# NCERT Solutions Class 12 Maths

**NCERT Solutions for Class 12 Maths** provide a solid conceptual base for all the topics of CBSE Board Class 12 Maths. We have covered all the important theorems and formulae with detailed explanations for students. NCERT Solutions for Maths are an essential asset for students of Class 12. These solutions of Maths include answers to all the questions as per latest CBSE Board Syllabus. These NCERT Solutions for Class 12 Maths have been designed by highly skilled teachers. Students can easily download NCERT Solutions free pdf.

These Maths NCERT Solutions will help students in the preparation of Competitive Exams like JEE (Mains and Advanced), VITEEE, other state level exams, etc. NCERT Solutions for Class 12 are designed by our subject-matter experts so that students can understand the concepts of Maths. These Class 12 Solutions of NCERT have step-by-step explanations of problems given in the Books.

Study Tutee provides CBSE Class 12 Maths Solutions in PDF format which can be downloaded for free. If you have trouble in understanding a topic related to Maths, you can verify the answer to the questions given in the exercise of the book. With these NCERT Solutions, you can score higher in your Board Exams.

Mathematics is one of the necessary subjects which not only decides the careers of many young students but also enhances their ability of analytical and rational thinking. Study Tutee provides chapter-wise NCERT Solutions to help students clear their doubts by giving in-depth knowledge of the concepts of the subject.

###### A list of chapters provided in Class 12 Maths NCERT Solutions **free **pdf

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##### 1) NCERT Solutions Class 12 Maths Chapter 1 Relations and Functions

An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input. NCERT Solutions Class 12 Maths Chapter 1 Relation and Function Download free pdf.

NCERT Solutions Class 12 Maths Chapter 1 Relations and Functions is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- Types of Relations
- Types of Functions
- Composition of Functions and Invertible Function
- Binary Operations

**Some important points in ‘Relations and Functions’ are as follows-**

**A relation R from set X to a set Y is defined as a subset of the Cartesian product X × Y. We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.****If n(A) = m and n(B) = n from set A to set B, then n(A × B) = mn and number of relations = 2**^{mn}.**Commutative Binary Operation: A binary operation * on set Y is said to be commutative, if a * b = b * a, ∀ a, b ∈ Y.****Associative Binary Operation: A binary operation * on set Y is said to be associative, if a * (b * c) = (a * b) * c, ∀ a, b, c ∈ Y.****For a binary operation, we can neglect the bracket in an associative property but in the absence of associative property, we cannot neglect the bracket.****Identity Element: An element e ∈ X is said to be the identity element of a binary operation * on set X, if a * e = e * a = a, ∀ a ∈ X. Identity element is unique.****Zero is an identity for the addition operation on R and one is an identity for the multiplication operation on R.**

Equivalence Classes: Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A, called partitions or sub-divisions of X satisfying

- all elements of Ai are related to each other, for all i.
- no element of Ai is related to any element of Aj, i ≠ j
- Ai ∪ Aj = X and Ai ∩ Aj = 0, i ≠ j. The subsets Ai and Aj are called equivalence classes.

** Types of Relation-**

**Empty Relation: A relation R in a set X, is called an empty relation, if no element of X is related to any element of X,****i.e. R = Φ ⊂ X × X****Universal Relation: A relation R in a set X, is called universal relation, if each element of X is related to every element of X,****i.e. R = X × X****Reflexive Relation: A relation R defined on a set A is said to be reflexive, if****(x, x) ∈ R, ∀ x ∈ A or****xRx, ∀ x ∈ R****Symmetric Relation: A relation R defined on a set A is said to be symmetric, if****(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or****xRy ⇒ yRx, ∀ x, y ∈ R.**

** Types of Functions-**

**One-One Function or Injective Function: A function f is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x**_{1}) = f(x_{2}) ⇔ x_{1}= x_{2}, ∀ x_{1}, x_{2}∈ X**A function which is not one-one, is known as many-one function.****Onto Function or Surjective Function: A function f is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.****In other words, a function is called an onto function, if its range is equal to the codomain.****Bijective or One-One and Onto Function: A function f is said to be a bijective function if it is both one-one and onto.**

**2) NCERT Solutions Class 12 Maths Chapter 2 ****Inverse Trigonometric Functions**

Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. These Inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions Download free pdf.

NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometry Functions is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- Basic Concepts
- Properties of Inverse Trigonometric Functions

**Some important points in ‘Inverse Trigonometric Functions’ are as follows-**

- The value of an inverse trigonometric function, which lies in the range of principal value branch, is known as the principal value of the inverse trigonometric function.
**Inverse Trigonometric Functions: Trigonometric functions are many-one functions but we know that inverse of function exists if the function is bijective. If we restrict the domain of trigonometric functions, then these functions become bijective and the inverse of trigonometric functions are defined within the restricted domain. The inverse of f is denoted by ‘f**^{-1}‘.**Let y = f(x) = sin x, then its inverse is x = sin**^{-1}y.- sin-1(sinθ) = θ ; sin-1 x should not be confused with (sinx)-1 = 1/sinx or sin-1 x = sin-1(1/x) and similarly for other trigonometric functions.

** FORMULAS-**

**sin**^{-1}(sinθ) = θ; ∀ θ ∈ [−π/2,π/2]**cos**^{-1}(cosθ) = θ; ∀ θ ∈ [0, π]**tan**^{-1}(tanθ) = θ; ∀ θ ∈ [−π/2,π/2]**cosec**^{-1}(cosecθ) = 0; ∀ θ ∈ [−π/2,π/2] , θ ≠ 0**sec**^{-1}(secθ) = θ; ∀ θ ∈ [0, π], θ ≠ π/2**cot**^{-1}(cotθ) = θ; ∀ θ ∈ (0, π)**sin(sin**^{-1}x) = x, ∀ x ∈ [-1, 1]**cos(cos**^{-1}x) = x; ∀ x ∈ [-1, 1]**tan(tan**^{-1}x) = x, ∀ x ∈ R**cosec(cosec**^{-1}x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)**sec(sec**^{-1}x) = x, ∀ x ∈ (-∞, -1] ∪ [1, ∞)**cot(cot**^{-1}x) = x, ∀ x ∈ R**sin**^{−1}x + cos^{−}^{1}**x = π/2****sec**^{−1}x + cosec^{−1}x = π/2**tan**^{−1}x + cot^{−1}x = π/2**2 cos**^{−1}x = cos^{−1}(2x^{2}– 1)**3 sin**^{−1}x = sin^{−1}(3x – 4x^{3})**3 cos**^{−1}x = cos^{−1}(4x^{3}– 3x)

**3) NCERT Solutions Class 12 Maths Chapter 3 ****Matrices**

A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. NCERT Solutions Class 12 Maths Chapter 3 Matrices Download free pdf

NCERT Solutions Class 12 Maths Chapter 3 Matrices are structured here for students to score higher in exam. The list of topics from this chapter are given below: –

- Introduction
- Matrix
- Types of Matrices
- Operations on Matrices
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Elementary Operation (Transformation) of a Matrix
- Invertible Matrices

