I will talk about the book Mathematics Volume-1 for Class 11 by R.D Sharma which will help you prepare for both your board exams (it says it is written for CBSE students, but it may be used for ISC as well) and competitives:
Chapter 1 is called SETS. It starts from the very basics by defining and showing you how to convert a set from roster form to set-builder form and vice versa. Then it talks about the different kinds of sets, subsets, universal sets, power sets followed by Venn Diagrams (important), operations on sets (important) and laws of algebra of sets along with proofs (important).
Chapter 2 is called RELATIONS. It talks about ordered pairs, cartesian products of sets, some useful theorems along with proofs (important) and Relations and related concepts. Make sure you understand Chapter 1 before you start this one. (More in Class 12)
Chapter 3 is called FUNCTIONS. It has interesting discussions on functions as special kinds of relations, functions as a correspondence, and a detailed explanation of real functions including their domain, range, graphs of important functions (very important) and operations on real functions. This is a very important chapter, so study it well. (More in Class 12)
Chapter 4 is called MEASUREMENT OF ANGLES. It contains some basic theory on angles and systems of measurement of angles and relations between them which you should understand.
Chapter 5 is called TRIGONOMETRIC FUNCTIONS. It explains trigonometric functions from the very basics and how their values for certain angles are obtained and then it moves onto trigonometric identities, explains how to obtain the signs of trigonometric functions, their values at allied angles and talks about their periodicity. At the end there is discussion on which trigonometric functions are even and which are odd.
Chapter 6 is called GRAPHS OF TRIGONOMETRIC FUNCTIONS. This is an extremely important chapter which you will need in other subjects as well. It contains detailed discussions on the graphs of all the trigonometric functions and I highly recommend that you study it in detail.
Chapter 7 is called VALUES OF TRIGONOMETRIC FUNCTIONS AT SUM OR DIFFERENCE OF ANGLES. It contains many formulae which you need to memorize along with proofs. Sections 7.4 and 7.5 are very important. Study them well.
Chapter 8 is called TRANSFORMATION FORMULAE. This chapter contains more important formulae which you need to know by heart and also how to apply to solve problems which the chapter demonstrates through the worked out examples. So, practice them well!
Chapter 9 is called VALUES OF TRIGONOMETRIC FUNCTIONS AT MULTIPLES AND SUBMULTIPLES OF AN ANGLE. This chapter again contains many important formulae which you will need to use alongside the formulae from Chapters 7 and 8 to solve problems. It will take sometime to know which formula to use and where, so make sure you get enough practice.
Chapter 10 is called SINE AND COSINE FORMULAE AND THEIR APPLICATIONS. It contains many formulae involving triangles which are very important because you will need them to solve problems in other subjects as well. So make sure you practice them until they are like second nature to you.
Chapter 11 is called TRIGONOMETRIC EQUATIONS. It lists some theorems along with proofs of general solutions of trigonometric equations which are very important. So, make sure you learn them and practice the problems.
Chapter 12 is called MATHEMATICAL INDUCTION and has some brief theory on mathematical statements and the first and second principles of mathematical induction and problems based on each. The main part of this chapter are the problems. It contains many different kinds of problems, both worked out and otherwise, so ensure that you get enough practice.
Chapter 13 is called COMPLEX NUMBERS. It explains the concepts of i (iota), imaginary quantities, complex numbers and their equality, addition, subtraction, multiplication, division, conjugate and modulus, reciprocal, square root, representations of a complex number (geometric, vectorial and trigonometrical or polar forms) along with representative problems on each. Everything in the chapter is presented in a logical and easily understandable way, so study everything thoroughly.
Chapter 14 is called QUADRATIC EQUATIONS. It introduces you to some general concepts and talks about quadratic equations with real and complex coefficients. Read the theory and solve the problems in the chapter before consulting other books for competitive exams.
