# NCERT SOLUTIONS FOR CLASS 11

## Introduction about NCERT Solutions for Class 11 Maths

NCERT Solutions for class 11 given here have been put together by our content analysts who have many years of experience of creating content for CBSE board, ICSE board and all Indian state boards. The solutions have been designed to simplify all class 11 Math problems, which are given in the textbooks. All the 16 chapters as prescribed by CBSE have been included in NCERT Maths class 11 solutions.

## Why students prefer NCERT Solutions for Class 11 Maths

Students prefer NCERT solutions to do their regular home assignment, mock tests and ace their final exams. In most of the common entrance tests conducted for admissions in Engineering institutes, the questions are designed as per the syllabus of textbooks. Therefore, the students prefer to practice these NCERT Solution exercises on a regular basis to score better marks in exams.

## Topics Covered in NCERT Solutions for Class 11 Maths & Brief Detail About Them

NCERT Solutions of class 11 maths contains all chapter solutions in pdf. Solutions can be downloaded chapter wise. NCERT solutions for class 11 cover all the chapters including Sets, Relations and Functions, Trigonometric Functions, Principle of Mathematical Induction, Complex Numbers and Quadratic Equations, Linear Inequalities, Permutations and Combinations, Binomial Theorem, Sequences and Series, Straight lines, Conic Sections, Introduction to Three Dimensional Geometry, Limits and Derivatives, Mathematical Reasoning, Statistics, Probability.

These all the chapters are fully comprehensive well explained. It can really help you with your exams, providing you with the easiest and fastest method to solve the question, additionally, the concept of solutions has kept so simple and easy to understand so that it can be remembered for a lifetime to students hence it will also help you cracking other higher-level exams.

Students can easily use the PDF of the chapter-wise solutions and gain conceptual knowledge to solve the problems according to the NCERT Maths textbook for Class 11. This helps students enhance their confidence, which is required to master concepts and perform well in exams.

### NCERT Solutions for Class 11 Maths Chapter 1 – Sets

1. Sets and Their Representations:
• A set is a well-defined collection of objects.
2. The empty set:
• A set that does not contain any element is called an empty set.
3. Finite and infinite sets
• A set that consists of a definite number of elements is called a finite set, otherwise, the set is called an infinite set
4. Equal sets
• Two sets A and B are said to be equal if they have exactly the same elements
5. Subsets
• A set A is said to be a subset of a set B if every element of A is also an element of B. Intervals are subsets of R.
6. Power set
• A power set of a set A is a collection of all subsets of A. It is denoted by P(A).
7. Universal set
• Basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.
8. Venn diagrams
• The relationships between sets can be represented by means of diagrams which are known as Venn diagrams.
9. Operations on sets
• The union of two sets A and B is the set of all those elements which are either in A or in B.
• The intersection of two sets A and B is the set of all elements which are common. The difference between two sets A and B in this order is the set of elements that belong to A but not to B.
10. Complement of a set
• The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.
11. Practical problems on union and intersection of two sets
• n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

### NCERT Solutions for Class 11 Maths Chapter 2 – Relations and Functions

1. Cartesian Poduct of Sets
• Cartesin Product: Given two non-empty sets P and Q. The cartesian product P×Q is the set of all ordered pairs of elements from P and Q.
2. Relations:
• Relation: A relation R from a set A to set B is a subset of the cartesian product A×B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A×
• Domain: The set of all first elements of the ordered pairs in a relation R from a set A to B is called the domain of the relation R.
• Range: The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The entire set B is called the codomain of the relation R. Note: Range ⊆
3. Functions
• Function: A relation from set A to set B is said to be a function if every element of set A has one and only one image in set B.
• Real-Valued Function: A function that has either R or one of its subsets as its range is called a real-valued function. If its domain is either R or a subset of R, it is called a real function.
• Types of functions:
• Identity Function: f:RR by y=f(x)=x for each xR is identity function.
• Constant Function: f:RR by y=f(x)=c,xR
• Polynomial Function: f:RR by f(x)=a0+a1x+a2x2+…+anxn, where n is a non-negative integer and a0,a1,a2,…,anR
• Rational Functions: Functions of the form f(x)/g(x), where f(x) and g(x) are rational functions and g(x)≠0.
• Modulus Function: f:RR by f(x)=|x|={xif x≥0if x<0
• Algebra of Real Functions:
• Addition of two real functions: Let f:XR and g:XR be any two real functions, where XR, then (f+g)(x)=f(x)+g(x), for all xX.
• Subtraction of two real functions: Let f:XR and g:XR be any two real functions, where XR, then (fg)(x)=f(x)−g(x), for all xX.
• Multiplication by a scalar: Let f:XR and α is a scalar. Then (αf)(x)=αf(x),xX

