The Quantitative Aptitude section in the CAT draws upon theory learnt in school up to the 12th grade, so it is theory that every one of us has learnt at some point of time or the other. Therefore, there is absolutely no need to pick up textbooks with advanced engineering mathematics in them. Textbooks I would recommend for theory are the NCERT textbooks for the 9th to 12th grades, and a personal favourite of mine - 'Higher Algebra' by Hall and Knight. All of these are very reasonably priced.
The most important part of preparation for QA in CAT, and in fact for CAT overall, are practice tests. I would recommend taking every practice test as seriously as the real CAT. Time yourself for every test that you take and have a target time in which you have to finish the test. For engineering students or students currently in college who may be familiar with the theory, it is okay to start off with tests immediately. However, for working professionals who may be out of touch with academics, I would recommend taking at least a couple of weeks to look at some of the formulae and theorems that you will require for your CAT.
Start off with topical tests in the initial stage of preparation. When you gain confidence in several topics, it is time to start giving a couple of full-length QA tests. After you have gained confidence in QA, merge individual sectional tests and start giving CAT-type full length test papers containing all three sections. Always analyse your performance after every test you give and use tests as a valuable feedback mechanism. If you feel the need, keep going back to topics which you feel require more work and take 1-2 more area-specific tests in that topic. An important thing you need to work on is the judicious selection of questions. Utilize practice tests for this purpose.
Most importantly, try and ensure that you do not have more than one really weak topic which you wish to avoid, as you never know which areas the CAT will test you on. It is okay to have one dodgy area, but you will still be taking a risk, and need to be that much better in the other topics. Therefore, I would recommend working hard in every area, so that you have the luxury of having all questions to select from in the real CAT.
Quantitative Aptitude for CAT can be broadly divided under three main heads:
1. Geometry, Coordinate Geometry and Mensuration: I have grouped these topics together since they deal with the portion of QA that can be visualized. Of the three, maximum weightage is given to geometry, although every CAT paper will have 3-4 questions on mensuration, as well as a couple of questions on coordinate geometry, totalling about 25-30% of questions in the QA section. Topics that need to be covered in geometry are basic theorems involving triangles, circles and parallel lines. A common type of question that is often asked in CAT is to find the value of certain angles or length of certain sides. Therefore, make sure that you cover topics such as congruency and similarity of triangles.
The only things that you need to do in coordinate geometry are straight lines and circles. Don't go into conic sections and other advanced topics. More importantly, do not try and solve IITJEE level questions in coordinate geometry. Given the equation of a circle, you should be able to comment on the centre and radius of the circle and draw it on a piece of graph paper, and nothing more. Similarly, you should know what the slope and y-intercept of a given straight line equation is, and be able to draw the line on a piece of graph paper.
For mensuration, flip through a school level textbook for basic formulae on areas, surface areas and volumes of triangles, circles, cylinders, cones, cuboids and spheres. Mensuration problems are calculation intensive, and require lots of practice.
NCERT textbooks will suffice for this head.
2. Algebra and Number Theory: Algebra and number theory provide the major chunk of questions in any CAT QA section - 55-60%. Topics that you need to look at are Permutations and Combinations, Probability (very basic, including die and card problems and perhaps Bayes' theorem), Functions, Progressions (A.P, G.P. H.P. and A.G.P), Logarithms, Equations (Quadratic and Linear/Simultaneous) and, most importantly, Number Theory.
Number Theory problems are usually very simple, if you know how to do them. They require certain tricks that you can pick up from any good textbook. Having said that, number theory contributes 3-4 questions to every CAT, and so it is a very important topic. You should be comfortable writing numbers in their algebraic form (e.g. a three digit number having digits xyz can be represented as 100x + 10y + z). You should also learn about divisibility tests and the 'modulo' notation and its applications (for programmers, 10%5==0 is also referred to as 10 modulo 5 is 0, that is, the remainder when 10 is divided by 5, is zero).
A textbook I would recommend for algebra and number theory is 'Higher Algebra' by Hall and Knight, which is available at any bookstore that sells textbooks for IITJEE.