Chapter 15 is called LINEAR INEQUATIONS. It starts by explaining basics regarding inequations in general (linear inequations in one variable, linear inequations in two variables, quadratic inequations, obtaining solutions etc) and it moves on to discussing solving different forms of inequations in one variable in detail. It also talks about solving systems of linear inequations in one variable and logically lists some theorems along with proofs which will come in handy while solving problems of this type. It also shows you how to apply these techniques to solve different kinds of word problems. The chapter ends by demonstrating how to graphically solve individual linear inequations and simultaneous linear inequations in two variables. This chapter contains very good theory on the topic and I highly recommend you go through it and practice the problems.
Chapter 16 is called PERMUTATIONS. It begins with a discussion on the concept of factorial followed by the fundamental theory of counting. It is a very important basic concept which you will use a lot in permutations. Then it starts explaining the concept of permutations with illustrations and theorems, followed by permutations under certain conditions and permutations of objects not all distinct along with lots of problems. The explanations and problems are very good and I recommend that you study them well.
Chapter 17 is called COMBINATIONS. It explains combinations, differences between a permutation and combination and proves the formula of all combinations of ‘n’ distinct objects taken ‘r’ at a time. Then it lists some properties along with their proofs which you will need to solve problems. After that there are problems on combinations as well as mixed problems on permutations and combinations. Practice them well.
Chapter 18 is on BINOMIAL THEOREM. It begins by introducing the reader to Pascal’s triangle. Then it moves on to proving the binomial theorem for positive integral index along with important conclusions which you can draw from it. The next section talks about general and middle terms in a binomial expression. There are a plethora of different kinds of problems in the chapter. Practice them: you are likely to see similar problems in your exams.
Chapter 19 is called ARITHMETIC PROGRESSIONS. It starts with a general discussion on sequences and then discusses Arithmetic Progression (A.P) in particular. It defines what an A.P is, followed by showing you how to find the general term of an A.P, methods of selecting terms in an A.P to solve problems, derives the formula for calculating the sum to ‘n’ terms of an A.P, lists some very important properties of A.Ps along with proofs which you will need to solve problems and shows you the technique of insertion of arithmetic means to form A.Ps. There are many different kinds of problems practising which will help you get used to the material.
Chapter 20 is called GEOMETRIC PROGRESSIONS. There is a brief introduction on Geometric Progressions (G.P), followed by discussions on a couple of theorems on finding general terms of a G.P, selection of terms in G.P to solve problems, sum of the terms of a G.P, sum of an infinite G.P, properties of geometric progressions with proofs, insertion of geometric means between two given numbers along with a couple of properties of arithmetic and geometric means with proofs and many different kinds of problems.
Chapter 21 is called SOME SPECIAL SERIES. This is a short but important chapter that shows you different techniques to find the sum of many different kinds of series along with problems.
I will discuss the salient features of Mathematics Volume-2 for Class 11 by R.D Sharma which you should start after you finish Volume-1:
Chapter 22 is called BRIEF REVIEW OF CARTESIAN SYSTEM OF RECTANGULAR CO-ORDINATES. It starts with a brief introduction and then talks about the distance between two points, area of a triangle, section formulae, centroid, in-centre and ex-centres of a triangle, locus and shifting of origin along with problems. Go through this chapter before you start the next one.
Chapter 23 is called THE STRAIGHT LINES. It has an interesting definition of straight lines, followed by discussions on slope, angle between two lines and related concepts, intercepts of a line on the axes, equations of parallel lines, different forms of the equations of line, how to transform the general equation of a line in different standard forms, point of intersection of two lines, concurrency of three lines, lines parallel and perpendicular to a given line, different formula for calculating angle between two straight lines and related concepts, how to find the positions of points with respect to a straight line, distance from a point, how to calculate distance of a point from a line, distance between parallel lines, how to use the forumlae you’ve already learnt to calculate the area of a parallelogram, how to find the equation of lines passing through a given point and making an angle with a given straight line and how to find the equation of a family of lines through the point of intersection of two given lines. This is an extremely important chapter and you will need these concepts later on and in other subjects as well, so study it and practice the problems well.
Chapter 24 is called THE CIRCLE. It derives the equations of circles in different positions with respect to the co-ordinate axes and then has some good theory on the general equation of a circle and how to find the equation of a circle drawn on a straight line joining two given points as diameter and problems on all of these.