**Some important points in ‘Matrices’ are as follows-**

** Types of Matrices-**

**Row Matrix – A matrix having only one row and any number of columns is called a row matrix.****Column Matrix – A matrix having only one column and any number of rows is called column matrix.****Rectangular Matrix – A matrix of order m x n, such that m ≠ n, is called rectangular matrix.****Horizontal Matrix – A matrix in which the number of rows is less than the number of columns, is called a horizontal matrix.****Vertical Matrix – A matrix in which the number of rows is greater than the number of columns, is called a vertical matrix.****Null/Zero Matrix – A matrix of any order, having all its elements as zero, is called a null/zero matrix. i.e., a**_{ij}= 0, ∀ i, j**Square Matrix – A matrix of order m x n, such that m = n, is called square matrix.****Diagonal Matrix – A square matrix A = [a**_{ij}]_{m x n}, is called a diagonal matrix, if all the elements except those in the leading diagonals are zero, i.e., a_{ij}= 0 for i ≠ j. It can be represented as A = diag[a_{11}a_{22}… a_{nn}]**Scalar Matrix – A square matrix in which every non-diagonal element is zero and all diagonal elements are equal, is called scalar matrix. i.e., in scalar matrix****a**_{ij}= 0, for i ≠ j and a_{ij}= k, for i = j**Unit/Identity Matrix – A square matrix, in which every non-diagonal element is zero and every diagonal element is 1, is called, unit matrix or an identity matrix.****Upper Triangular Matrix – A square matrix A = a[**_{ij}]_{n x n }is called a upper triangular matrix, if a[_{ij}], = 0, ∀ i > j.**Lower Triangular Matrix – A square matrix A = a[**_{ij}]_{n x n }is called a lower triangular matrix, if a[_{ij}], = 0, ∀ i < j.**Submatrix – A matrix which is obtained from a given matrix by deleting any number of rows or columns or both is called a submatrix of the given matrix.****Equal Matrices – Two matrices A and B are said to be equal if both having same order and corresponding elements of the matrices are equal.****Principal Diagonal of a Matrix – In a square matrix, the diagonal from the first element of the first row to the last element of the last row is called the principal diagonal of a matrix.****Singular Matrix – A square matrix A is said to be singular matrix, if determinant of A denoted by det (A) or |A| is zero, i.e., |A|= 0, otherwise it is a non-singular matrix.**

** Algebra of Matrices-**

**Addition of Matrices – Let A and B be two matrices each of order m x n. Then, the sum of matrices A + B is defined only if matrices A and B are of same order.****If A = [a**_{ij}]_{m x n}, A = [a_{ij}]_{m x n }**then, A + B = [a**_{ij}+ b_{ij}]_{m x n}**Subtraction of Matrices – Let A and B be two matrices of the same order, then subtraction of matrices, A – B, is defined as A – B = [a**_{ij}– b_{ij}]_{n x n};**where A = [a**_{ij}]_{m x n}, B = [b_{ij}]_{m x n}

**Multiplication of a Matrix by a Scalar – Let A = [a**_{ij}]_{m x n}be a matrix and k be any scalar. Then, the matrix obtained by multiplying each element of A by k is called the scalar multiple of A by k and is denoted by kA, given as kA= [ka_{ij}]_{m x n}

**Properties of Addition of Matrices: If A, B and C are three matrices of order m x n, then**

**Commutative Law: A + B = B + A****Associative Law: (A + B) + C = A + (B + C)****Existence of Additive Identity: A zero matrix (0) of order m x n (same as of A), is additive identity, if****A + 0 = A = 0 + A****Existence of Additive Inverse: If A is a square matrix, then the matrix (- A) is called additive inverse, if****A + ( – A) = 0 = (- A) + A****Cancellation Law****A + B = A + C ⇒ B = C (left cancellation law)****B + A = C + A ⇒ B = C (right cancellation law)**

**Properties of Scalar Multiplication – If A and B are matrices of order m x n, then**

**k(A + B) = kA + kB****(k**_{1}+ k_{2})A = k_{1}A + k_{2}A**k**_{1}k_{2}A = k_{1}(k_{2}A) = k_{2}(k_{1}A)**(- k)A = – (kA) = k( – A)**

##### 4)** NCERT Solutions Class 12 Maths Chapter 4 ****Determinants**

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. NCERT Solutions Class 12 Maths Chapter 4 Determinants **Download free pdf.**

NCERT Solutions Class 12 Maths Chapter 4 Determinants is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- Determinant
- Properties of Determinants
- Area of a Triangle
- Minors and Cofactors
- Adjoint and Inverse of a Matrix
- Applications of Determinants and Matrices

**Some important points in ‘Determinants’ are as follows-**

**Read |Z| as determinant Z not absolute value of Z.****Difference between determinant and matrix is that determinant gives numerical value but matrix do not give numerical value.****A determinant always has an equal number of rows and columns, i.e. only square matrix have determinants.****For easier calculations of determinant, we shall expand the determinant along that row or column which contains the maximum number of zeroes.****While expanding, instead of multiplying by (-1)**^{i+j}, we can multiply by +1 or -1 according to as (i + j) is even or odd.

