How to prepare for JEE Advanced from class 11th

So you got 96% in your 11th test conducted by your school. You are entering in class 12th. The usual clamor for the preparation for JEE has started in the family and in your friend circle. You don’t care much because you have done wonderful with your 11th. So you start solving problems of JEE related to the syllabus of class 11th. The first question you encounter is finding the possible number of real roots of a polynomial with degree 4. You are stumped. This is not what you studied in class 11th. All you did was few equations of degree 2. Then what is this?

The reason for this is difference between the “depth” of some of the topics in 11th versus the syllabus of JEE Advanced comprising the 11th topic. Since 12th syllabus is pretty vast, you do not have time to go through 11th chapters once again. This creates problems because not doing 11th topics well will reduce your chances further.

Had it not been better if you knew the topics which are hardly touched by your NCERT books? Let me list some of the topics which you need to go through in depth before you can call them “complete”. The chapters in NCERT hardly touch the surface of these topics.

Theory of Equation

In your 11th book, there is a chapter on quadratic equation. This is such a small chapter that you can finish it in 2-3 hours irrespective of your intelligence. Yes, try that if you haven’t.

However, JEE is notorious for going deep into this. Theory of equations is a very important topic both from calculus and algebra point of view. It literally has no bound. The rules governing nature of roots, sum and product of roots, sign of roots, sum of series of various forms of roots, relation between the roots of an equation and its derivative, and nature of equation between two values of x, convergence or divergence of equation with x tending to infinity, division of an expression, finding highest common factors (HCF), and many more phenomenon are huge in number. The syllabus does great injustice to this one of the most important topics for JEE main and advanced.

Complex number

The NCERT book for 11th combines quadratic equation and complex number in one chapter which can be studies in one sitting. The real scope of complex number in JEE is humongous. This is one of the toughest topics for students. JEE question setters just love this chapter. Playing with arguments of a complex number with different sign of real and imaginary part, using geometry with complex numbers, and twisting all-pervading and unpronounceable De-Moivre’s theorem are some of the ways JEE questions complicate the already complex topic.

Conic Section

The content in conic section in NCERT book is pretty small. They cover just the basic of conic sections. It doesn’t cover tangents, normal, pole & polar, combination of various conics, other forms of conics etc. Once again, JEE loves conic sections, especially JEE Advanced. They have almost abandoned the poor straight line and circle to favour conic section.

What should you do?

While nobody expects CBSE to go deeper into such topics as they are not meant for coaching you for JEE but treatment of theory of equation, complex number, and conic sections given in the 11th textbook is too shallow. There are few more chapters but they cover about 50% or more (at least concept wise).

Hence to prepare well for JEE and do not get overburdened with your 11th topics in your 2nd year, focus on theory of equation and complex number in great details. You must make it sure to cover the gap between CBSE syllabus and JEE standard. You can go through any of the JEE books (Cengage, Arihant, Eduwiser etc.) and study complex numbers and theory of equation in details, solve each and every question, and note down the lessons learnt, important formula, important concepts etc. so that this comes handy for last minute preparation.

Conic section is time consuming. It will require discipline and hours of practice to get expertise. Thankfully since the syllabus of 11th is small compared to 12th, you get more time to study. Use this time to achieve a certain level of expertise before you go to class 12th. Class 12th syllabus is huge. Coupled with the pressure of board, it is highly unlikely that you will have to time to study the 11th syllabus in details in your second year.

Work through these topics and do better in JEE.

~ Pankaj Priyadarshi

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What is the relation between the roots of f(x) and its derivative

Very often students come across questions where they have to deduce the number of roots of a function based on the number of roots of its derivative and vice versa.  Even though there are exact rules regarding the relation between the roots of a function and its derivatives, students get confused. Here is some of the concepts defining the relation. Let’s take some of the typical scenarios.

In all the cases, assume that the function f(x) is differentiable for all x on the set of real numbers, i.e. R. Naturally; f(x) will be continuous for all x on R too.

  1. If a function f(x) is a polynomial of degree n has p real roots and n-p imaginary roots, what can be said about the number of real roots of its derivative, i.e. f’(x).
  2. If f(x) is a polynomial of degree n has n real roots and no imaginary roots, what can be said about the number of real roots of its derivative, i.e. f’(x).
  3. If f’(x) is a polynomial of degree n has q real roots, what can be said about the nature of roots of f(x).
  4. If f(x) and f’(x) has common roots, how to know about this.

