NCERT Solutions for Class 12 Maths  Math Olympiad For Class 12  Olympiad Preparation Online  IMO  Math Olympiad Books
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NCERT Solutions for Class 12 Maths
Read Notes and Videos,Practice,Mock Tests for Class 12 Maths
Practice questions and NCERT Solutions for Class 12
Maths is one of the vital subject and it is also of very much importance while preparing for various competitive exams and hence it is essential to prepare it from the very beginning. Understanding the concepts and applying them in the provided situation can improvise your analytical as well as logical thinking.
Maths is a very important resource for students of CBSE Class 12. Maths is an interesting subject for the enhancement of analytical and problem solving skills.
Unfortunately, many students find it difficult because of no interest or lack of better knowledge to comprehend. Maths is really a difficult subject but very interesting when comprehended well and practice with easy to hard problems and students find it difficult because most of them do not have their basics clear for the subject. Lack of sufficient practice tool is a major reason why solving Mathematical problems seems a tough nut to crack.
School Connect Online students are not only solving and understanding questions from Maths from NCERT or Syllabus but also they are solving questions to practice Maths Olympiad this helps students to become expert
To help you with this, we have learning notes and videos, Practice questions for NCERT,JEE Main and Advance,IIT JEE Main and Advance, with Solutions for Class 12 Maths.
Practice Solutions for Class 12 Maths
NCERT solutions of Class 12 Maths includes solutions of all questions and more to help and support practice with more relevant and important questions. These solutions have been provided and prepared by the most experienced teachers and important resources. A very simple approach has been followed while solving the questions and designing the NCERT Solutions for Class 12 Maths. Students will find it extremely easy to understand the problems and how to go about solving them.
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CBSE Class 12 Deleted Portion of Syllabus for 20202021
Check subjectwise details of the deducted portion of CBSE Class 12 syllabus from the following links:
IIT JEE Main And Advance Paper Analysis for Maths
Maths
Maths was the easiest paper amongst the three. Most questions were calculative. There were 10 questions asked from the 12th standard syllabus and 20 questions from the 11th standard syllabus. This section had 3 difficult questions, 14 moderate questions and 13 easy questions.
This is a breakup of chapters which had the toughest and easiest questions:
Tough Questions

Easy questions (11th Standard)

Easy questions (12th Standard)

Hyperbola
Parabola
Sets
Relations and Functions

Mathematical Induction and Reasoning
Parabola
Permutations and Combinations
Quadratic Equations
Sequences and Series
Statistics
Trigonometry

3D Geometry
Determinants
Differential Equations

This is the breakup of marks between different topics of the Maths section:
Topic

Marks Allotted

Algebra

56

Calculus

28

Trigonometry

8

Conics

16

Vector Algebra

12

Maths Syllabus for Class 12
Unit

Topic

Marks

I.

Relations and Functions

10

II.

Algebra

13

III.

Calculus

44

IV.

Vectors and 3D Geometry

17

V.

Linear Programming

6

VI.

Probability

10


Total

100

Unit I: Relations and Functions
1. Relations and Functions
 Types of relations: reflexive, symmetric, transitive and equivalence relations.
 One to one and onto functions, composite functions, inverse of a function.
 Binary operations.
2. Inverse Trigonometric Functions
 Definition, range, domain, principal value branch.
 Graphs of inverse trigonometric functions.
 Elementary properties of inverse trigonometric functions.
UnitII: Algebra
1. Matrices
 Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices.
 Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication.
 Non Commutativity of multiplication of matrices and existence of nonzero matrices whose product is the zero matrix.
 Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse.
2. Determinants
 Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix.
 Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables using inverse of a matrix.
Unit III: Calculus
1. Continuity and Differentiability
 Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions.
 Concept of exponential and logarithmic functions.
 Derivatives of logarithmic and exponential functions.
 Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
 Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.
2. Applications of Derivatives
 Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals.
 Use of derivatives in approximation, maxima and minima.
3. Integrals
 Integration as inverse process of differentiation.
 Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
 Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).
 Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses Area between any of the two above said curves (the region should be clearly identifiable).
5. Differential Equations
 Definition, order and degree, general and particular solutions of a differential equation.
 Formation of differential equation whose general solution is given.
 Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree.
 Solutions of linear differential equation of the type.
Unit IV: Vectors and ThreeDimensional Geometry
1.Vectors
 Vectors and scalars, magnitude and direction of a vector, direction cosines and direction ratios of a vector.
 Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio.
 Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.
2. Three – dimensional Geometry
 Direction cosines and direction ratios of a line joining two points.
 Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines, Cartesian and vector equation of a plane.
 Angle between (i) two lines, (ii) two planes, (iii) a line and a plane, distance of a point from a plane.
Unit V: Linear Programming
Linear Programming
 Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming problems.
 Mathematical formulation of L.P. Problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three nontrivial constraints).
Unit VI: Probability
Probability
 Conditional probability, multiplication theorem on probability
 Independent events, total probability, Baye’s theorem, Random variable and its probability distribution, mean and variance of random variable.
 Repeated independent (Bernoulli) trials and Binomial distribution.
IIT JEE Syllabus for Class 12 Maths
Algebra

Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.
Logarithms and their properties.
Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients.
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skewsymmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.