** Properties of Determinants-**

**If all the elements of any row/column of a determinant are zero then the value of a determinant is zero.****If each element of any one row/column of a determinant is a multiple of scalar k then the value of the determinant is a multiple of k.****If in a determinant any two rows/columns are interchanged then the value of the determinant obtained is negative of the value of the given determinant. If we make n such changes of rows (columns) in determinant ∆ and obtain determinant ∆ , then ∆**_{1}= (-1)^{n}∆.**If all corresponding elements of any two rows/columns of a determinant are identical or proportional, then the value of the determinant is zero.****The value of a determinant remains unchanged on changing rows into columns and columns into rows. It follows that, if A is a square matrix, then |A’| = |A|. Note: det(A) = det(A’), where A’ = transpose of A.****If some or all elements of a row/column of a determinant are expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.****In the elements of any row/column of a determinant, if we add or subtract the multiples of corresponding elements of any other row/column, then the value of determinant remains unchanged. In other words, the value of determinants remains the same, if we apply the operation R**_{i}–> R_{i}+ kE_{j}or C_{i}–> C_{j }–> kC_{j}.

** Properties of a Inverse Matrix-**

**(X**^{-1})^{-1}= X**(X**^{T})^{-1}=(X^{-1})^{T}**(XY)**^{-1}= Y^{-1}X^{-1}**(XYZ)**^{-1}=Z^{-1}Y^{-1}X^{-1}**adj (X**^{-1}) = (adj X)^{-1}

**5) NCERT Solutions Class 12 Maths Chapter 5 ****Continuity and Differentiability**

Continuity and Differentiability is one of the most important topics which help students to understand the concepts like, continuity at a point, continuity on an interval, derivative of functions and many more. It implies that this function is not continuous at x=0. One is to check the continuity of f(x) at x=3, and the other is to check whether f(x) is differentiable there. First, check that at x=3, f(x) is continuous. It’s easy to see that the limit from the left and right sides are both equal to 9, and f (3) = 9. Next, consider differentiability at x=3. NCERT Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Download free pdf.

NCERT Solutions Class 12 Maths Chapter 5 Continuity and Differentiability are structured here for students to score higher in exam. The list of topics from this chapter are given below: –

- Introduction
- Continuity
- Differentiability
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Mean Value Theorem

**Some important points in ‘Continuity and Differentiability’ are as follows-**

**A function f(y) is said to be continuous at a point y = a, if****Left hand limit of f(y) at(y = a) = Right hand limit of f(y) at (y = a) = Value of f(y) at (y = a)****i.e. if at y = a; LHL = RHL = f(a)****Continuity in an Interval: A function y = f(x) is said to be continuous in an interval (a, b), where a < b if and only if f(x) is continuous at every point in that interval.**

**Every identity function is continuous.****Every constant function is continuous.****Every polynomial function is continuous.****Every rational function is continuous.****All trigonometric functions are continuous in their domain.**

**Algebra of Continuous Functions:****Suppose f and g are two real functions, continuous at real number c. Then,**

**f + g is continuous at x = c.****f – g is continuous at x = c.****f.g is continuous at x = c.****cf is continuous, where c is any constant.****(f/g) is continuous at x = c, [provide g(c) ≠ 0]**

**6) NCERT Solutions Class 12 Maths Chapter 6 ****Applications of Derivatives**

With the help of the derivative, one can solve such problems as investigation of functions and sketching their graphs, optimization of various systems and modes of operations, simplifying algebraic expressions, approximate calculations, and much more. NCERT Solutions Class 12 Maths Chapter 6 Applications of Derivatives **Download free pdf.**

NCERT Solutions Class 12 Maths Chapter 6 Applications of Derivatives is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- Rate of Change of Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima

**Some important points in ‘Application of Derivatives’ are as follows-**

**Rate of Change of Quantities: Let y = f(x) be a function of x. Then, dy/dx represents the rate of change of y with respect to x.****If two variables x and y are varying with respect to another variable t, i.e. x = f(t) and y = g(t),****then dy/dx=(dy/dt)/(dx/dt) [By Chain Rule]****In other words, the rate of change of y with respect to x can be calculated using the rate of change of y and that of x both with respect to t.****Marginal revenue represents the rate of change of total revenue with respect to the number of items sold at an instant.****If R(x) is the revenue function for x units sold, then marginal revenue (MR) is given by MR = d/dx{R(x)}**

