As per the educational system in India 10+2 is in effect currently. We all know that class 12 is a very important year as it decides the future path for the students. This is the year that is considered the turning point in one’s life. For years it has been the prime year for students and to get rich success in class 12 exams remains the goal of every student. Though with the introduction of CET, the situation has changed a bit but the importance of the 12^{th} has not reduced. Ambitious students cut themselves off from the outer world and concentrate on their studies.

In this condition, they try to get as much as possible study material to enhance their talent. Bright students will always be on a hunt to grab the material like **R D Sharma Solutions for Class 12 **to perform best in the exams. Here, we will discuss in brief the Maths solutions chapter-wise.

**Mathematics**

**Chapter wise summary**

**Chapter 1 – Relations Functions**

In this chapter, the students will understand the concepts of relations and function in mathematics. They will learn relations, types of relations (empty, universal, reflexive, symmetric, transitive, and equivalence), functions, types of functions (one-to-one, onto, one-one & onto), the composition of functions, invertible functions, and binary operations.

**Chapter 2 – Binary Operations**

Addition, subtraction, multiplication, and division are the basic operations of mathematics and are performed on two operands. While adding three numbers, we first add two numbers and then add a third in the total of two numbers. Thus the mathematical operations are done on two numbers and are called binary operations. In this chapter, the students will know what is a binary operation, the types of binary operations, and examples of binary operations. The types of binary operations are distributivity, associativity, and commutativity.

**Chapter 3 – Inverse Trigonometric Functions**

Inverse trigonometric functions are defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. They are also known as antitrigonometric functions, arcus functions, or cyclometric functions. The students will also get to know about inverse trigonometric functions graphs, arcsine function, arccosine function, arctangent function, arccotangent (arccot) function, arcsecant function, arccosecant function, and inverse trigonometric functions properties.

**Chapter 4 – Algebra of Matrices**

Algebra of matrices is a branch of mathematics that deals with the vector spaces between different dimensions. Because of the presence of n-dimensional planes in coordinate space, the innovation of matrix algebra came into existence. A matrix is an arrangement of numbers, expressions, or symbols in a rectangular array. The arrangement is done in horizontal rows and vertical columns with the order of the number of rows x the number of columns. The students will learn the algebra of matrix and the rule of matrix algebra.

**Chapter 5 – Determinants**

To determine the uniqueness of the solution associated with matrices we find the determinant. It is widely used in Science, Economics, Engineering, etc. in this chapter the students will learn the determinants up to order 3 with real entries, properties of determinants, the area of a triangle using determinants, and minors & cofactors.

**Chapter 6 – Adjoint and Inverse of Matrix**

In this chapter, the students will learn the definition of the adjoint of a square matrix, the inverse of a matrix, some useful results on invertible matrices, an elementary transformation of elementary operations of a matrix with an example, and properties of adjoint & inverse of a matrix.

**Chapter 7 – Solutions of Simultaneous Linear Equations**

In this chapter, the details of equations starting with definition, consistent system, homogeneous & non-homogeneous system, matrix method for the solution of the non-homogeneous system, and final solution of a homogeneous system of linear equations. Of the two types, a homogeneous equation does not zero on the right-hand side of the equality sign, while a non-homogeneous equation has a function of the independent variable on the right-hand side of the equal sign.

**Chapter 8 – Continuity and Differentiability**

Here, the students will know the important concepts of continuity and differentiability and the relation between them. The topics covered are; introduction to continuity & differentiability, algebra of a continuous function, derivatives of composite functions, derivatives of implicit functions, derivatives of inverse trigonometric functions, exponential & logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second-order derivative, and mean value theorem.

**Chapter 9 – Differentiation**

In calculus, differentiation is one of the two important concepts apart from integration. Differentiation is used to find the derivative of a function. In differentiation, the instantaneous rate of change in function is based on one of its variables. Velocity is an example of differentiation. From this chapter, the students will learn what is differentiation, a derivative of a function as limit, notations, linear & non-linear functions, differentiation formulas, differentiation rules, sum or difference rule, product rule, quotient rule, chain rule, and real-life applications of differentiation.

**Chapter 10 – Higher Order Derivatives**

This chapter talks about the higher-order derivatives. The chapter covers proving relations involving various order derivatives of Cartesian functions, proving relations involving various order derivatives of parametric functions, and proving relations involving various order derivatives via illustrations.

**Higher order derivative n parametric form**: The derivative of the first order in parametric equations is given by: dy/dx = dy/dt x dt/dx = y’(t)/x’(t)

**Second order derivative**: It is the derivative of the first order derivative of the given function. If y=f(x) then dy/dx = f’(x). If f’(x) is differentiable, then differentiating dy/dx again w.r.t. x the 2^{nd} order derivative is d/dx (dy/dx) = d^{2}y/dx^{2} = f’(x)

**Chapter 11 – Derivative as a Rate Measurer**

This chapter speaks about the derivative as a rate measurer. It covers how to find the rate measurer of derivative and related rates in which the rate of change of one of the quantities involved is required, corresponding to the given rate of change of another quantity.