Chapter 25 is called THE PARABOLA. It starts with a general discussion on conic sections and focuses on parabolas: definition, derivation of the equation of a parabola in it’s standard form, various terms and results related to the parabola (very important) and other standard forms of parabola along with problems.
Chapter 26 is called ELLIPSE. It provides the definition and then discusses equation of the ellipse in standard form, various terms and equations (very important) and equations of the ellipse in other forms along with related problems .
Chapter 27 is called HYPERBOLA. It again begins with the definition followed by the equation in standard form, various important terms and equations and problems.
Chapter 28 is called INTRODUCTION TO THREE DIMENSIONAL COORDINATE GEOMETRY.
The topic is explained well starting from the very basics followed by the distance formula, section formulae and various problems based on them (More in Class 12).
Chapter 29 is called LIMITS. It has a very good discussion on the basics of limits, the algebra of limits (important), different methods of evaluation of algebraic, trigonometric exponential and logarithmic limits along with exercises which are all very important. Make sure you master the techniques.
Chapter 30 is called DERIVATIVES. It defines derivatives using the concept of limits along with the physical and geometric interpretations (important), followed by discussion of a derivative of a function and many theorems which derive the derivatives of many standards functions (very important) which you will need to memorize, followed by more theorems on differentiation which you will need to have at your fingertips as well. Practicing the problems are essential to get used to applying all the different formulae and concepts.(More in Class 12)
Chapter 31 is called MATHEMATICAL REASONING. It contains a lot of good theory on statements, negation of a statement, compound statements, basic connectives such as ‘and’ and ‘or’, quantifiers, different kinds of implications, validity of statements with ‘and’, ‘or’, ‘if-then’, ‘if and only if’, by contradiction and invalidity of statements by counter examples along with problems. It will take a lot of practice to get used to this material, so keep that in mind.
Chapter 32 is called STATISTICS. It starts with a good discussion on dispersion, measures of dispersion, range, how to calculate mean deviations of ungrouped data, discrete frequency distributions, grouped or continuous frequency distributions. That is followed by the concepts of variance and standard deviation: how to calculate the variance of individual observations, discrete frequency distributions and grouped or continuous frequency distributions. The chapter ends with a brief discussion on the analysis of frequency distributions. There are a lot of problems, both worked out and otherwise. It is essential that you get enough practice, so practice them all.
Chapter 33 is called PROBABILITY. There are good discussions on the all-important concepts of random experiments, sample spaces, events, algebra of events, types of events, axiomatic approach to probability (important) and some theorems along with proofs (which are also very important). There are a myriad of different kinds of problems and it is essential that you practice them. (More in Class 12)
I will talk about Mathematics Volume-1 for Class 12 by R.D Sharma, so read on!
Chapter 1 is called RELATIONS. It starts with a brief recap of what you learnt in Class 11 followed by types of relations and some theorems along with proofs and many different kinds of problems.
Chapter 2 is called FUNCTIONS. Once again there is a recap of Class 11 material, followed by good explanations on different kinds of functions (with illustrations), composition of general functions along with properties, compositions of real functions, inverse of an element, inverse of a function along with properties and how to draw the graphs of inverses along with problems. Practising these will clear your concepts and are similar to what you will see in your exams.
Chapter 3 is called BINARY OPERATIONS. This is not in the syllabus, so I will not talk about it here.
Chapter 4 is called INVERSE TRIGONOMETRIC FUNCTIONS. It describes inverses of different trigonometric functions along with their graphs (very important) and properties. This is a long chapter with many different kinds of problems which will go a long way in preparing you for your exams.
Chapter 5 is called ALGEBRA OF MATRICES. It starts with a brief introduction followed by types of matrices, equality of matrices, addition of matrices along with properties, scalar multiplication along with properties, subtraction of matrices, matrix multiplication along with properties, transpose of a matrix along with properties and symmetric and skew-symmetric matrices along with different kinds of problems on each.