**7) NCERT Solutions Class 12 Maths Chapter 7 ****Integrals**

In Calculus, an integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of Calculus. NCERT Solutions Class 12 Maths Chapter 7 Integrals **Download free pdf.**

NCERT Solutions Class 12 Maths Chapter 7 Integrals is provided here in the simplest and understanding pattern for students in getting more efficiency. The list of topics from this chapter are given below: –

- Introduction
- Integration as an Inverse Process of Differentiation
- Methods of Integration
- Integrals of Some Particular Functions
- Integration by Partial Fractions
- Integration by Parts
- Definite Integral
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Some Properties of Definite Integrals

**Some important points in ‘Integrals’ are as follows-**

**Integration using Trigonometric Identities**

**2 sin A . cos B = sin( A + B) + sin( A – B)****2 cos A . sin B = sin( A + B) – sin( A – B)****2 cos A . cos B = cos (A + B) + cos(A – B)****2 sin A . sin B = cos(A – B) – cos (A + B)****2 sin A cos A = sin 2A****cos**^{2}A – sin^{2}A = cos 2A**sin**^{2}A + cos^{2}A = 1

**Integration by Parts****For a given functions f(x) and q(x), we have****∫[f(x) q(x)].dx = f(x)∫g(x).dx – ∫{ f'(x) ∫g(x).dx }.dx****Here, we can choose the first function according to its position in ILATE, where****I = Inverse trigonometric function****L = Logarithmic function****A = Algebraic function****T = Trigonometric function****E = Exponential function**

** [the function which comes first in ILATE should taken as first junction and other as second function]**

**8) NCERT Solutions Class 12 Maths Chapter 8 ****Applications of Integrals**

The application of integrations in real life is based upon the industry types, where this calculus is used. Like in the field of engineering, engineers use integrals to determine the shape of building constructions or length of power cable required to connect the two substations etc. NCERT Solutions Class 12 Maths Chapter 8 Applications of Integrals Download free pdf.

NCERT Solutions Class 12 Maths Chapter 8 Applications of Integrals is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- The area of the region bounded by a curve and a line
- Area under Simple Curves
- Area between Two Curves

**Some important points in ‘Application of Integrals’ are as follows-**

**Symmetry**

**If powers of y in a equation of curve are all even, then curve is symmetrical about X-axis.****If powers of x in a equation of curve are all even, then curve is symmetrical about Y-axis.****When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.****If x and y are interchanged and equation of curve remains unchanged then the curve is symmetrical about line y = x.**

**Nature of Origin**

**If point (0,0) satisfies the equation then curve passes through origin.****If curve passes through origin then equate lowest degree term to zero and get equation of tangent. If there are two tangents, then origin is a double point.**

**Asymptotes**

**Equate coefficient of highest power of x and get asymptote parallel to X-axis.****Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.**

**The Sign of (dy/dx) –****Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tangent is parallel or perpendicular to the X-axis.**

**9) NCERT Solutions Class 12 Maths Chapter 9 ****Differential Equations**

A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x). Here ‘x’ is an independent variable and ‘y’ is a dependent variable. For example, dy/dx = 5x. NCERT Solutions Class 12 Maths Chapter 9 Differential Equations Download free pdf.

NCERT Solutions Class 12 Maths Chapter 9 Differential Equations is provided here in the simplest and understanding pattern for students in getting more efficiency in Maths. The list of topics from this chapter are given below: –

- Introduction
- Basic Concepts
- General and Particular Solutions of a Differential Equation
- Formation of a Differential Equation whose General Solution is given
- Methods of Solving First Order, First Degree Differential Equations

