# NCERT solutions for class 11 Maths

## About NCERT solutions for class 11 Maths

Class 11th is the beginning stage of one’s technical career. The students can get the assistance of the best faculty of mathematics. The faculties have years of experience in a teaching career. The **NCERT solutions for class 11 maths** is prepared as per the latest syllabus. The very idea of preparing this study material is to ensure that students do not miss anything.

This NCERT solution for class 11 maths is prepared some very experienced teachers. If students are stuck at any topic, they right away consult our experts. Since class 11th maths is considered to a grade of advance level, so students need to learn it accordingly. Our team has segregated NCERT solutions for class 11 maths on the basis of chapters. This solution is arranged in a systematic manner, which helps students.

## Why NCERT solutions for class 11 maths are important?

The **NCERT solution** to each and every sum is provided in a very judicious manner. Special care has been taken on difficult topics. The team has taken efforts to break down the most typical numerical as well into simpler type. We have also provided students with more practical based questions in NCERT solutions for class 11 maths, which help them enhance their thinking horizon. All we want from students is un-deviated attention and perseverance.

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The study material comprises of various competitive level questions as well which helps in understanding competitive examinations. The very speciality of NCERT solutions for class 11 maths provided by Entrancei is easy to read. The NCERT solutions for class 11 maths has been very effective for students when revising just before the examination. All the questions are chosen very selectively.

## Class 11 Maths NCERT Chapter Wise Solutions

### Chapter 1 - Sets

A set is a well-defined collection of distinct objects. Well-defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally sets are denoted by capital letters A, B, C, X, Y, Z etc.

REPRESENTATION OF A SET-Usually, sets are represented in the following ways

ROASTER FORM OR TABULAR FORM-In this form, we list all the member of the set within braces (curly brackets) and separate these by commas. For example, the set of all even numbers less than 10 and greater than 0 in the roster form is written as: A = {2,4, 6,8}

SET BUILDER FORM OR RULE FORM-In this form, we write a variable (say x) representing any member of the set followed by a property satisfied by each member of the set. A = {x| x < 5, x < N} the symbol ‘|’ stands for the words” such that”.

### Chapter 2 - Relations and Functions

Function can be classified into two categories:

- One-one or Many-one
- Into or Onto

Each of above category comprises entire gamut of function. Thus if a function is not one-one, it has to be many-one and vice-versa.

Similarly if function is into then it cannot be onto and converse.

- (i) One-one functions (Injective)
- A function is said to be one - one if for every value of x in the Domain, there exists one and only one value in co-domain. Two different values of x do not have same value of y.
- (ii) Many-one functions
- A function is said to be many-one if there exists at least two elements in domain which have same image in co-domain.
- (iii) Into functions
- If there exist at least one element in co-domain which doesn't have a pre-image then such functions are said to be into.
- (iv) Onto functions (Surjective)
- If there doesn't exist any element in co-domain which does'nt have a pre-image then such functions are said to be onto. Note that for onto function co-domain = Range.

Based on above two categories of function, we can classify functions into four type

- One-one and Onto (Bijective)
- One-one and Into
- Many-one and Onto
- Many-one and Into.

### Chapter 3 - Trigonometric Functions

Literally, the word 'trigonometry' means the study of the properties of triangles. This involves the measurement of angles and lengths. An angle is defined as the amount of rotation of a revolving line with respect to a fixed line. If the rotation is in anticlockwise sense, the angle is taken to be positive, otherwise negative. Basically, we use two types of systems to measure an angle, viz., the sexagesimal system (or English system) and the circular system. In the former, a right angle is divided into 90 equal parts and each part is called one degree (1°). So, a straight angle is 180° and complete angle is 360°. Each degree is further divided into 60 equal parts and each part is called one minute (1'). Each minute again has 60 equal parts, each part being called one second (1).

- Thus, 1 right angle = 90°; 1° = 60; 1° = 60°.
- e.g., 40.32° = 40°(0.32 * 60)’ = 40°19.2’ = 40°19’(0.2 *60) = 40°19 -12'
- In circular system, the unit of measurement is radians. One radian is defined as the
- angle subtended at the center of a circle by an arc of length equal to the radius of that circle. It is written as 1c.

