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In this post, I will be telling you about my preparation journey for all the entrance exams like JAM, GATE, TIFR and NBHM. For those who wanted to know about my journey from engineering to mathematics can have a look (Journey).
First, starting with the list of books. For every book below, you will also find the reason why I chose that particular book.
1. A Basic Course in Real Analysis by S.Kumarsen : This book is written in a way that students of different caliber, students preparing for different exams and interviews can refer it and this book will serve their purpose. Each question in this book comprises of 2 parts : First is strategy which deals with how to approach the question and the second is the formal mathematical solution of the question which is motivated by the strategy. For engineers it is like you develop an algorithm for your code.
Let me give you an example : suppose you want to find limit x as x tends to a.
In strategy part you would find something like when x is getting closer and closer to a, the function f(x) = x is also getting closer to a.
Now in the formal method, you would find that they would relate closeness(used in strategy) with epsilon-delta definition of limit to give the formal proof.
Also, they will illustrate both the strategy and Formal proof using graphs which is very important for better understanding of the concept.
2. Topology without tears by Sidney A. Morris: I know if you ask anyone about topology, first book they will suggest would be Munkres. But I think you should be familiar with the subject (Because this is by far the most difficult subject I have studied till now) before directly going for standard books. Sometimes, These standard books may kill the interest in the subject because of how they are written. These books are mainly written for Masters or above Masters level students. That’s why I feel, one should start with Topology without tears because it is written in a very concise way that even undergrads like me can understand the concepts and develop the feel of the subject.
3. Contemporary Abstract Algebra by Joseph Gallian: It is the most suggested book for this subject. You won’t find a lot of theory and solved examples in this book. This is the reason I liked this book a lot because it develops your concepts through questions.
4. Linear Algebra done right by Sheldon Axler: One of the most suggested book for Linear Algebra.
5. Calculus(Vector, Differential, and Integral): I didn’t follow any particular book for these topics because I am an engineer and these topics were there in our syllabus. So, I just practiced lots of questions from these topics.
Although I was preparing for Ph.D. level exams including GATE, NBHM, and CSIR I didn’t focus on topics like Complex Analysis, Functional Analysis, Measure Theory, PDE, etc. That’s why I haven’t mentioned any books for these topics here.
Apart from these books I bought question banks of each topic and solved a lot of questions to prepare for these exams.
Also, before starting a topic I used to watch some videos about it to get some motivation. Also, I used to find some of its interesting applications.
Like the 3Blue1Brown’s(Channel link) lecture series on Linear Algebra, Who cares about Topology? How Group theory is used in Cryptography? etc.
Now, Let’s begin the Journey
Starting with Topology: I started my preparation in the month of June and like most of the engineers, I started preparing for both GATE and JAM. So, I downloaded the syllabus and decided to prepare for GATE because I found that J is a subset of G, where J denotes Jam syllabus and G denotes Gate syllabus.
In my 6th semester, I had taken NPTEL courses on Linear Algebra and Multivariable Calculus. And I wanted to start my preparation with a new topic.
So, I started with Topology because it had 2 advantages. One, it was a part of G, and second, it also helps in building concepts of Real Analysis (which belongs to J intersection G).
Coming to Algebra: After learning most of the concepts from Topology, I started Abstract Algebra. While working on Abstract Algebra, I was practicing questions from Topology and metric spaces.
By the end of July, I had some idea of Group Theory, Topology, Multivariable Calculus, and Linear Algebra.
When I was about to finish some topics and move to another, I used to try most of the previous years GATE/JAM questions from that topic to test whether I was able to solve them or not.
Let’s Analyse now: Next I moved to Real Analysis. This topic is in my opinion the most important topic for all the exams including JAM, TIFR, GATE and NBHM. So, I wanted to give it a lot more time compared to others.