Let’s take these questions one by one.

  1. If the function f(x) of degree n has p real roots, then its derivative has at least p-1 real roots. The keyword here is at least. There are cases when the derivatives can have more real roots than the function. For example, take a simple quadratic equation f(x) = x2 + x + 1 has 0 real roots while its derivative f’(x) = 2x + 1 has one real root. There are other cases such as f(x) = x3 – 3x + 5 and f’(x) = 3x2 – 3. The function f(x) has just one real root while f’(x) has 2 real roots.
  2. If the function f(x) of degree n has n real roots, then f’(x) will have n-1 real roots and f’’(x) will have n-2 real roots etc. The degrees of f’x) and f’’(x) are n-1 and n-2 respectively. This means all the roots are real. This means if all the roots of a function f(x) are real, then all the roots of f’(x), f’’(x), f’’’(x) etc. will be real.
  3. If any of the derivative of f(x), i.e. f’(x), f’’(x), f’’’(x) and more do not have all real roots, then f(x) doesn’t have all real roots. So it is useful test if you have to find if a polynomial has all real roots. Keep differentiating and checking if the derivative has got any complex roots. If yes, then the polynomial doesn’t have all real roots.
  4. If f(x) and f’(x) have common roots, you can take HCF (highest common factor) between f(x) and f’(x). If the HCF is an expression in x, then the roots of that expression are also the common roots of f(x) and f’(x). If the HCF is 1, then there is no common expression between f(x) and f’(x). This means there is no common root.

By the way, do you know how to find HCF of two expressions?

~ Pankaj Priyadarshi

How to take competitive tests and use your time effectively

In pen and paper test format, you have the ability to take a look at all the questions in one go. This provides you with certain advantages that online test takers do not have. Online test takers can see only one question at a time. The comprehensive view of the entire question paper helps you plan your strategy better.

Despite the flexibility offered by pen & paper test, students tend to do the same mistakes. This is not surprising though as each year the test takers change and for them the same mistake is new mistake. Here are some of the most common mistakes in tests that students make. These mistakes play havoc with their chances of clearing the test and scoring good marks.

  1. Spending excessive time in solving a difficult question thus leaving less amount of time for easier ones.
  2. Leaving few easy questions because of lack of time
  3. Making silly mistakes in easy questions because of lack of time
  4. Leaving out revision because of lack of time

All these mistakes are common. You may not make all of them but some of them. If someone is not making these mistakes, he or she is well on the path of success. Good luck.

However, if you are one of those who can identify with these mistakes, here we are giving some thoughts on how you can avoid them and use your time more effectively. These are simple, yet powerful strategies known to improve the score as well as provide framework for disciplined way of taking tests.

Here is what can be done to improve the efficiency and thus score in the tests.

Spend few minutes in going through the question paper

As soon as you get the question paper, give some time to it rather than start solving from the start. Go through the questions. This will give you a fair idea about the “toughness” of the paper.

Categorize the problems

Categorize the questions in 3 types; namely easy, medium, and difficult. Mark these problems on the question paper in any symbol that you can use. For example, you can use tick mark for easy, circle for medium, and star for difficult ones. Now the question is how you categorize the questions under these categories. The easy ones are those which are easier to solve for you. You know the process and you also know the result that the process generates.

The medium ones are those which you know how to approach initially but depending on the interim results, you will use further concepts to solve. For example, a question on functional equations may come under medium for you because you may know that partial differentiation is way to go, but whether this produces the equation in right form for further work is not clear.

Difficult ones are those which you either cannot solve or have a vague idea about how to approach.

Go in the order of easy, medium, and difficult

Start with solving the easy ones. Solving them first not only gives you a solid start but also initial confidence. The initial confidence is crucial in any competitive test. Many careers are wasted because students lose confidence in the first few minutes of the tests. So start with the ones you can solve.

