# I.S.I. & C.M.I. Entrance Exam Preparation

# How to prepare for I.S.I. and C.M.I.?

This course is for those students who are passionate about Mathematics. Firstly analyze whether you qualify yourself fit to study in ISI or CMI. By being fit to study, I mean to say if you love math. If your target is just to get a brand name or just qualify the exam for “a” college, please don’t take the exams.

Now that you are sure that you are a math lover, you can proceed to take the course.

The course is for high school students who want to give the ISI & CMI entrance exam after their 10+2.

Recommended Books:

- Problem Solving Strategies (pdf available) (which I’ll refer to as PSS),
- Test of mathematics at the 10+2 level (published by ISI) (which I’ll refer as TOMATO),
- An excursion in mathematics (which I’ll refer to as Excursion).
- Challenges and Thrill of pre-college Mathematics.
- Also refer to some standard textbooks for calculus definitions (like I.A. Maron, Bartle-Sherbert or Apostol).

The syllabus is just high school math. I’d broadly classify the syllabus into the following topics:

- Number Theory: My personal favorite. Questions seem to attack the common sense (after all, they’re beautiful numbers, which are close to one’s conscience). Read Excursion for basics. Solve a few exercise problems. Try out the problems on TOMATO. If comfortable, do PSS exercises (examples are hard).
- Combinatorics: You can expect beautiful problems from here. CMI lives up to this expectation. In ISI, the combi problem(s) will certainly be tricky. Know very well when to add and when to multiply. Know P&C of 11th grade like the back of your hand. Read Excursion for basics. The only mantra to master this topic is to practice lots and lots of problems. Again, Excursion for basics. Read through PSS as well (easy questions from chapters 1,3,4,5 are very very important). Do all questions from TOMATO without help (if you stumble, think later, but solve on your own).
- Algebra: Know your high school algebra very well. Besides this, you might want to know (and understand) the triangle inequality, the AM-GM-HM inequality, and the Cauchy-Schwartz inequality. It’ll be your added advantage if you some further tricks in solving problems for algebra. But it’s pretty fun to analyze algebraic structures and draw analogies from other fields (which makes the topic super interesting). Polynomial is an important subtopic. Theory from Excursion and practicing problems from TOMATO is enough.
- Geometry: Yet another treasure (Coordinate geometry must be your last option). Even if you love Euclidean geometry, nobody can assure you that you can answer a question in the paper. Revise all the geometry theorems and lemmas that you had done in your 9th and 10th grade. Ask your coaching institute guide to help you out with topics and problems. Nothing much to read from Excursion. But practice from TOMATO and the easier problems from Excursion.
- Calculus: Very much weighted for ISI paper. Know everything very precisely. Before applying a theorem or a lemma, make sure you can prove it. Requirements: Definition of sequence, limit (of sequence and function), continuity, differentiability. Extreme value theorem (proof not needed), Intermediate value theorem(proof not needed), sandwich theorem, mean value theorems (Rolle, Lagrange’s MVT, Cauchy’s MVT), MVT for integral, Darboux theorem, one-one-ness of a continuous monotone function etc. Please ask your coaching teacher to guide you, or ask help from someone experienced. Read Apostol or Bartle-Sherbert for theorem statements snd definition. For practice, try out TOMATO.

After knowing the basics of these topics, you might want to try out many A1 and A2 problems from Putnam contest. It’s okay if you can’t solve all by yourself. See the solutions and learn. You might face many questions which will cover concepts from multiple topics. These test your actual ability to think. But if you can’t solve a problem, don’t get discouraged.

For MCQ in ISI, practice mathematics papers of JEE mains (avoid solving too many questions from coordinate geometry and indefinite integration). CMI questions have an added Olympiad flavour to them. So practicing Excursion and PSS helps.

Before sitting for the exams, make sure that you complete TOMATO (each and every question, with or without help) and all the past year questions of ISI and CMI. Keep away a few MCQ papers (of ISI) as unseen ones so that you can use them for time-bound practice a couple of weeks before the D-day.

If you think you’re pretty comfortable with a topic, please go for PSS (Don’t start with the mentality that you’ll complete it, save for a few exceptional ones). Even solving the first 15 problems (in each exercise) by yourself is good enough.