3. Arithmetic and Miscellaneous: 15-20% of questions in any CAT paper fall under this head. Major topics that you need to cover are Set Theory (especially Venn diagrams) and problems on Time, Speed and Distance, both of which are always asked. Both of these topics are covered as part of the school syllabus, but may need some brushing up on. Sometimes, questions on topics such as Linear Programming are also asked. An NCERT textbook is enough to study from for this head.
Miscellaneous problems are those problems which do not fall under any head. They are rarely asked, and even when they do appear in a CAT paper they do not number more than one or two. They are purely tests of mathematical aptitude, and you cannot learn how to solve them. The only advice I can give for dealing with these problems is to try back-substitution of answer choices, or to avoid these problems altogether.
An area that had a high concentration of questions in CAT 2007 was Data Sufficiency. Data Sufficiency problems can come from any of the three heads, and are in the form of a question followed by two statements. You need to answer whether you can solve the problem using the statements individually, or using both, or whether you cannot solve the problem using the information provided. The key to answering such problems is to pretend like one statement does not exist, try solving the problem, then pretend like the other statement does not exist and try solving the problem again. These problems are generally tricky, and I would recommend lots of practice and perhaps solving them near the end of your QA section, after you have solved the other problems.
Although there is absolutely no substitute for knowing your theory, and practice, in your QA section, there are some question-solving strategies that you may use. They are:
- Substitution of numbers for variables in algebraic problems, which may make the problem simpler. Remember, however, that this usually does not work when the answer choices are also in terms of variables.
- Back-substitution of answers into the problem in order to solve it, i.e. assume one of the answer choices to be the answer and then solve the problem. If the problem cannot be solved or reduces to the trivial case, repeat for another answer choice until you stumble on the correct answer choice.
- Substituting variables for numbers in the answer choices. This usually works for progression problems. Lets say the nth term of a progression is given in terms of n and some other terms. You are then asked to find the 100th term in the progression. The answer choices are of the form 2100, 299 - 1 etc. (say). Then, you can start with the first answer choice, assume that the nth term will be 2n, solve the first few terms of the progression and find if this is indeed the case (lets say it's easy calculating the 3rd term, which you find to be 8 or 23. Hence the 100th term will be 2100). If it is not, assume that the nth term is 2n-1 - 1 and repeat, until you get to the correct answer choice.
- Solving coordinate geometry algebraically, or vice-versa. Often a complicated algebra problem involving several equations can be solved very easily if you draw the corresponding figures on an imaginary graph paper. Similarly, coordinate geometry problems can often be solved by writing corresponding algebraic equations. Always remember the correspondence between algebra and coordinate geometry.
- If you can eliminate all options except two, guess. The CAT rewards educated guessing. Look at it this way: If you have two questions, probability states that you will get one of these wrong and the other right. The expected number of marks you will get is +4 and -1, which translates to +3 for both the questions combined, or +1.5 per question answered. If you do this for a significant number of questions, unless you are exceptionally unlucky, the benefits of not wasting time solving every problem completely will far outweigh the loss of marks due to some incorrect answers.
To summarize, the most important part of your QA preparation is PRACTICE. The theory is not too tough, so practice as much as you can. QA has been an area where students have done well in the last two CATs, so you should look at it as an area where you can also improve your overall score.
The total quant area can be divided into 4 main parts:
Part 1: Numbers, and Geometry and Mensurations -- These two topics have to be considered a separate area, because the maximum number of questions come from here.
Part 2: Arithmetic -- Equations, Ratio Proportion and Variation, Percentages Profit and Loss, Averages Mixtures and Alligations, Simple Interest and Compound Interest, Time and Work, and Time and Distance.
Part 3: Algebra -- Indices, Logs and Surds, Quadratic Equations, Progressions, Special Equations and Inequalities.
Part 4: Pure Maths --
Permutations and Combinations, Functions and Graphs, Number
Systems, Coordinate Geometry.
1. Number Systems and Number Theory
3. Profit and Loss
4. Simple & Compound Interest
5. Speed, Time and Distance
6. Pipes and Cisterns
9. Ratio, Proportion
10. Mixtures and Alligations
1. Linear and Quadratic Equation
3. Progressions - AP, GP, HP
4. Binomial Theorem
6. Permutation & Combination
9. Set Theory
2. Co ordinate Geometry