The Complex Number came into existence because of our inability to solve some quadratic equations such as x^2 = -4. Complex Number is one of the feared topics by competitive exam aspirants, but it’s not that difficult from JEE point of view. All it requires is an indirect use of vectors and coordinate geometry. It’s best to practice more and more so that you are able to categorize different types of questions yourself. In most of the cases, you will have linked questions so all you need is a good understanding of the ‘concept’ rather than the tricks that can be applied in some specific situations only.
Quadratic Equations, on the other hand, being the most important and basic chapter in JEE Mathematics, is an easy topic of algebra. It is of importance largely because multiple questions in the other chapters are based on quadratic equations, which you cannot solve until and unless you have a good hold on this chapter. Quadratic equation inequalities form a major part of the problems in JEE.
- NCERT mathematics textbook: This book contains all the relevant matter along with basic and easy questions being more than enough as far as JEE Mains is concerned.
- R. D. Sharma: For practicing more and more questions and building a strong foundation.
- IIT Mathematics by M. L. Khanna: (highly subjective book for JEE) For being thorough with the theory.
- Algebra for JEE Mains & Advanced by S. K. Goyal (Arihant Publications): You will need this book for both reading theory and practicing advanced level questions. It contains a large number of solved and unsolved examples.
Some basic tips
- Don’t refer to a solution without trying the problem, because most of the questions of complex number asked in JEE Mains as well as JEE Advanced are based on a totally new concept and approach. You don’t have to spend much time on each question but at least 70 – 80% questions should be attempted yourself first and then the solution is to be referred (give at least 15 minutes to each question).
- Complex number is basically divided into two parts:
- Algebra of Complex Number
- Geometrical Interpretation of Complex Number
Algebra part mainly focuses on the concepts of conjugate, modulus, argument and their properties. Sometimes question from this topic appear to be tricky, but by applying properties it can be reduced to the simplest form.
Geometrical interpretation covers rotation theorem (a bit tricky so may take some time), practice enough questions so that you know where and how to apply, it makes various solutions easy.
- Cube root of unity; its properties and Nth root of unity and its properties are of utmost important. Mainly series type of questions are asked in JEE from these topics as they appear tricky that way but are eventually easy.
- Locus: Questions asked are either Modulus based or Argument based, and so the answer may result into the general equation of a straight line, ray, circle, ellipse, or hyperbola. All these cases are discussed in Algebra for JEE Mains & Advanced (S.K. GOYAL) hence you can refer the book.
- De-moivre’s theorem: Most important topic of complex number. Though it appears a bit easy, but questions asked in JEE are twisted and tested indirectly. So, practice lots of questions on it and be confident.
- It is important to know the formula for sum and product of roots. Most of the questions asked in JEE can be solved by using these two values only. Secondly, if there is a question on solution of an equation, you should try to substitute the given options in the parent equation, i.e. find the value or values which satisfy the equation and are solutions. This saves time and helps in reaching the solution easily.
- Graphically quadratic equation represents a parabola. This representation is very useful. Always try to solve the question by the graphical method as this will consume ‘less time and efforts’. Solving/Practicing questions this way saves not only time but also helps you prepare better for coordinate geometry.
- Most problems of the Quadratic equation can be solved by using few identities. For example – Putting arbitrary values x1 & x2 and then using f(x1).f(x2) < 0, which means the root lies between x1 and x2.
- The best way to deal with quadratic equations is to take it as a ‘polynomial of second degree’ and apply the techniques of calculus. Concept of maxima and minima can be applied to find the roots of the equation, location of roots and nature of roots.
- Discriminant method of finding the nature of roots is one of the easy method used to solve problems. Most of the questions in JEE can be solved using this.
- Most problems on cubic polynomials are solved by finding one of its root by inspection and then reducing it to a quadratic form by using division.
- Whenever you get stuck in any problem of quadratic equation, just write down the general equation [P(x)=a(x^2)+bx+c] and start plugging in some random values of ‘x’. With each value try to find some common relation between a, b and c. With this the question becomes easier to solve.
Some important points
- Every year at least two questions are asked in JEE Mains and three to four questions in JEE Advanced from Complex Number. Also, many times paragraph or matching type questions are based on complex number. So, a thorough understanding of the topic is very important.
- Questions from Quadratic Equation will always be clubbed with a question from another topic in JEE Advanced. But for JEE Main, which is relatively simple, there will be direct questions from the topic.
- Solving previous year JEE Mains/JEE Advanced Questions from both the chapters is of utmost importance so as to get familiar with the type/level of questions asked in both the exams.