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The solutions are concise, and also discuss alternative methods to solve some problems. We make it a point to provide the students the best help they can get, and the solutions reflect just that.
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The chapter builds up on the concepts of relations, functions, domain and codomain introduced in Class The students explore several real valued functions, and their graphs.
The chapter comprises of four exercises that consolidate the concepts explained in the chapter. This chapter plays in the restricted range of trigonometric functions that enables them to be one on one, and hence makes inverse of the functions visible.
This chapter explores the inverse trigonometric through two exercises that teach the students the behavior of the functions, their graphical representations, and the elementary properties. These prove to be of more value in integration later. This chapter takes you through the properties of one of the most important tools of Mathematics, namely Matrices.
The students learn the basics of Matrices and Matrix Algebra.. Correctly preceded by Matrix, this chapter deals with an important part of Matrices, namely Determinants. The chapter deals with determinants of orders upto 3, and their cofactors and inverses. This chapter also talks about the applications of determinants, like calculation of area of triangles, adjoint of matrices, consistency of a set of linear equations.
This chapter picks up after the Differentiation of Functions introduced in Class The chapter explores the important concepts of Continuity, Differentiability and establishes the relationship between them. It also deals with the continuity and differentiability of Inverse Trigonometric Functions. It also introduces exponential and logarithmic functions. With Continuity and Differentiability marking the start of Calculus, this chapter deals with the differentiability of functions, and introduction to derivatives.
It delves into the derivatives of specific functions, and their applications, namely the tangents and normal to curves, and the rate of change in some quantities. It also talks about increasing and decreasing functions. This is one of the most important and time consuming chapters in the whole Class 12 Mathematics. It covers all the solutions based on 12th Class Maths Book Solution including the degree of a differential equation, order of a differential equation, general and particular solutions of a differential equation, homogeneous differential equations, linear differential equations, procedure involved formation of a differential equation etc.
And also it covers some miscellaneous examples. And you can know how to find position vector, some basic concepts related to vector algebra etc. You can also learn about algebraic as well as geometric properties. And multiplication of a vector by a scalar, r, vector joining two points, components of a vector, section formula, product of two vector etc. This chapter 10 has four exercises. NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry, you can get know about Three Dimensional Geometry, the student can constitute topics related to three dimensional geometry, direction cosines and direction ratios of line, equation of a line through a given point and parallel direction cosines of a line passing through two points, relation between the direction cosines of a line, equation of a line in space, plane passing through the intersection of two given planes, angle between two planes, coplanarity of two lines and many more.
And last it contains miscellaneous examples. There are three exercises in this chapter which is based on 12th Class Maths Book Solution. In this chapter. Here this chapter covered linear programming problem and its mathematical formulation, the mathematical formulation of the problem, different types of all linear programming problems graphical method, feasible and infeasible solutions and at last there are some miscellaneous examples given.
Multiplication rule of Probability. You can understand here the concept of random variables and its probability in this chapter. Other than this chapter also covers mean of random variable, variance of random variable, and binomial distribution and Bernoulli trials with miscellaneous examples.
As Class 12 Maths is an important subject that plays a vital role in your future career. Mathematics is a subject you cannot get good marks in the exam by reading and memorizing the formulae. So you need to practice well to succeed. It mathematical sense the two different words having a different meaning. Students get confused with these terms before going deep into this topic. Remember that all functions are relations, but not all relations are functions.
In this topic, you can find a definition of functions and relations, and its Special functions, different types of relations. Summary: Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, the inverse of a function. Binary operations. In this NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions, we are going to know about Inverse trigonometric functions, which are simply defined as the inverse functions of the basic trigonometric functions which include sine, cosine, tangent, cotangent, secant, and with its cosecant functions.
In this topic, we know how to use inverse functions in trigonometry to get the angle with any of the trigonometry ratios. Trigonometric Functions is very important and mostly used in the field of engineering, physics, geometry and navigation. Trigonometric functions are especially applicable to the right angle triangle. There are a total of six inverse trig functions including Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant, Arccosecant.
Summary: Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.
Matrix is a rectangular array of real or complex numbers in the form of m horizontal lines which is called row and n vertical lines which are called columns. The matrix is just an arrangement of certain quantities. The elements which are used in a matrix may be real or complex numbers. When the elements of a matrix are real, then it is known as a real matrix. Summary: 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. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix restrict to square matrices of order 2. Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; Here all matrices will have real entries.
The determinant is a scalar value which can be calculated from the elements of a square matrix. When a system of simultaneous linear equations is started to solve, then Development of determinants took place.
The shape of every determinant is square. If a determinant is of order n then it contains n rows and n columns. For every square matrix A of order m x n, there exists a number associated with it known as the determinant of a square matrix where the number of elements in a determinant of order n is n2. Summary: Determinant of a square matrix up to 3 x 3 matrices , properties of determinants, minors, co-factors 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 a system of linear equations by examples, solving system of linear equations in two or three variables having unique solution using the inverse of a matrix. Continuity and Differentiability is one of the most important topics here you can understand the concepts like continuity at a point, continuity on an interval etc. Differentiability of functional parameters is very difficult to learn.
It can be seen that a function is continuous at a point if we are able to graph it without lifting the pen. If the function is undefined, then the function is discontinuous. Summary: Continuity and differentiability, a 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. Derivative can be defined as the rate of change of one quantity with respect to another. The concept of derivatives has been used in many ways, such as when a matter of change of temperature or rate of change of shapes and sizes. Where the derivative function is used to calculate the highest and lowest point of the curve in a graph, with the help of the graph, you can also know its turning point as well.
Simple problems that illustrate basic principles and understanding of the subject as well as real-life situations. This is used to find our many useful quantities such as areas, volumes, displacement, etc. Integrals are related to usually definite integrals.
Integration is one of the two major calculus in Mathematics. The idea of limit is used where algebra and geometry are implemented. Integration is a step of adding or summing up the parts to find the whole, and the opposite step is known as differentiation. An integral that contains the upper and lower limits where Indefinite integrals are defined without upper and lower limits.
Summary: Integration as the inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the types given in the syllabus 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.
Applications of the Integrals is applied in various fields to the calculation of areas like Mathematics, Science, and Engineering etc. An integral is a function which is a given function is the derivative. It is used to find the areas of the two-dimensional and volumes of three-dimensional objects. The integral is also known as anti-derivative as it is the opposite process of differentiation.
And the function of Differential Equations is the rate of change of a function at a point. Basically, Differential Equations is used in physics, engineering, biology, and another field. The main study of solutions that satisfy the equations, and the properties of the solutions. The derivatives represent a rate of change, where differential equation describes a relationship between the quantities and the speed of change.
Summary: Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given.
The solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of the first order and first degree. Solutions of the linear differential equation of the type given in the syllabus. There are numbers of quantities that involve magnitude as well as direction. When the quantity has magnitude and direction, then it is known as vectors. And such types of quantities are known as Vector Quantities.
The main use of vectors is to describe the physical quantities having directions and magnitude associated with them. Summary: 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, the 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, a scalar triple product of vectors.
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