Multiplication of two real functions: Let f:XR and g:XR be two real functions. Then fg:XR by (fg)(x)=f(x)g(x), for all xX

### NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions

1. Angles
• Angle: Angle is a measure of rotation of a given ray about its initial point.
• Degree measure: If a rotation from the initial side to terminal side is 1/360 th a revolution, the angle is said to have a measure of one degree.
• Radian measure: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.
• Relation between radian and real numbers: Radian measures and real numbers can be considered as one and the same.
• Relation between degree and radian
2. Trigonometric Functions
• Sign of trigonometric functions
• Domain and range of trigonometric functions
3. Trigonometric Functions of Sum and Difference of Two angles
• Trigonometric functions of sum of two angles: The basic results in this connection are called trigonometric identities.
• Trigonometric functions of a difference of two angles; The basic results in this connection are called trigonometric identities.
4. Trigonometric Equations
• Trigonometric Equations: Equations involving trigonometric functions of a variable are called trigonometric equations.

### NCERT Solutions for Class 11 Maths Chapter 4 – Principle of Mathematical Induction

1. The principle of mathematical induction
• The principle of mathematical induction is one such tool that can be used to prove a wide variety of mathematical statements.
• Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.

### NCERT Solutions for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations

1. Complex Number
• Complex Number: Any number of the form a+ib, where a,b∈R and i2=−1″i”iscalled”iota”, is a complex number. A complex number is denoted by z.

Here, a is called the real part and denoted by Re z, and b is called the imaginary part and denoted by Imz.

• Equality of complex numbers: Two complex numbers z1=a+ib and z2=c+idare equal if a=c and b=d.
1. The Modulus and the Conjugate of a complex number.
• Modulus of a complex number: Let z=a+ib be a complex number. Then, modulus of z, denoted by |z|.
• Conjugate of a complex number: For any complex number z=a+ib, the conjugate, represented by z¯, is given by z¯=a−ib.
2. Argand Plane and Polar Representation.
• The complex number x+iy corresponds to the ordered pair (x,y) can be shown geometrically as the unique point P(x,y) in the XY− plane and vice-versa.
• The plane that has a complex number assigned to each of its points is called the complex plane or the Argand plane.
• In the Argand plane, the modulus of the complex number is the distance between the point P(x,y) and the origin O(0,0).
• The points on the x-axis correspond to the complex numbers of the form a+i0 and the points on the y-axis corresponds to the complex numbers of the form 0+ib.
• The x-axis and y-axis in the Argand plane are called the real axis and the imaginary axis.
• Representation of complex numbers and their conjugate.
• Polar form of the complex number.
3. Square root of the complex number.
• Square root of the complex number.

### NCERT Solutions for Class 11 Maths Chapter 6 – Linear Inequalities

1. Inequalities
• Two real numbers or two algebraic expressions related by the symbols <, >, ≤ or ≥ form an inequality.
• Equal numbers may be added to (or subtracted from ) both sides of an inequality.
• Both sides of an inequality can be multiplied (or divided ) by the same positive number. But when both sides are multiplied (or divided) by a negative number, then the inequality is reversed.
• The values of x, which make an inequality a true statement, are called solutions of the inequality
2. Algebraic solutions of linear inequalities in one variable and their graphical representation
• To represent x < a (or x > a) on a number line, put a circle on the number a and a dark line to the left (or right) of the number a. 
• To represent x ≤ a (or x ≥ a) on a number line, put a dark circle on the number a and dark the line to the left (or right) of the number x. 
• If inequality is having ≤ or ≥ symbol, then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by a dark line that satisfies an arbitrary point in that part. 
• If inequality is having < or > symbol, then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies to the left (below) or right (above) of the graph of the corresponding equality represented by a dotted line that satisfies an arbitrary point in that part.
3. Graphical solution of linear inequalities in two variables
• The solution region of a system of inequalities in the region satisfies all the given inequalities in the system simultaneously.