### Chapter 4 - Principle of Mathematical Induction

Principle of Mathematical InductionPrinciple of Mathematical Induction deals with topics such as the process of proving the induction its application in natural numbers as the least inductive subset of real numbers. Numerical related to the Principle of Mathematical Induction.

### Chapter 5 - Complex Numbers and Quadratic Equations

Very often we come across many equations involving several powers of one variable. If the indices of all these powers are integers then the equation is called a polynomial equation. If the highest index of a polynomial equation in one variable is two, then it is a quadratic equation. A quadratic equation is a second degree polynomial in x usually equated to zero. In other ‘words, for an equation to be a quadratic, the coefficient of x2 should not be zero and the coefficients of any higher power of x should be 0.

Roots of the equation-Just as a first degree equation in x has one value of x satisfying the equation, a quadratic equation in x has two values of x that satisfy the equation. The values of x that satisfy the equation are called the ROOTS of the equation. These roots may be real or complex.

### Chapter 6 - Linear Inequalities

In earlier classes, we have studied equations in one variable and two variables and also solved some statement problems by translating them in the form of equations. Now a natural question arises: ‘Is it always possible to translate a statement problem in the form of an equation? For example, the height of all the students in your class is less than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), (less than or equal) and (greater than or equal) which are known as inequalities. In this Chapter, we will study linear inequalities in one and two variables. The study of inequalities is very useful in solving problems in the field of science, mathematics, statistics, optimisation problems, economics, psychology, etc.

### Chapter 7 - Permutations and Combinations

In real life we are often encountered with problems of counting. Many a times the number of ways of happening of an event is so large that we can not count it on our hands. Permutation and Combinations are techniques to simplify the process of counting and are very useful when the number of ways of happening of some event are very large.

THE SUM RULE-Suppose a work A can occur in m ways and B can occur in n ways and both cannot occur simultaneously. Then A or B (at least one of them) can occur in (m + n) ways. This rule is also applicable for two or more exclusive events. THE PRODUCT RULE-Suppose there are two works A and B. Let A can occur in m ways and B in n ways. Suppose that the ways for A and B are not related in the sense that B occur in n ways regardless the outcome of A. Then both A and B can occur in mn ways. For example, let there be two questions A and B which can be solved by two methods and 3 methods respectively. Then both A and B can occur in 2 3 = 6 ways. The above rules are very useful and many problems of this chapter can be solved by using these fundamental concepts without using any formula.

### Chapter 8 - Binomial Theorem

Binomial Theorem was introduced by Newton for the purpose of calculating or approximating expansion of two terms raised to any power.

- Binomial Expression : Algebraic expressions which have two terms are called Binomial Expressions.
- Binomial Expansions :Binomials when raised to different powers are called Binomial Expansions.
- Binomial Index :It is the power to which a binomial is raised to give a Binomial Expansion.In this chapter, we are considering only positive integral index.
- Coefficient : The constant or the number associated with each term of a binomial expansion is known as the coefficient

### Chapter 9 - Sequences and Series

A sequence is a set of terms in a definite order. The sequence which obey a definite rule and its general term is always expressible in terms of*n*, is called progression.

### Chapter 10 - Straight Lines

Important Formulas

### Chapter 11 - Conic Sections

### Chapter 12 - Introduction to Three Dimensional Geometry

### Chapter 13 - Limits and Derivatives

The idea of limits is one of the basic concepts that distinguishes calculus from the remaining branches of mathematics. Limits are used to check the behaviour of functions close to certain points. This use paves the way to define continuity and differentiability of functions which are extremely important concepts to analyse functions. We also introduce derivatives in this lesson and discuss their geometrical significance. Problems involving the concepts of limit, continuity and differentiability simultaneously are very common, which are explained elaborately in this lesson. Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L’Hospital rule of evaluation of limits of functions, continuity of composite functions, intermediate value property of continuous functions.

### Chapter 14 - Mathematical Reasoning

### Chapter 15 - Statistics

One of the most important objectives of statistical analysis is to get one single value that describes the characteristics of entire mass of unwieldy data Such a value is called central value or the average. It is a single value, which represents a group of values.

Classification of Data:

- Discrete series: The observations along with their frequency of occurrence are given.
- Continuous series: The class intervals along with their frequency are given.

