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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You can then go through the detailed, step-wise solutions for each and every question for that chapter.
CBSE Class 12 Deleted Portion of Syllabus for 2020-2021
Check subject-wise 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:
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Tough Questions
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Easy questions (11th Standard)
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Easy questions (12th Standard)
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Hyperbola
Parabola
Sets
Relations and Functions
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Mathematical Induction and Reasoning
Parabola
Permutations and Combinations
Quadratic Equations
Sequences and Series
Statistics
Trigonometry
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3D Geometry
Determinants
Differential Equations
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This is the breakup of marks between different topics of the Maths section:
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Topic
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Marks Allotted
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Algebra
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56
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Calculus
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28
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Trigonometry
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8
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Conics
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16
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Vector Algebra
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12
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Maths Syllabus for Class 12
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Unit
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Topic
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Marks
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I.
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Relations and Functions
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10
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II.
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Algebra
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13
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III.
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Calculus
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44
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IV.
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Vectors and 3D Geometry
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17
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V.
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Linear Programming
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6
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VI.
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Probability
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10
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Total
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100
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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.
Unit-II: 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 non-zero 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 Three-Dimensional 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 non-trivial 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
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Algebra
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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 skew-symmetric 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.
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Trigonometry
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Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.
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Analytical Geometry
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Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).
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Two dimensions
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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.
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Three dimensions
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Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
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Differential calculus
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Real valued functions of a real variable, into, onto and one-to-one 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.
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Integral calculus
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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.
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Vectors
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Addition of vectors, scalar multiplication, dot and cross products, scalar triple products and their geometrical interpretations.
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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 subject-wise 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
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Chapter Name
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Important Topics and Tips
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Relations and Functions
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- Types of Relations
- Composite of two functions
- Invertible Functions
- Frequently asked question are from ‘Equivalence relations’ and composite of functions
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Inverse Trigonometric Functions
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- 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
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Matrices
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- 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
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Determinants
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- 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
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Continuity and Differentiability
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- Continuity of a function
- Logarithmic Differentiation
- Second order derivatives
- Differentiation of Parametric form of functions
- Logarithmic, trigonometric functions, exponential should be at tip
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Application of Derivatives
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- Rate of change
- Increasing and decreasing functions
- Tangents and Normal to Curves
- First and Second Derivatives Test for finding Local Maxima and Minima
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Integrals
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- 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
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Application of Integrals
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- 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
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Differential Equations
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- Formation of differential equation
- Method of Solving Differential Equation with variable separable
- Homogeneous Differential Equation
- Linear differential equation
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Vector Algebra
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- Scalar Product of Vectors and Projection of Vectors on a line
- Vector Product of Vectors
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3-D Geometry
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- 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 non-collinear 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
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Linear Programming
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- Graphical Solution to linear problems
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Probability
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- 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
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- 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 re-creation 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
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AI Olympiad
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International Artificial Intelligence Olympiad
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Coding Olympiad
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International Coding Olympiad
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IMO
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International Maths Olympiad
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ISO
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International Science Olympiad
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KVPY
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Kishore Vaigyanik Protsahan Yojana
|
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