**Some important points in ‘Differential Equation’ are as follows-**

**Differential Equation:****An equation involving independent variable, dependent variable, derivatives of dependent variable with respect to independent variable and constant is called a differential equation.****e.g.****x (dy/dx) + xy (d**^{2}y/dx^{2}) + 5 = 0 ; ¶z/¶x + ¶z/¶y = 0**Ordinary Differential Equation: An equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.****e.g.****(dy/dx) + (d**^{2}y/dx^{2}) -3 = 0**If any given relationship between the dependent and independent variables then a differential equation can be formed by differentiating it with respect to the independent variable and eliminating arbitrary constants involved.****Order of a Differential Equation:****It****is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.****Order of the differential equation cannot exceed the number of arbitrary constants in the equation.****Degree of a Differential Equation: The highest exponent of the highest order derivative is called the degree of a differential equation if exponent of each derivative and the unknown variable appearing in the differential equation is a non-negative integer.****Order and degree of a differential equation are always positive integers.****The differential equation is a polynomial equation in derivatives.****The degree of a differential equation is not defined if it is not a polynomial equation in its derivative.**

**10) NCERT Solutions Class 12 Maths Chapter 10 ****Vector Algebra**

An algebra for which the elements involved may represent vectors and the assumptions and rules are based on the behavior of vectors. NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Download free pdf.

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra is provided here in the simplest and understanding pattern for students in getting more efficiency in Maths. The list of topics from this chapter are given below: –

- Introduction
- Some Basic Concepts
- Types of Vectors
- Addition of Vectors
- Multiplication of a Vector by a Scalar
- Product of Two Vectors

**Some important points in ‘Vector Algebra’ are as follows-**

**Those quantities which have magnitude as well as direction are called vector quantities or vectors.****Those quantities which have only magnitude and no direction are called scalar quantities.****Direction Cosines: If α, β and γ are the angles which a directed line segment OP makes with the positive directions of the coordinate axes OX, OY and OZ respectively then cos α, cos β and cos γ are known as the direction cosines of OP and are generally denoted by the letters l, m and n respectively.**

** Types of Vectors-**

**Null vector or zero vector:****A vector whose initial and terminal points coincide and magnitude is zero, is called a null vector.****Unit vector:****A vector of unit length is called unit vector.****Collinear vectors: Two or more vectors are said to be collinear, if they are parallel to the same line, irrespective of their magnitudes and directions.****Coinitial vectors:****Two or more vectors having the same initial point are called coinitial vectors.****Equal vectors:****Two vectors are said to be equal, if they have equal magnitudes and same direction regardless of the position of their initial points.****Negative vector: Vector having the same magnitude but opposite in direction of the given vector, is called the negative vector.**

**11) NCERT Solutions Class 12 Maths Chapter 11 ****Three Dimensional Geometry**

Everything in the real world is in a three-dimensional shape. You can simply look around and observe! Even a flat piece of paper has some thickness if you look sideways. A strand of your hair or a big-sized bus, all of them have a three dimensional geometry. NCERT Solutions Class 12 Maths Chapter 11 Three Dimensional Geometry Download free pdf.

NCERT Solutions Class 12 Maths Chapter 11 Three Dimensional Geometry is provided here in the simplest and understanding pattern for students in getting more efficiency in Maths. The list of topics from this chapter are given below: –

- Introduction
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Angle between Two Lines
- Shortest Distance between Two Lines
- Plane
- Co-planarity of Two Lines
- Angle between Two Planes
- Distance of a Point from a Plane
- Angle between a Line and a Plane

**Some important points in ‘Three Dimensional Geometry’ are as follows-**

**Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis and Z-axis respectively then cos α, cos β, and cos γ are called direction cosines of a line. They are denoted by l, m and n. Therefore, l = cos α, m = cos β and n = cos γ.****Sum of squares of direction cosines of a line is always 1,****i.e. l**^{2}+ m^{2}+ n^{2}= 1 or cos^{2}α + cos^{2}β + cos^{2}γ = 1**Direction cosines of a directed line are unique.****Direction Ratios of a Line: Number proportional to the direction cosines of a line are called direction ratios of a line.****If a, b and c are direction ratios of a line then l/a = m/b = n/c****Direction ratios of two parallel lines are proportional.****Direction ratios of a line are not unique.**

**12) NCERT Solutions Class 12 Maths Chapter 12 ****Linear Programming**

Linear programming is an optimization technique for a system of linear constraints and a linear objective function. An objective function defines the quantity to be optimized, and the goal of linear programming is to find the values of the variables that maximize or minimize the objective function. NCERT Solutions Class 12 Maths Chapter 12 Linear Programming Download free pdf.