TYPES OF AVERAGES

(a)Mean

- Arithmetic Mean
- Weighted Arithmetic Mean
- Geometric Mean
- Weighted Geometric Mean
- Harmonic Mean
- Weighted Harmonic Mean

(b)Median

(c)Mode

### Chapter 16 - Probability

RANDOM EXPERIMENT-An experiment, whose all possible outcomes are known in advance but the outcome of any specific performance can not predicted before the completion of the experiment, is known as random experiment.

Sample space and Sample Point-A set of all possible outcomes associated with same random experiment is called sample space and is usually denoted by ‘S’. Each element of a sample space is called a sample point.

- Event -An event is a subset of sample space.
- Simple Event and Compound Event
- If an event is a set containing only one element of the sample space, then it is called a
- simple event.
- A compound event is one that can be represented as a union of simple events.
- PROBABILITY
- If a random experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes favour A, then the probability of event A, P (A) =n/N

## How to Study NCERT solutions for class 11 maths effectively?

The team has totally removed the un-necessary studying part from the study material. We wish students have sharp question-solving skills for examination. Students referring NCERT solutions for class 11 maths have an extra cutting edge over the rest students. These NCERT solutions for class 11 maths are helpful for a student looking to score maximum marks. The complete study material has been prepared after extensive research of previous year question papers.

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## Key Benefit for NCERT solutions for class 11 Maths

NCERT solutions for class 11 Maths is prepared by expert of Entrancei to gives you errorless and accurate solutions of all questions given in the NCERT.

- We have uploaded solution of class 11 maths of all questions given in the exercise of NCERT text book.
- All NCERT solutions for class 11 Maths are prepared carefully to provide you errorless solutions with all required steps.
- All solution of class 11 maths are prepared and solved as per the new guideline of CBSE so that you can score full marks in your exam. With Entrancei NCERT solutions for class 11 Maths you can score full marks in class 11 maths.
- Based on new pattern of CBSE class 11 according to which now MCQ based questions weightage will be increased and lots of school are adding this in their internal exam too to give students additional advantage and practice for class 12 upcoming board exam. Entrancei consist of additional questions which are MCQ based and added several solved examples apart from NCERT solutions for class 11 Maths.
- NCERT solutions for class 11 Maths in the present subject consist of solution to all questions given in exercise of NCERT in exercise wise so that you can get solutions of your questions which you are facing problem quickly.
- All the solutions consist of detail information with each and every step which are required.
- Step by step solution of numerical will help you to understand the method of solving the questions and clear your concepts of class 11 maths.

### FAQ For NCERT Solutions for class 11 Maths

**Q-1. How to score full marks in class 11 Maths with the help of NCERT solutions of class 11 Maths ?**

Ans- To score good marks in class 1 Maths you need to have good foundation of concepts and its application in the numerical. Conceptual celerity can be develop by reading and understanding the theory part given in the chapter of NCERT and application can be understood through solved examples and questions given in Exercise.

**Q-2. What are the most important chapters of class 11 Maths ?**

Ans- In unit test and final exam of class 11 maths all most all chapters are covered and questions are asked form all chapters. Few topics are needed to be solved more frequently like Trigonometric Functions, Modulus and Conjugate of a Complex Number, Binomial Theorem, Permutations and Combinations, Linear Inequalities.

**Q-3. What are the tips and trick to do final revision for class 11 Maths ?**

Ans- class 11 Maths consist of 16 chapters so make sure you have prepared a detail notes of all 16 chapters which consist of all important points and formulas used. These notes are to be prepared form NCERT text book and having all bullet points which are needed to solve the questions. Notes prepared by you is highly helpful for the final revision of class 11 Maths. Do solve few sample papers of class 11 Maths which help you to find out the mistake you do in the exam and work on these mistakes.

**Q-4. Should we use some additional book apart form NCERT text book of class 11 Maths ?**

Ans- If you are preparing for class 11 Maths than no need to follow any reference book NCERT class 11 Maths is sufficient for class 11 Maths. But if you are preparing for competitive entrance exam than you must follow few additional book of Maths and solve MCQ based questions for class 11 Maths.

**Q-5. Solving Exercise question by the help of NCERT solutions for class 11 Maths is good practice ?**

Ans- No, NCERT solutions for class 11 maths is only for reference one shouldn’t use it as helping guide. Use these solutions only for the numerical which you can’t solve despite of several attempts.