Once you complete easy ones, attack the medium ones. Some of these questions will open up more as you start working on them and you will find your answer. Some of them will give you hard time and you fail to get the answer. In such cases, try the process of elimination. Usually, you will be able to eliminate 2 guesses. For the remaining two, you will have to work little more. Use values, work backward etc. to eliminate the last wrong answer and select the right. Of course, the process is elimination is more useful in case of single choice. For multiple choice and other types, you have no option but to solve. This can be extremely useful in JEE Main as the questions are all single choice.

For difficult ones, try with the ones which come from your relatively stronger subject. For example, if your Algebra is relatively stronger than your calculus, start with difficult questions from Algebra. You will have higher chances of getting it right than if you start with calculus.

Leave 15-25 minutes in the end for marking, revising etc.

This is the most crucial step. Many students circle the options in the OMR sheet at the end of the test. If you are one of them, leave ample time for marking and revision. If you did it before the time, you can go back to solving the remaining questions.

At all cost, avoid rush in filling up the options in OMR sheet. Any mistake in sequence of questions and wrong filling the answers will do irreparable damage.

Take these steps and see your results getting better.

~ Pankaj Priyadarshi

Use these topics to increase focus on Mathematics

Mathematics is quite scoring subject for board and IITJEE if students do enough practice with focus. What is important in mathematics is the ability to focus on a problem long enough to solve it. There are questions which can be solved easily while there are ones which require your utmost focus and ability to look at the problem from all angles possible. Especially JEE Advanced problems require you to think from various perspective. For example, many questions in coordinate geometry can be solved using complex number concepts. Many in 3D can be solved by suing the concepts learned in 2D. Vectors can not only help you solve coordinate geometry, both 2D and 3D but also problems in physics. Similarly concepts in theory of equations in Algebra can be used effectively to solve problems in calculus. We all anyway know between huge overlap between permutation and combination with probability.

Distractions cause irreparable damage to your career

The question here is how to overcome students’ propensity to check their devices every now and then. Even when they are switched off, the focus span is too low to really solve long questions. A long question requires focussed work for 20-30 minutes if you are doing it for the first time. Once you get accustomed to such questions, the time reduces to 5-10 minutes. However, the journey from 30 minutes to 10 minutes is crucial. This comes out of nothing but focus and practice, focus and practice, and once more focus and practice. There is no escape.

How do we develop this skill? I am sure many students are really serious about increasing their focus span but fail to do so because of sheer volume of distractions in today’s life. Here is what I think students should try to increase the focus on problem solving. Remember that focus will only come when you solve long questions which require your brain to think and your hands to work. These are the topics which you should solve to increase your focus span. This would enable you tackle challenging problems and also avoid silly mistakes that most of students make while solving a lengthy question.

Here are 5 topics that you can study in details and solve questions to increase your focus span.

Do the elementary operation to find inverse of a matrix

You know what you have to do. You also know the elementary operations that you can do and cannot do. Basically, you know the rules and the objective. All you have to do is apply your common sense and get 1 and 0 in order to get unit matrix out of any given matrix and apply the same in unit matrix to get the inverse. Do it for 3*3 and 4*4 matrices. Get it right the first time.

Solve challenging determinant equality questions

The other way is to solve the questions on determinant where a determinant must be modified to get a given answer. Some of them are really tricky and require you to think from various angles to get it into the right form. In this case too, the rules and clear and objective are given.

Draw graph to solve calculus problems on functions, continuity, and differentiability

Drawing graph requires you to follow few steps and use common sense such as finding the increasing or decreasing nature of function, shape of curve (concave up or down), behaviour of function at infinity, vertical and horizontal asymptotes if any, common points which can help use connect etc. When you think through all these points, you get a complete view of the graph and in turn learn to focus on problems for longer.

Do Bayes’ theorem problems

This too requires you to frame the problem to fit the formula used in Bayes’ theorem. Framing part is important and it requires you to think through the problem and premises given.

Try word problems & transportation problems in linear programming

These types of problems require you to translate the problems from English to mathematics. This requires you to study the problems for objective, premises, decision variables, and constraints. Looking at problem in a wholesome way improves your focus span and concentration.

There are other ways too to help you increase the focus on study of mathematics. I have mentioned few of them which can be used as a stepping stone to start building your ability to focus and work through a problem for longer duration. Solve these topics and see your scores in exam.

~ Pankaj Priyadarshi