### NCERT Solutions for Class 11 Maths Chapter 7 – Permutations and Combinations

1. Fundamental Principle of Counting
• Fundamental principle of counting If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m × n.
2. Permutations
• Permutations: A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.
• Permutations when all the objects are distinct: The number of permutations of n different objects taken r at a time, where 0 < r ≤ n and the objects do not repeat  is n ( n – 1) ( n – 2). . .( n – r + 1)
• Factorial notation
• Derivation of formula
• Permutations when all the objects are not distinct objects
3. Combinations
• The number of combinations of n different things taken r at a time, denoted by nCr.

### NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem

1. Binomial Theorem for positive integral indices
• Coefficients are known as binomial coefficients.
• Number of terms in binomial expansion = n + 1
• In every term of the expansion, a sum of indices of a and b is n.
2. General Term and Middle Terms
• General term: The (r + 1)th term is called the general term of the expansion (a + b)n.
• Middle terms: If n is even then the number of terms in the expansion will be n + 1. If n is odd then there will be two middle terms in the expansion.

### NCERT Solutions for Class 11 Maths Chapter 9 – Sequences and Series

1. Sequences
• Sequence
• Finite Sequence: A sequence containing a finite number of terms is called a finite sequence.
• Infinite Sequence: A sequence containing an infinite number of terms is called a finite sequence.
• Fibonacci Sequence: A sequence generated by the recurrence relation is called the Fibonacci sequence.
2. Series
• Series: The sum expressed as a1 + a2 + a3 + … is called series.
• Finite Series: A series is called finite series if it has got a finite number of terms.
• Infinite Series A series is called finite series if it has got an infinite number of terms.
3. Arithmetic Progression
• Arithmetic Progression: An arithmetic progress is a sequence in which terms increase or decrease regularly by the same constant.
• Standard form of an AP
• General form of an AP
• Properties of AP
• Sum of first n terms of an AP.
• Arithmetic Mean: The sequence a, A, b is in A.P.
4. Geometric Progression
• Geometric progression: A sequence is said to be geometric progression if the ratio of any term to its preceding term is the same throughout.
• General term of a GP
• Finite or infinite GP
• Finite or infinite geometric series
• Sum to n terms of a GP
• Geometric mean: The sequence a, G, b is G.P.
5. Relation between AM and GM
• Relation between AM and GM
6. Sum to n terms of special series
• Sum of first n natural numbers
• Sum of squares of first n natural numbers
• Sum of cubes of first n natural numbers
7. Infinite GP and its sum
• Infinite GP.
• Formula for finding sum to infinity

### NCERT Solutions for Class 11 Maths Chapter 10 – Straight Lines

1. Slope of a line:
• Slope of a line
• Slope of a line when coordinates of any two points on the line are given.
• Conditions of parallelism and perpendicularity of lines in terms of their slopes.
• Two lines are parallel if and only if their slopes are equal.
• Two lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
• Angle between two lines.
• Collinearity of three points.
2. Various forms of the equation of a line:
• Horizontal and vertical lines
• Point slope form of a line.
• Two-point form of a line
• Slope-intercept form of a line
• Intercept form of a line
• Normal form of a line
3. General Equation of a line
• Any equation of the form Ax+By+C=0, where A and B are not zero simultaneously is called a general linear equation or general equation of a line.
• Different forms of Ax+By+C=0

(i) Slope-intercept form

(ii) Intercept form

(iii) Normal form

### NCERT Solutions for Class 11 Maths Chapter 11 – Conic Sections

1. Sections of a cone
• Circle, ellipse, parabola, and hyperbola
• Degenerated conic sections
2. Circle
• A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.
• The equation of a circle with center (h, k) and the radius r
3. Parabola
• A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.
• The equation of the parabola with focus at (a, 0) a > 0 and directrix x = – a is y2 = 4ax.
• Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose endpoints lie on the parabola.
• Length of the latus rectum of the parabola y2 = 4ax is 4a.
4. Ellipse
• An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant
• Equation of an ellipse with foci on the x-axis
• Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose endpoints lie on the ellipse.
• Length of the latus rectum of the ellipse
• The eccentricity of an ellipse is the ratio between the distances from the center of the ellipse to one of the foci and to one of the vertices of the ellipse.
5. Hyperbola
• A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.
• The equation of a hyperbola with foci on the x-axis.
• Latus rectum of a hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose endpoints lie on the hyperbola.
• Length of the latus rectum of the hyperbola.
• The eccentricity of a hyperbola is the ratio of the distances from the center of the hyperbola to one of the foci and to one of the vertices of the hyperbola.

### NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

1. Coordinate axes and coordinate planes in three-dimensional shapes
• In three dimensions, the coordinate axes of a rectangular Cartesian coordinate system are three mutually perpendicular lines. The axes are called the x, y, and z-axes.
• The three planes determined by the pair of axes are the coordinate planes, called XY, YZ, and ZX-planes.
• The three coordinate planes divide the space into eight parts known as octants.
2. Coordinates of a point in space
• The coordinates of a point P in three-dimensional geometry are always written in the form of triplet-like (x, y, z). Here x, y, and z are the distances from the YZ, ZX, and XY-planes.
• Any point on x-axis is of the form (x, 0, 0)
• Any point on y-axis is of the form (0, y, 0)
• Any point on z-axis is of the form (0, 0, z)
3. Distance between two points
• Distance formula
4. Section formula
• Section formula
• Mid-point formula

### NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives

1. Intuitive Idea of Derivative
• The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit
2. Limits
• Limits: Limit of a function at a point is the common value of the left and right hand limits, if they coincide.
• Algebra of limits
• Limits of polynomials and rational functions
3. Limits of Trigonometric Functions
4. Derivatives
• Algebra of derivative of functions
• Derivative of polynomials and trigonometric function

### NCERT Solutions for Class 11 Maths Chapter 14 – Mathematical Reasoning

1. Statements
• A sentence is called a mathematically acceptable statement if it is either true or false but not both
2. New Statements from old:
• Negation of a statement: Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p
• Compound statements: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement.
3. Special Words/Phrases
• The word “And”
• The word “Or”
• Quantifiers
4. Implications
• Contrapositive and converse: The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p. The converse of a statement p ⇒ q is the statement q ⇒ p ⇒ q together with its converse, gives p if and only if q.
5. Validating Statements
• The following methods are used to check the validity of statements: (i) direct method (ii) contrapositive method (iii) method of contradiction (iv) using a counterexample.

### NCERT Solutions for Class 11 Maths Chapter 15 – Statistics

1. Measures of Dispersion
• Range
• Quartile deviation
• Mean deviation
• Standard deviation
2. Range
• Range of a series = Maximum value – Minimum value
3. Mean Deviation
• Mean deviation for ungrouped data
• Mean deviation for grouped data
• Limitations of mean deviation
4. Variance and Standard Deviation
• Standard deviation
• Standard deviation of a discrete frequency distribution
• Standard deviation of continuous frequency distribution
• Shortcut method to find variance and standard deviation
5. Analysis of frequency distribution
• Comparison of two frequency distributions with same mean

### NCERT Solutions for Class 11 Maths Chapter 16 – Probability

1. Random Experiments
• Outcomes: The set of all possible outcome.
• Sample space: Elements of sample space
2. Event
• Occurrence of an event
• Types of events
• Impossible and sure events: The empty set. The whole sample space is a sure event.
• Simple events: If an event E has only one sample point of a sample space.
• Compound event: If an event has more than one sample point.
• Algebra of events
• Complementary event: The set A′ or S – A
• The event A or B: The set A ∪ B
• Mutually exclusive events: A and B are mutually exclusive if A ∩ B = φ
• Exhaustive events: Events E1, E2,…, En are mutually exclusive and exhaustive if E1 ∪ E2 ∪ …∪ En = S and Ei  ∩ Ej = φ  V i ≠ j
3. Axiomatic Approach to Probability
• Probability of an event
• Probabilities of equally likely outcomes
• Probability of the event A or B
• Probability of event ‘not A’

## Benefit of NCERT Solutions for Class 11 Maths

NCERT solutions for class 11 Maths will create better and effective learning for students. It will help in building a strong foundation of all the concepts for higher-level classes and also for competitive exams.