Chapter 6 is called DETERMINANTS. There is a discussion on determinants of square matrices of different orders, singular matrix, minors and cofactors, properties of determinants and teaches you the different techniques of evaluation of determinants through many different kinds of problems (very important).The chapter ends with a brief discussion on applications of determinants to co-ordinate geometry and in solving homogeneous and non-homogeneous systems of linear equations along with problems.
Chapter 7 is called ADJOINT AND INVERSE OF A MATRIX. It explains the concepts of adjoint and inverse along with several theorems and elementary operations of a matrix with related problems (very important).
Chapter 8 is called SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS. This is a short chapter which shows you how to use matrices to solve non-homogeneous and homogeneous systems of linear equations.
Chapter 9 is called CONTINUITY. It starts with a very interesting introduction on continuity and then moves on to discussing algebra of continuous functions (contains important theorems with proofs), continuity on an interval and properties of continuous functions (again with very important theorems with proofs) and problems.
Chapter 10 is called DIFERENTIABILITY. It starts by explaining differentiability at a point followed by differentiability in a set along with problems.
Chapter 11 is called DIFFERENTIATION. There is a recap of what you learnt in Class 11 followed by different theorems on how to derive formulae of differentiation of inverse trigonometric functions from first principles. That is followed by a very good discussion on chain rule and several theorems on how to derive formulae of differentiation of inverse trigonometric functions, this time using chain rule. After that, there are a ton of problems on differentiation using trigonometric substitutions, logarithmic differentiation, differentiation of infinite series, differentiation of parametric functions, differentiation of a function with respect to another function and differentiation of determinants which are all very important for your exams.
Chapter 12 is called HIGHER ORDER DERIVATIVES. It mostly contains different kinds of problems on higher order derivatives. You will learn a lot of vital problem-solving techniques by practicing the problems in this chapter.
Chapter 13 is called DERIVATIVE AS A RATE MEASURER. This chapter shows you how to apply derivatives to solve different kinds of problems involving rates.
Chapter 14 is called DIFFERENTIALS, ERRORS AND APPROXIMATIONS and shows you how to use the concept of differentials to solve problems involving errors and approximations.
Chapter 15 is called MEAN VALUE THEOREMS. It talks about Rolle’s theorem and Lagrange’s mean theorem. This is a short but very important chapter and the concepts and problems in this chapter are of utmost importance for your exams.
Chapter 16 is called TANGENTS AND NORMALS. It explains how to find slopes of tangents and normals, equations of tangent and normal, angle of intersection of two curves and related problems.
Chapter 17 is called INCREASING AND DECREASING FUNCTIONS. It demonstrates how to solve inequations known as rational algebraic inequations followed by the concepts of strictly increasing and strictly decreasing functions, monotonic functions, necessary and sufficient conditions for monotonicity and problems thereon.
Chapter 18 is called MAXIMA AND MINIMA. It has very good explanations on maximum and minimum values of a function in its domain, local maximum and local minimum and first derivative test to find them, higher order derivative test and related concepts, maximum and minimum values in a closed interval and problems on the aforementioned concepts as well as applied problems on maxima and minima which are very important.
Chapter 19 is called INDEFINITE INTEGRALS. It lists all the different integration formulae which you will need (which you can easily derive based on differentiation formulae), some standard theorems (with proofs), geometric interpretation of indefinite integrals (important) and many different methods of integration along with problems on each method which are extremely important. This is a very long and very important chapter which contains pretty much everything you need to know on indefinite integrals. So, take your time to study it and practice the problems until you are comfortable with all the techniques.
Here is a discussion on Mathematics Volume-2 for Class 12 by R.D Sharma:
Chapter 20 is called DEFINITE INTEGRALS. It starts with the fundamental theorem of integral calculus followed by problems, a discussion on the evaluation of definite integrals by substitution, properties of definite integrals along with proofs (very important), integration as the limit of a sum (also very important)along with problems. You will see similar problems in your exams, so it essential that you practice them.
Chapter 21 is called AREAS OF BOUNDED REGIONS. It shows you how to calculate areas of bounded regions using vertical and horizontal strips, area between two curves using vertical and horizontal strips along with problems.
