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## NCERT Solutions For Class 12 Maths

Class: | 12 |

Subject: | Maths |

Contents: | NCERT Solutions |

### CBSE NCERT Solutions For Class 12 Maths

Here you can freely download * NCERT Solutions for class 12 Maths*, We have listed all the chapters PDF’s. All the Solutions available here are latest. There is no Login, nor Sign Up, neither Redirects, just click the downloading link, you can view your PDF and download accordingly.

**NCERT Solutions For Class 12 Maths PDF Free Download**

### Class 12 Maths Important Points Related To All Chapters.

#### Chapter 1: Relations And Functions

Before doing Exercise 1.1 of Class 12 Relations and Functions, we should know about Empty and Universal relations. A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive. Knowledge of One-One and onto needed for Exercise 1.2 of Class 12 Maths. These functions lead to finding an invertible function.

Properties and functions for Binary Operations are useful for the CBSE Board Exams. NCERT Solutions For Class 12 Maths Chapter 1 Miscellaneous is one of the best exercises for final exams in CBSE and UP Board. We have prepared sols based on the latest CBSE Syllabus for 2023.

**NCERT Sols for Class 12**

NCERT Solutions for Class 12 Maths, Phy, Chem, Bio with study material and notes for revision are given below. You can also download contents for Business studies, P. E. and English in PDF.

#### Chapter 2: Inverse Trigonometric Functions

Inverse Trigonometric Functions of NCERT Solutions For Class 12 Maths deals with principle values of trigonometric functions. We should also know the branches other than the principal values. Practice all questions given in Exercise 2.2 and Miscellaneous for exams or tests. To practice well for Board Exams, one should do the NCERT Exemplar books also. Must do questions number 16 of Miscellaneous Exercise for class tests or Board.

#### Chapter 3: Matrices

Matrices are one of the most powerful tools in mathematics to solve multiple Linear Equations. We solve the system of linear equations of two or three variables. Matrix notation and operations are helpful in electronic spreadsheet programs for different areas of business and science. There are two methods for solving the system of linear equations. The Matrix Method, using the inverse of Matrix, is applicable in CBSE Board exams of class 12. Crammer’s rule or Determinant Method is not allowed for Boards.

#### Chapter 4: Determinants

Determinant helps in solving a system of linear equations with multiple variables. In NCERT Solutions For Class 12 Maths, chapter 4, we can explore only a square matrix. Using cofactors and determinants, we find an inverse of The Matrix. It helps to find out values of unknown variables in the equations. In the last ten years, word problems frequently come in CBSE Board Papers and School Exams. This chapter is easy to score in Board Exams.

#### Chapter 5: Continuity And Differentiability

The first exercise of NCERT Solutions For Class 12 Maths Chapter 5, briefly explains about Continuity and Differentiability. Exercises from 5.2 to 5.6 are easier than the others. Exercise 5.8 contains the concepts of Rolle’ and Mean Value theorems. Miscellaneous Exercise 5 have questions for practice in CBSE Board exam. Must go through this chapter to do Integration comfortably. This chapter is also helpful to do Chapter 6 AOD. We apply the same formulae and technique for solving Exercise 6.1 to 6.5.

#### Chapter 6: Application Of Derivatives

Exercise 6.1, Rate of Change, is suitable for school tests as well as Boards. It is easy to learn and understand. Exercise 6.2 and 6.3 need more practice for understanding. Exercise 6.4 is easy to explore the concepts. Exercise 6.5 covers the topic of Maxima and Minima. Questions must come from this exercise in Boards. So, this exercise, including miscellaneous, is strenuous but scoring one.

#### Chapter 7: Integrals

We consider NCERT Solutions For Class 12 Maths Chapter 7 Integration as a tricky chapter. It frequently needs more practice to solve the sums. If we give regular time to practice integration, it appears as the simplest one. Exercise 7.11 and Miscellaneous are the important ones for the Board exams. Never learn, always try to derive the sums from being confident. Miscellaneous 7 is the most import exercise for the exams. It contains various sums on different concepts of integration. If one has done integration confidently, mean he has done the 12 maths.

