Physics and chemistry have standard prototype solutions for most problems, but in math, it often happens that you start out on a particular path to answer a question and end up arriving nowhere.

You face a trigonometry problem or a series summation and you don’t know where to start, which angle to consider first, whether to solve the series directly, or to invoke the differences of consecutive terms. The single-step solution to all these troubles is to keep practicing.

**Algebra**can be made easier if you have the ability to picture functions as graphs and are good at applying vertical and horizontal origin shifts carefully as zeroes of functions and other specific values can be done in much less time using these techniques.

**Differential calculus**again relates well to roots of equations, especially if you use the Rolle and Lagrange theorems.

**Complex numbers**can be used to solve questions in co- ordinate geometry too. Trigonometric questions require applications of De Moivre’s theorem.

Then, another very important topic in algebra is

**permutation-combination and probability**.

You need to be thorough with the basics of Bayes theorem, derangements and various ways of distribution, taking care of cases where objects are identical and when they are not.

**Matrices**can be related to equations, hence a 3x3 matrix can actually be visualised as being three-planed in 3D geometry. Determinants have some very nice properties, for instance, the ability to break them into two using a common summand from a row/ column, which should be made use of in tougher questions.

**Integral calculus**can be simplified using tricks and keeping in mind some basic varieties of integrable functions, lists of which can be found in the Arihant book authored by Amit M. Aggarwal. It has many useful tricks in its pages, which if kept in mind can help you step up your speed up in competitive exams.

Achieving upper and lower bounds and then verifying the options also works sometimes.

**Coordinate geometry**requires a good working knowledge of the parametric forms of various conic sections and an ability to convert the other, tougher ones (like when the major axis is at an angle to the x- axis) to these basic forms and then interpret the solutions accordingly.

*— Vipul Singh ranked fifth in the IIT- JEE 2010 and topped the AIEEE 2010.*
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