Chapter 22 is called DIFFERENTIAL EQUATIONS. It starts by defining order and degree of a differential equation followed by discussions on formation of differential equations, solution of a differential equation (general solution and particular solution), initial value problems, general form of a first-order first-degree differential equation, methods of solving different types of first order first degree differential equations (important) along with applications of differential equations. Practicing the problems will prepare you for your exams.
Chapter 23 is called ALGEBRA OF VECTORS. It starts from the very basics and talks about addition of vectors along with properties, subtraction, multiplication of a vector by a scalar along with properties, position vectors, theorems on section formulae along with proofs, linear combination of vectors, components of a vector in 2-D and 3-D, collinearity (along with a couple of theorems), coplanarity (along with some theorems) and direction cosines and direction ratios (along with some important theorems).
Chapter 24 is called SCALAR OR DOT PRODUCT. It defines scalar product and moves on to the geometric interpretation, properties (along with proofs) and derives some important formulae based on scalar product.
Chapter 25 is called VECTOR OR CROSS PRODUCT. It again starts with the definition followed by the geometric interpretation, properties (along with proofs), followed by some formulae (along with proofs).
Chapter 26 is called SCALAR TRIPLE PRODUCT. It once again contains the definition, geometric interpretation and properties along with proofs.
You may have seen some of the material on Chapters 23, 24, 25 and 26 before, nevertheless study everything, specially the problems as preparation for your exams.
Chapter 27 is called DIRECTION COSINES AND DIRECTION RATIOS. You were introduced to three dimensional geometry in Class 11. This chapter starts with a recap of what you learnt in Class 11 followed by explanations on direction cosines and direction ratios of a line, angle between two vectors and many important problems.
Chapter 28 is called STRAIGHT LINE IN SPACE. This chapter talks about vector and cartesian equations of a line with several theorems along with proofs (important), angle between two lines, intersection of two lines, perpendicular distance of a line from a point, shortest distance between two straight lines along with related problems.
Chapter 29 is called THE PLANE. It has very good explanations on the general equation of a plane, equations of a plane passing through a given point, intercept form of a plane, vector equation of a plane passing through a given point as well as normal to a given vector, equation of a plane in normal form, vector equation of a plane passing through three given points, angle between two planes, equation of a plane passing through a given point and parallel to two given vectors, equation of a line parallel to a given plane, equation of a plane through the intersection of two planes, distance of a point from a plane, distance between the parallel planes, different formulae relating lines and planes and image of a point in a plane.
The chapters on three dimensional geometry are long, but I would suggest learning how to derive the formulae on three-dimensional geometry instead of just memorizing them. You will find it much easier to solve problems if you do that.
Chapter 30 is called LINEAR PROGRAMMING. It has an introduction followed by definitions of some terms, mathematical formulation (transforming verbal description into mathematical form), concepts relating to the solutions of linear programming problems, graphical methods of solving linear programming problems, corner-point method, ISO-Profit or ISO-Cost method and explains the formulation and solution of different types of linear programming problems with examples. Practicing the problems in the chapter will help you get used to solving these types of problems and quickly.
Chapter 31 is called PROBABILITY. It starts with a recap of what you learnt in Class 11 followed by excellent theory on conditional probability along with various theorems (very important), independent events, the law of total probability, Baye’s theorem (very important) and many problems solving which will prepare you for your exams.
Chapter 32 is called MEAN AND VARIANCE OF A RANDOM VARIABLE. It starts with the concept of discrete random variable, followed by probability distribution, mean and variance of a discrete random variable and related problems.
Towards the end of the book is an Appendix and a Supplement for Volume 1 and 2 of the Class 12 book.
What I like most about both the Class 11 and 12 books is that the theory is brief and easily understandable and there are solved examples and exercises of different levels after each section and also at the end of the chapter along with hints to NCERT and selected problems. What’s more, there are Multiple Choice Questions (MCQs), Fill In The Blanks Type Questions (FBQs) and Very Short Answer Questions (VSAQs) along with answers to everything and a summary. The problems are representative of what you will see in your exams (including competitives).
Good luck!