Trigonometry

Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and submultiple angles, general solution of trigonometric equations.

Analytical Geometry

Relations between sides and angles of a triangle, sine rule, cosine rule, halfangle formula and the area of a triangle, inverse trigonometric functions (principal value only).

Two dimensions

Cartesian coordinates, distance between two points, section formulae, shift of origin
Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle. Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Locus Problems.

Three dimensions

Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.

Differential calculus

Real valued functions of a real variable, into, onto and onetoone functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L’Hospital rule of evaluation of limits of functions.
Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions. Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, Rolle’s Theorem and Lagrange’s Mean Value Theorem.

Integral calculus

Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, Fundamental Theorem of Integral Calculus. Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves. Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, linear first order differential equations.

Vectors

Addition of vectors, scalar multiplication, dot and cross products, scalar triple products and their geometrical interpretations.

Tips and Tricks to prepare for Class 12 Maths
You need to know what to study, how to study, how to manage time and where to look for references. Below is a list of subjectwise pointers for your preparation.
Regular studies and completing your CBSE Class 12 Maths Syllabus help you achieve your academic goal on time wins you excel in your final exam and competitive exam.
Important Topics to focus in class 12 Maths Exam Chapterwise
Chapter Name

Important Topics and Tips

Relations and Functions

 Types of Relations
 Composite of two functions
 Invertible Functions
 Frequently asked question are from ‘Equivalence relations’ and composite of functions

Inverse Trigonometric Functions

 Properties of Inverse Trigonometric functions
 Write down domain and range of all trigonometric function and related graphs on paper and revise them on daily basis

Matrices

 Multiplication of Matrices
 Symmetric and Skew Symmetric Properties
 Finding Inverse of a Matrix using elementary transformation
 Most of the students get stuck in elementary transformation

Determinants

 Properties of determinants
 Adjoint and Inverse of a Matrix
 Solution of system of linear equations
 Always mark the question while you get stuck, because all questions cannot be practised before exams

Continuity and Differentiability

 Continuity of a function
 Logarithmic Differentiation
 Second order derivatives
 Differentiation of Parametric form of functions
 Logarithmic, trigonometric functions, exponential should be at tip

Application of Derivatives

 Rate of change
 Increasing and decreasing functions
 Tangents and Normal to Curves
 First and Second Derivatives Test for finding Local Maxima and Minima

Integrals

 Integration by method of Substitution
 Integration by Method of Partial Fractions
 Integration by Parts
 Definite Integral as Limit of a sum
 Properties of Definite Integrals

Application of Integrals

 Area under curves
 Area bounded by a curve and a line
 Area bounded by 2 Curves
 It always better to draw the curve and shade the area to be calculated

Differential Equations

 Formation of differential equation
 Method of Solving Differential Equation with variable separable
 Homogeneous Differential Equation
 Linear differential equation

Vector Algebra

 Scalar Product of Vectors and Projection of Vectors on a line
 Vector Product of Vectors

3D Geometry

 Direction Cosines and Direction Ratio of line
 Equation of line
 Coplanarity of line
 Angle between 2 lines
 Shortest distance between 2 skew lines
 Equation of plane in normal form
 Equation of plane perpendicular to given vector and passing through a given point
 Equation of plane passing through 3 noncollinear points
 Plane passing through the intersection of two planes
 Angle between 2 planes
 Distance of a point from a plane
 Angle between a line and a plane

Linear Programming

 Graphical Solution to linear problems

Probability

 Multiplication Theorem of Probability
 Independent Events
 Bayes’ Theorem
 Random Variable and its probability distribution
 Mean and Variance of Random Variable
 Binomial Distribution
 Try to revise the concepts of permutation and combination before getting into the chapter

 Attempt as many conceptual questions on those derivations as you can. This will give you a better understanding of the subject.
 Look for new questions on similar context from previous year’s question papers, sample papers and other model test papers.
 Always time yourself while solving sample papers as it will help you better manage time during the exam.
 Revision of notes and memorizing the formulas and the definition of difficult terms helps a lot while last minute preparation. Keep a few lists of topics separately for quick revision and it must contain the important formulae, derivations and important definitions.
 Solve as many class 12 sample papers you can,Practice past year question papers & model test papers.
 Time management is a must; take out time for recreation as well along side preparation.
 Try and finish all the topics and questions in the NCERT books.CBSE board questions are majorly based on the question in this book only.
Important Links for reference
More Reference Links –
Other National and International Level Olympiads
AI Olympiad

International Artificial Intelligence Olympiad

Coding Olympiad

International Coding Olympiad

IMO

International Maths Olympiad

ISO

International Science Olympiad

KVPY

Kishore Vaigyanik Protsahan Yojana

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