Overall, TOMATO is very important and must be solved completely. Excursion is for reference. PSS is for further problem-solving practice to gain experience.

The interview in ISI is very important. The interview panel will sit with your subjective copy and they’ll ask abt any conceptual errors that you made in your paper. Be prepared to face that. Not even God can help if your concepts aren’t crystal clear. It’s okay if they ask you to solve a problem and you solve it with the hints they provided (or don’t solve it completely). I repeat they don’t tolerate basic mistakes. But don’t worry, even if you make such errors that may not appeal to your eye, they’ll point it out and you’ll get a chance to correct yourself. Please think aloud. They want to know your thought process (it won’t matter even if you don’t solve a problem fully).

Another very important point is to find yourself a math community. This keeps your zeal intact. One way is to (mathematically) socialize with your friends at the coaching center. If there aren’t enough enthusiastic candidates, you can always find an online community. The best one that I know so far is Art of Problem Solving (aops.com). Every year, an ISI/CMI preparation group is created. You can keep yourself updated with math everyday on such a forum, besides discussing “other” beautiful ways to approach a particular sum. There are pr0 people (like IMO medalists) there to help you too.

2 years should be enough to prepare for ISI and CMI, provided you are sincere enough. There’s no need to sacrifice sleep. Proper sleep enhances your performance. Just remember to brush up your concepts and practice problems regularly.

## Exam Pattern and Syllabus

### For ISI:

Test Codes: UGA (Multiple-choice Type) and UGB(ShortAnswerType)

Questions will be set on the following and related topics.

##### Algebra:

Sets, operations on sets. Prime numbers, factorization of integers and divisibility. Rational and irrational numbers. Permutations and combinations, basic probability. Binomial Theorem. Logarithms. Polynomials: Remainder Theorem, Theory of quadratic equations and expressions, relations between roots and coefficients. Arithmetic and geometric progressions. Inequalities involving arithmetic, geometric and harmonic means. Complex numbers. Matrices and determinants.

##### Geometry:

Plane geometry. Geometry of 2 dimensions with Cartesian and polar coordinates. Equation of a line, angle between two lines, distance from a point to a line. Concept of a Locus. Area of a triangle. Equations of circle, parabola, ellipse and hyperbola and equations of their tangents and normals. Mensuration.

##### Trigonometry:

Measures of angles. Trigonometric and inverse trigonometric functions. Trigonometric identities including addition formulae, solutions of trigonometric equations. Properties of triangles. Heights and distances.

##### Calculus:

Sequences – bounded sequences, monotone sequences, limit of a sequence. Functions, one-one functions, onto functions. Limits and continuity. Derivatives and methods of differentiation. Slope of a curve. Tangents and normals. Maxima and minima. Using calculus to sketch graphs of functions. Methods of integration, definite and indefinite integrals, evaluation of area using integrals.

### For CMI:

The entrance examination is a test of aptitude for Mathematics at the 12th standard level, featuring both objective questions and problems drawn mostly from the following topics: arithmetic, algebra, geometry, trigonometry, and calculus.

### For ISI:

UGA (Multiple-choice Type)

Time: 2 hours

Instructions. UGA is a multiple choice examination. Each of the following questions will have exactly one of the choices to be correct. You get four marks for each correct answer, one mark for each unanswered question, and zero marks for each incorrect answer.

### UGB(ShortAnswerType)

Time: 2 hours

This section will contain 8 short answer type questions where you have to show all your work and prove what has been asked. Credit will be given to a partially correct answer. Do not feel discouraged if you cannot solve all the questions. This paper will be checked only when you clear the cut-off of the UGA paper.

### For CMI:

Time allowed is 3 hours. Total points: 100 = 40 for part A + 60 for part B.

The question paper usually contains two parts, the first part contains 10 questions and the second paper contains 6 questions.

Part A will be used for screening. Part B will be graded only if you score a certain minimum in part A. However your scores in both parts will be used while making the final decision. Specific instructions for each part are given below.

• Advice: Please ensure that you have about 2 hours left for part B.