Sacrifices one has to make: After a month and half of understanding concepts of real analysis and solving problems, I started Measure Theory. But after sometime, I felt that there was no need to study these topics like Functional Analysis, Measure Theory and Complex Analysis. These topics were quiet deep and their weightage in all the exams were quite low. So, I decided to leave them and focus more on topics like Real Analysis, Linear Algebra, Group Theory and Calculus.
1st real test: In October, I started preparing for NBHM MSc exam(although I was not eligible but I applied just to see whether I was able to solve these questions during exam) by solving most of the previous year question papers. I was very confident before the exam because I had practised some Analysis and Algebra. And there used to be a Miscellaneous section in the exam(Most of the questions in this section were similar to IIT JEE questions) which was my strong point. 
Surprise(good or bad)? : But guess what, NBHM changed the question paper pattern completely (Of all the years, they found this year to do it. Why?). There were no seperate sections, no questions like miscellaneous ones and lots of questions from Vector Calculus etc. My exam didn’t go well as per my expectations, but later when results were out, I found that I had cleared the exam and was one of the few candidates selected for interview.
Please let me prepare: I didn’t prepare much in November because of my college exams and final year project. In December, I gave both the TIFR and CSIR exam without any practice which was reflected in the results as I couldn’t clear CSIR JRF cutoff by just 3 marks and also, couldn’t secure enough marks in TIFR GS to get call from TIFR Bombay (but got the interview call from TIFR BNG).
Focus Shift: In January, I decided to focus completely on JAM because I realized that even if clear GATE with a very good rank, it’s not of much use. Hardly any IIT would take a BE student without any pre-requisites in their doctorate program(After the exam, I applied in IIT Delhi PhD program and my rank was good enough to get an interview call but I was rejected because of my branch in engineering). So, I practised a lot of questions from Calculus and ODE.
3 in 2: I had 3 very important exams in 2 weeks. First one was NBHM PhD(25th Jan), next one was GATE(2nd Feb) and at last the most important exam of all 3, JAM on 9th Feb.
NBHM PhD exam was a nightmare. I wasn’t well that day and they changed the pattern in this too and the exam was very very tough. After that, I gave GATE without because in my mind, I was thinking only about 9th Feb.
A week before D-Day: Few days before JAM I bought an online test series to practise for the exam under time pressure. I gave 4 to 5 tests with an atmosphere like in a exam center and it helped me a lot in the final exam. I think this online test series helped me on how to solve problems quickly under time pressure and how to approach the question paper.
I didn’t prepare anything a day before JAM (8th Feb) and I feel that it is very important to relax and have a good sleep before such an important day.
D-Day: I was confident but little bit tensed also before the exam. But all my nervousness was over as soon as I saw the question paper on my screen and I started solving one problem after the other. My target was to score around 80 marks and I ended up scoring 79.67/100 which was enough to get me a rank under 20. I was quite happy with my performance because with this score I can get the IIT I want and also a interview calls from IISc Bangalore, IISER Pune and IISER Mohali.
The last words: I would like to say that solving a lot of problems is the key to perform well in these exams and a thorough and deep understanding of concepts is needed to perform well in interviews. Also, I would like to thank my advisor(He is currently pursuing a PhD from IIT Delhi and he used to teach me during JEE preparation days) for guiding and helping me throughout this journey.
About the author:


Mr. Tushar Arora has secured an All India Rank of 18 in JAM & 54 in GATE mathematics in the year 2020! He also has cleared NBHM (MSc & Ph.D.). TIFR-GS and was selected to IISER I-PhD, all this while still in the final year of engineering. We, the MAE(Mathematics after Engineering) and PAE (Physics After Engineering) community congratulate him on this achievement and thank him for the inspiration & hope he has given us with it.  He is currently looking forward to joining the IISER Pune I-PhD program. We wish him the best in all his future endeavors.
Also, do visit all other useful content of our blog 
Mathematics after Engineering.

Here is the link for our WhatsApp group of members all of whom are engineers who actively discuss and help each other out in entering mathematics stream after engineering. 

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