**Chapter 12 – Differentials, Errors, and Approximations**

In this chapter, the concepts of differentials and errors are discussed.

**Differentials**: It deals with the rate of change of one quantity with another.

**Absolute error**: If ‘x’ is the actual value of a quantity and X0 is the measured value of the quantity, then the absolute error value can be calculated.

**Relative error**: It is defined as the ratio of the absolute error of the measurement to the actual measurement. With this method, one can determine the magnitude of the absolute error in terms of the actual size of the measurement.

**Percentage error**: It is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

**Chapter 13 – Mean Value Theorem**

A mean value theorem is a crucial tool in calculus. This theorem helps to analyze the behavior of functions. As per this theorem, if ‘f’ is a continuous function on the closed interval [a,b] and it can be differentiated in the open interval (a,b), then there exists a point ‘c’ in the interval (a,b). From this chapter, the students will understand what is mean value theorem is, mean value theorem proof, mean value theorem for integrals, mean value theorem for derivatives, mean value theorem applications, and mean value theorem examples. There are two known theorems; Rolle’s Theorem and Lagrange’s Mean Value Theorem.

**Chapter 14 – Tangents and Normals**

The applications of derivatives are determining the rate of change of quantities, finding the equations of tangents & normals to a curve at a point, and finding turning points on the graph of a function which in turn will help us to locate points at which the largest or smallest value (locally) of a function occurs. In this chapter, the students will learn tangent & normal equations and problems on them. It also deals with the slope of a line, slopes of tangent and normal, finding the slope of tangent and normal at a given point, and equations of tangent and normal.

**Chapter 15 – Increasing and Decreasing Functions**

In this chapter, the topics of increasing and decreasing functions, solution of rational algebraic inequations with logarithms, strictly increasing functions, strictly decreasing functions, monotonic functions, monotonic increasing & monotonic decreasing functions, necessary & sufficient conditions for monotonicity, finding the intervals in which a function is increasing or decreasing and proving the monotonicity of a function on a given interval.

**Increasing function**: For a function y = f(x) to be increasing dy/dx >/= 0 for all such values of interval (a,b) and equality may hold for discrete values.

**Decreasing function**: For a function y = f(x) to be monotonically decreasing dy/dx </= 0 for all such values of interval (a,b) and equality may hold for discrete values.

**Monotonic function**: It is defined as any function which follows one of the four cases mentioned above.

**Chapter 16 – Maxima and Minima**

The concept of maxima and minima are applicable to both maths and science. Maxima and minima deal with the maximum and minimum values of a function in its domain, definition of maximum, local maxima & local minima, definition & meaning of local maximum, first derivative test for local maxima & minima along with algorithm, higher-order derivative test, point of inflection, properties of maxima & minima, maximum & minimum values in the closed interval, and applied problems on maxima & minima.

**Chapter 17 – Indefinite Integrals**

An indefinite integral is an integral which is not have any upper or lower limit. This chapter will explain indefinite integral, primitive & antiderivative, fundamental integration formulae, some standard results on integration, integration of trigonometric functions, integration of exponential functions, geometrical interpretation of indefinite integral, comparison between differentiation & integration, methods of integration, integration by substitution, integration by parts, integration of rational algebraic functions by using partial fractions and integration of some special irrational algebraic functions.

**Why Maths is an important subject?**

Maths has been the base of Science and it is also used everywhere in daily life. We are blessed to have the opportunity of learning such truths about our Earth, the atmosphere, and the infinite treasure hidden in nature. Lives will be insufficient to explore all that nature possesses. We must grab every possibility to learn more about science and math.

Maths is exceptionally wide and we even can’t measure the depth of knowledge it contains. We use Maths knowingly and unknowingly in daily life. We count numbers, we calculate costs, and we estimate the work and a lot more. For the class 12 students, more study material is available with the likes of **R D Sharma Solutions for Class 12** and they can take full advantage to refer to them.

**Conclusion: – **

Class 12 is a very important stage in a student’s life as it is the end of schooling but the start of a big, new, and unknown world for them. Having performed well in studies is an advantage to students as they can cope with the world they are going to explore, in a much better way. Studies not only give knowledge but boost confidence a lot.

It is not easy for a student to come out of their comfort zone and protected life as the parents ensure they get what they want. Things start changing once the schooling is over and the layer of protection starts disintegrating. The students who have become adults now, face different situations and different people who are not all good. **R D Sharma Solutions for Class 12** help support the students to achieve their goals as the solutions are designed as per the need of the students and cover all syllabuses.