’s totient function. Since it is far simpler and easy to remember stick it in some part of the brain. Stick it in the same part of the brain just before the part where you have stored Wilson’s theorem. Title this part of the brain “unlikely to be tested”.
Now the most interesting of Fermat’s creations, rules and theorems is Fermat’s Last Theorem. Now Fermat’s Last Theorem is totally worth digging into. Fermat noted this theorem down on the margins of his notebook, scribbled that he had just got the most wonderful proof for this and died soon after. This remained unsolved for centuries – prizes were announced, competitions were held, careers were put on hold, wild furious debates were held about whether Fermat really did know the proof. Finally, at long last Andrew Wiles proved it, more than 300 years after Mr. Fermat had written it down.
So, why the hell am I asking you to dig into it. Oh, Fermat’s Last Theorem is a wonderful book by a gent called Simon Singh. Reading that would be really delightful prep for VARC.
Fermat’s 2IIM connection, or rather 2IIM’s Fermat connection
In a moment of madness, quite probably in the aftermath of having read something about the last theorem, yours truly (Rajesh Balasubramanian) decided to name the beloved company that owns the 2IIM brand as Fermat Education. So, being closely associated with Fermat is definitely good for your CAT prep – just take the 2IIM route and not the Last Theorem route.
Chinese Remainder Theorem needs a mention
From Wikipedia – “The Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.”
In plain speak, if we know the remainder on dividing N by 5 and on dividing N by 6, we can write N as some 30K + R. How exactly do we do this? More or less by listing possible remainders and seeing where both conditions are satisfied. Put another way, simple old-fashioned method of observing patterns.
For instance, let us say N divided by 5 gives remainder of 3 and N by 6 gives remainder of 1.
We can write N = 5K + 3 and N = 6p + 1.
A number of the form 5K + 3 can be of the form 30K +3 or 30K + 8 or 30K + 13 or 30K + 18 or 30K + 23 or 30K + 28. Similarly a number of the form 6p + 1 can be of the form 30K + 1, 30K + 7, 30k + 13, 30K + 19 or 30K + 29.
If you notice 30K + 13 sits in both cases, or our number has to be of the form 30K + 13. If you know how to do this, then all that remains is to say the word ‘Chinese Remainder Theorem’ every time some question like this appears. It might not help at all, but it does intimidate people around you and that has got to count for something. If you want to sound really in, you should probably say CRT.
The other approach would be more algebraic:
N = 5K + 3 or 6N = 30K + 18
and N = 6p + 1. Or, 5N = 30P + 5
Subtracting one equation from another N = 30(K – N) + 13 = 30q + 13. Again, CRT!
Verbal can also take you down time-sinks
Vocabulary is not tested, grammar rules are not important. We do not need to be able to distinguish between present participle and gerund. Even if you are not able to articulate the technical distinction between the usage of swimming in “I went swimming yesterday” and “Swimming is good exercise”, chances are the exam will go swimmingly well.
Laundry list of interesting-sounding vague unnecessary stuff plus some pure voodoo
Ceva’s theorem, Menelaus Theorem, Ptolemy’s Theorem, Diophantine Equations, Roots of a cubic equation, Cauchy Schwarz inequality, Koningsberg bridge problem, Schrodinger’s Cat and our own beloved Vedic math. And before I forget, CRT! – this is a nice non-exhaustive list of fancy-sounding irrelevant guff.
There will always be the best friend of the niece of the uncle of your second-cousin once removed (by marriage) who swears that these ideas are really important.
So, what rules and theorems should we study?
Firstly, read the proofs of Angle Bisector Theorem, Perpendicular bisector theorem, triangle inequality, triangle angle property, tangent-secant theorem, why area of a triangle is rs and abc/4R. Secondly, prove Congruence and Similarity rigorously. Thirdly, learn about the basis for test of divisibility by 4, 9, and 11. Finally, ponder why the formula giving number of factors of a number works. Have your own intuitive explanation for why (n + r – 1) C (r – 1) works.
In Conclusion, these are the rules and theorems to avoid during your CAT Preparation. If you’re naturally curious do, check them out at your leisure but for CAT Prep, they are guff.
A Final Note
On September 5th, we celebrated Teachers’ Day. 🙂
Teaching is a wonderful method to consolidate an idea in your own mind.
Teaching imposes the pressure of conveying the right idea in an engaging fashion; this makes us search for some juicy analogy. This process of creating analogous frameworks is frightfully useful for getting clarity. Beyond all this is the joy of explaining something to someone else, the pleasure of possibly demystifying something that might have been a pain for the recipient.
God knows our Country (and the world) needs as many teachers as it can lay its hands on. We also need teachers to be creative, irreverent, innovative, and free-spirited. We are too conditioned to teaching being strait-jacketed. If ideas need to stick in the brains, they need to be pushed through with loads of passion. Just some thoughts. 🙂
In terms of CAT Preparation, you should definitely be taking Mocks now. Check out this wonderful article about Handling Mocks Better with around 86 days to go!
Cheers, Stay Safe and Best Wishes for CAT preparation.
Rajesh Balasubramanian takes the CAT every year and is a 4-time CAT 100 percentiler. He likes few things more than teaching Math and insists to this day that he is a better teacher than exam-taker.