#### Chapter 8: Application Of Integrals

Application of Integrals o fNCERT Solutions For Class 12 Maths Maths, involves the area bounded by the curves. These curves may be parabola, circles, ellipse or line. First of all, we have to identify the curves in which the area occurs. There may be more than one methods to find the same bounded region. Always write the answer in square units in exams. Most of the time, the combination of two or three curves appears in the question. Solve these curves to get intersection point and then plot them.

#### Chapter 9: Differential Equations

We should be aware of the Degree and Order of a differential equation. Order means how many time s it is differentiated, and Degree is the power of the highest derivative. In class 12 Maths Chapter 9, most of the questions are following degree 2. There is a total of four types of questions in the chapter. First, the formation of the differential equation and rest the solutions of these equations.

#### Chapter 10: Vector Algebra

Vector Algebra of Class 10 Maths includes simple operations of vectors, Cross Product and Dot Product. Initially, the first exercise introduces the sum of vectors in 3-D and standard terms like unit vectors, magnitude. One section covers dot product or scalar product of two vectors. In the end, the cross product or vector product helps us to find the area of triangle and parallelogram. Vector triple product is now in the course of CBSE Board for 2023. Generally, a question happens to come from this section.

#### Chapter 11: Three Dimensional Geometry

What is direction cosine and how they are related to direction ratio? These are the main concept of Chapter 11 3-D Geometry of Class 12 Maths. It also helps to derive the equation of a line using a given point and direction as well. In Chapter 11, we learn the concepts of a straight line in the 3-D plane. Exercise 11.3 describe the equation of the plane and the relation between a line and a plane. Distance between line and plane or point and a plane is quite important for the CBSE Board Exams.

#### Chapter 12: Linear Programming

Do you ever think, what is the use of LPP? It is the most important tool for analysing production profitably. One never ignore this chapter. Think why? Because this chapter continues to higher classes also. It provides the perfect solutions for managing goods, labour and space in the factories. It is also one of the main subjects in MBA, a degree for Management. So, keeping its importance in mind, do the chapter by heart. This chapter is scoring, as well as easier to understand.

#### Chapter 13: Probability

The concept of probability starts from the middle classes. The level of difficulty increases gradually. In class 12 Probability, we are keen to know about Distribution, Binomial and Bays theorem. However, in higher classes, it extents to Poisson Distribution, Normal Distribution for the prediction in many fields. The concepts of probability remain the same as we have already learnt in earlier classes. It contains about ten marks weightage in class 12. So, this chapter can make a significant change in % of one’s results. Do you think that it is a secure chapter? Just think about the questions of Miscellaneous. You will realise that probability is easy but tricky also.

**Summary & Key Points Of Chapters Of 12th Maths**

**1. Relations And Functions**

Relations and Functions – summary: 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**

Inverse Trigonometric Functions – summary: Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

**3. Matrices**

Matrices – 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).

**4. Determinants**

Determinants – 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 system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

**5. Continuity And Differentiability**

Continuity and Differentiability – summary: 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.

**6. Applications Of Derivatives**

Applications of Derivatives – summary: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

**7. Integrals**

Integrals – summary: 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 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.

**8. Applications Of The Integrals**

Applications of the Integrals – summary: Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area Bounded between any of the two above said curves (the region should be clearly identifiable).

**9. Differential Equations**

Differential Equations – summary: 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 given in the syllabus.

**10. Vector Algebra**

Vectors – 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, 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.

**11. Three – Dimensional Geometry**

3 – D Geometry – summary: 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.

**12. Linear Programming**

LPP – Linear Programming Problems – summary: Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) 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).

### What is Multiplication Theorem on Probability?

Let E and F be two events associated with a sample space of an experiment.

Then

P (E ∩ F) = P (E) P (F | E), P (E) ≠ 0.

### What is a linear objective function in LPP?

Linear function Z = ax + by, where a and b are constants, which has to be maximised or minimised is called a linear objective function.

### What are Skew lines in 3-D geometry?

Skew lines are lines in the space which are neither parallel nor intersecting. They lie in the different planes.

### What is the triangle law of vector addition?

The triangle law of vector addition states that “If two vectors are represented by two sides of a triangle taken in order, then their sum or resultant is given by the third side taken in opposite order”.

### What do you meant by a partial differential equation?

A differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation.

### Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b^2} is neither reflexive nor symmetric nor transitive.

R = {(a, b): a ≤ b^2}

It can be observed that (1/2,1/2)∉R, since, 1/2>(1/2)^2

∴ R is not reflexive.

Now, (1, 4) ∈ R as 1 < 42 But, 4 is not less than 12. ∴ (4, 1) ∉ R ∴ R is not symmetric. Now, (3, 2), (2, 1.5) ∈ R [as 3 < 22 = 4 and 2 < (1.5)^2 = 2.25] But, 3 > (1.5)^2 = 2.25

∴ (3, 1.5) ∉ R

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

### Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

The functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} are defined as

f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}.

gof(1) = g[f(1)] = g(2) = 3 [as f(1) = 2 and g(2) = 3]

gof(3) = g[f(3)] = g(5) = 1 [as f(3) = 5 and g(5) = 1]

gof(4) = g[f(4)] = g(1) = 3 [as f(4) = 1 and g(1) = 3]

∴ gof = {(1, 3), (3, 1), (4, 3)}

### Find the principal value of tan−1 (−1).

Let tan−1 (−1) = y. Then, tan〖y=-1=-tan(π/4)=tan(-π/4)〗

We know that the range of the principal value branch of tan−1 is (-π/2,π/2) and tan(-π/4)=-1

Therefore, the principal value of tan−1 (−1) is -π/4.

### If A, B are symmetric matrices of same order, then what type of matrix AB – BA is?

( − )′=( )′ − ( )′ [∵ ( − )′= ′− ′]

= ′ ′ − ′ ′ [∵ ( )′= ′ ′]

= − [∵ Given: ′= , ′= ]

=− ( − )

⇒( − )′

=−( − ),

Therefore,the matrix (AB − BA) is a skew symmetric matrix.

### Which type of matrices have determinants?

Only square matrices have determinants.

### Find the second order derivatives of x^2+3x+2.

Let y=x^2+3x+2, therefore,

dy/dx = d/dx (x^2+3x+2)

=2x+3

⇒(d^2 y)/(dx^2 )

= d/dx (2x+3)=2

### The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Let the radius of circle =r cm

Area of circle with radius r is given by A=πr^2

Therefore, the rate of change of A with respect to t

= dA/dt = 2πr.dr/dt

= 2πr.3

= 6πr

When r=10 cm, then dA/dt=6π.(10)=60π cm^2/s

Hence, the area of circle is increasing at the rate of 60π cm^2/s.

### Find both the maximum value and the minimum value of 〖3x〗^4-8x^3+12x^2-48x+25 on the interval [0, 3].

Let f(x)=3x^4-8x^3+12x^2-48x+25

f^’ (x)=12x^3-24x^2+24x-48 =12(x^3-2x^2+2x-4) =12[x^2 (x-2)+2(x-2)]

=12(x-2)(x^2+2)

Now, f^’ (x)=0 gives x =2 or x^2+2=0 for which there are real roots

Therefore, we consider only x=2∈[0,3]

Now, we evaluate the value of f at critical point x = 2 and at the end points of the interval [0, 3].

f(2)=3〖(2)〗^4-8〖(2)〗^3+12〖(2)〗^2-48(2)+25

=3(16)-8(8)+12(4)-48(2)+25=48-64+48-96+25 =-39

f(0)=3〖(0)〗^4-8〖(0)〗^3+12〖(0)〗^2-48(0)+25=3(0)-8(0)+12(0)-48(0)+25=25

f(3)=3〖(3)〗^4-8〖(3)〗^3+12〖(3)〗^2-48(3)+25 F(3)=3(81)-8(27)+12(9)-48(3)+25

=243-216+108-144+25 =16

Hence, we can conclude that the absolute value of f on [0, 3] is 25 occurring at x=0 and the absolute minimum value of f at [0, 3] is – 39 occurring at x = 2.

### What is integration?

The formula that gives all anti derivatives is called the indefinite integral of the function and such

process of finding anti derivatives is called integration.

### Fill in the blank: The area of the region bounded by the curve y = x2 + x, x-axis and the line x = 2 and x = 5 is equal to ________.

297/6 square units.

**13. Probability**

Probability – summary: Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.