NCERT Class 11th Maths solutions will help students to clear their doubts by offering an in-depth understanding of the concepts. Through detailed explanations, students can learn the concepts which will enhance their thinking and learning abilities.

• It is very easy for students to access the solutions for every application from the chapters available.
• There are graphs and illustrations provided to students that help them understand the concepts better and retain them.
• These solutions are prepared by an expert team of Fliplearn that focuses on accuracy.
• The study material helps students to bridge the knowledge gap with the exercises and solutions.
• Simple and easily understandable solutions are provided.
• Detailed study material is available for class 11 students

Fliplearn is here to simplify your problems and make learning easy and enjoyable. Download the Fliplearn app to find more help and a personalized experience.

### How Fliplearn’s NCERT solutions assist us in scoring good marks in the exam?

The content analysts at Fliplearn have designed the NCERT Solutions in accordance with the syllabus designed by the CBSE board. The essential explanation is provided for major points to make the concepts easier for the students while learning. NCERT solutions are designed with the aim of helping students ace the exam without fear. The solutions mainly help students to improve their problem-solving abilities which are important for the exam.

### What are the important concepts in chapter 11 of conic sections?

The chapter covers topics like circle, parabola, ellipse, hyperbola, and degenerate conic sections. In parabola, you will learn standard equations of parabola and latus rectum. In the case of an ellipse, you will learn the relationship between the semi-major axis, semi-minor axis, and the distance of the focus from the center of the ellipse, Special cases of an ellipse, and latus rectum.

In this chapter, you will also learn eccentricity, standard equation, and latus rectum of Hyperbola. A conic section is one of the important chapters of class 11 maths.

### What will I study in Chapter 4 Mathematical Induction of NCERT Solutions for Class 11 Maths?

Mathematical Induction is a technique that is used to prove whether a given statement is true or false. The important fact is that the theorem should be true for every given natural number. After covering the chapter students will be familiar with the following points.

1. Proving a statement is the main motive.
2. The proof should be true for all the values of natural numbers.
3. For initial value, the statement should be true.
4. For values till nth, the statement should be true.
5. All the steps in the proof should be true and justified.

### How many chapters are there in the class 11 Maths NCERT textbook?

There is a total of 16 chapters in class 11 Maths NCERT textbook. The name of the chapters are Sets (Chapter 1), Relations and Functions (Chapter 2), Trigonometric Functions (Chapter 3), Principle of Mathematical Induction (Chapter 4), Complex Numbers and Quadratic Equations (Chapter 5), Linear Inequalities (Chapter 6), Permutations and combination (Chapter 7), Binomial Theorem (Chapter 8), Sequence and Series (Chapter 9), Straight Lines (Chapter 10), Conic Sections (Chapter 11), Introduction to Three Dimensional Geometry (Chapter 12), Limits and Derivatives (Chapter 13), Mathematical Reasoning (Chapter 14), Statistics (Chapter 15), Probability (Chapter 16)

### Is it required to solve each and every problem of NCERT Solutions for Class 11th Maths?

In the CBSE board, each and every problem irrespective of their understanding level are important for the exam. So the students are recommended to solve the NCERT textbook on a daily basis to gain a grip on the fundamental concepts. By regular practice, students will be able to analyze their areas of weakness and work on them for a better academic score. The step-wise explanation under each problem will help students to perform well in the annual exams.

### NCERT Solutions for other Classes

NCERT Solutions is also available for other classes. Students preparing for class 6th, 7th, 8th, 9th, 10th and 12th will find it the best way to score good marks. NCERT Solutions for class 6th, 7th, 8th, 9th, 10th and 12th provides a plethora of study material and references that help students trust it the most.

NCERT Solutions for Class 6 Maths RS Aggarwal Solutions for Class 8 Maths
NCERT Solutions for Class 7 Maths RS Aggarwal Solutions for Class 9 Maths
NCERT Solutions for Class 8 Maths RS Aggarwal Solutions for Class 10 Maths
NCERT Solutions for Class 9 Maths
NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 12 Maths

## NCERT Solution for Class 11 Maths

 Addition 2020-21 Board CBSE Curriculum Subject Math PDF Download Yes Textbook Solutions Class 11 Total Chapters 16

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