NCERT Solutions Class 12 Maths Chapter 12 Linear Programming is provided here for students to learn better and for the help of the students in problem solving. The list of topics from this chapter are given below: –

- Introduction
- Linear Programming Problem and its Mathematical Formulation
- Different Types of Linear Programming Problems

**Some important points in ‘Linear Programming’ are as follows-**

**A linear programming problem is one in which we have to find optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to certain conditions that the variables are non-negative and satisfying by a set of linear inequalities with variables, are sometimes called division variables.****A linear function z = px + qy (p and q are constants) which has to be maximised or minimised, is known as an objective function.****The maximum or minimum value of an objective function is called its optimal value.****The linear inequalities or equations or restrictions on the variables of the linear programming problem are known as constraints. The conditions x ≥ 0, y ≥ 0 are known as non-negative restrictions.****A problem, which seeks to maximise or minimise a linear function subject to certain constraints as determined by a set of linear inequalities, is known as an optimisation problem.****The common region determined by all the constraints including non-negative constraints x,y>0 of a linear programming problem is called the feasible region for the problem. The feasible region is always a convex polygon.****The region other than the feasible region is called an infeasible region.****Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is known as an infeasible solution.****Any point in the feasible region that gives the optimal value of the objective function is called the optimal feasible solution.****A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a circle else it is called unbounded.****The corner point method says that if a maximum or minimum value exists, then it will occur at a corner point of the feasible region.**

##### 13)** NCERT Solutions Class 12 Maths Chapter 13 ****Probability**

Probability is the branch of Mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where 0 indicates impossibility of the event and 1 indicates certainty. NCERT Solutions Class 12 Maths Chapter 13 Probability Download free pdf.

NCERT Solutions Class 12 Maths Chapter 13 Probability is provided here in the simplest and understanding pattern for students in getting more efficiency in Maths. The list of topics from this chapter are given below: –

- Introduction
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Random Variables and its Probability Distributions
- Bernoulli Trials and Binomial Distribution

**Some important points in ‘Probability’ are as follows-**

**A subset of the sample space associated with a random experiment is called an event or a case.****e.g. Getting either head or tail in an event, tossing a coin, rolling a ‘4’ on dice, drawing a card from the suits of club, etc.****The given events are said to be equally likely if none of them is expected to occur in preference to the other.****e.g. Each numeral on a die is equally likely to occur when the die is rolled.****A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other i.e. if A and B are mutually exclusive, then (A ∩ B) = Φ****e.g. In throwing a die, all the 6 faces numbered 1 to 6 are mutually exclusive since if any one of these faces comes, then the possibility of others in the same trial is ruled out. Also, turning left and turning right are Mutually Exclusive because you can’t do both at the same time.****A set of events is said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them.****If E**_{1}, E_{2}, …, E_{n}are exhaustive events then E_{1}∪ E_{2}∪……∪ E_{n}= S.**e.g. In throwing of two dice, the exhaustive number of cases is 6**^{2}= 36. Since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die.**Let A be an event in a sample space S then the complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by A’.****i.e. A’ = {x : x ∈ S, n ∉ A]****An operation which results in some well-defined outcomes is called an experiment but an experiment in which the outcomes may not be the same even if the experiment is performed in an identical condition is called a random experiment.****0 ≤ P(A) ≤ 1****Probability of an impossible event is zero.****Probability of certain event (possible event) is 1.****P(A ∪ A’) = P(S)****P(A ∩ A’) = P(Φ)****P(A’)’ = P(A)****P(A ∪ B) = P(A) + P(B) – P(A ∩ B)****P[(E ∪ F)/G] = P(E/G) + P(F/G) – P[(E ∩ F)/G], P(G) ≠ 0****P[(E ∪ F)/G] = P(E/G) + P(F/G), if E and F are disjoint events.****P(F’/G) = 1 – P(F/G)****Mean(μ) = Σ x**_{i}p_{i}= np**Variance(σ**^{2}) = Σ x_{i}^{2}p_{i}– μ^{2}= npq**Standard deviation (σ) = √Variance = √npq****Mean > Variance**

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