NCERT Solutions for Class 9 Political Science Chapter 2

August 8, 202239 Insightful Publications

**NCERT Solutions for Maths Class 11** is one of the most useful study materials for Class 11 students as it helps them understand complex Maths concepts easily and score more marks. Here we have provided Maths NCERT solutions for class 11 , which covers all 16 chapters from the Maths book prescribed for CBSE Class 11. NCERT Class 11 Maths Solutions provided here are 100% accurate and prepared as per the latest CBSE curriculum.

Students looking for CBSE Class 11 Maths NCERT Guide PDF can download the same from this article. The Class 11 Maths Chapter-wise Solutions can be downloaded from the links included in this article. Scroll down for direct links to CBSE Class 11 Maths PDF Download.

**CBSE NCERT Solutions** for Class 11 Maths acts as the best study material for students as it helps them in their studies as they advance through the syllabus. We have arranged the **Maths NCERT Solutions for Class 11** in a chapter-wise manner so that students can download the solutions for the chapter they are preparing or solving questions for. You can check the chapter list along with NCERT Guide for Class 11 Maths PDF download links below:

**Chapter-Wise class 11 Maths NCERT book solutions** **are as follows**:

**NCERT Solutions for Class 11 Maths Chapter 1**: Sets**NCERT Solutions for Class 11 Maths Chapter 2**: Relations and Functions**NCERT Solutions for Class 11 Maths Chapter 3**: Trigonometric Functions**NCERT Solutions for Class 11 Maths Chapter 4**: Principle of Mathematical Induction**NCERT Solutions for Class 11 Maths Chapter 5**: Complex Numbers and Quadratic Equations**NCERT Solutions for Class 11 Maths Chapter 6**: Linear Inequalities**NCERT Solutions for Class 11 Maths Chapter 7**: Permutations and Combinations**NCERT Solutions for Class 11 Maths Chapter 8**: Binomial Theorem**NCERT Solutions for Class 11 Maths Chapter 9**: Sequences and Series**NCERT Solutions for Class 11 Maths Chapter 10**: Straight Lines**NCERT Solutions for Class 11 Maths Chapter 11**: Conic Sections**NCERT Solutions for Class 11 Maths Chapter 12**: Introduction to Three Dimensional Geometry**NCERT Solutions for Class 11 Maths Chapter 13**: Limits and Derivatives**NCERT Solutions for Class 11 Maths Chapter 14**: Mathematical Reasoning**NCERT Solutions for Class 11 Maths Chapter 15**: Statistics**NCERT Solutions for Class 11 Maths Chapter 16**: Probability

Also, check our NCERT solutions for other subjects from Class 11:

NCERT Solutions for Class 11 Physics |

NCERT Solutions for Class 11 Chemistry |

NCERT Solutions for Class 11 Biology |

CBSE NCERT Guide for Class 11 Maths is the best source of exam preparation for Class 11 students. The benefits of using Maths NCERT solutions for Class 11 are listed below:

- NCERT solutions include all fundamental concepts for each subject. This helps in a thorough understanding of the subject.
- These solutions are presented in a step-by-step and detailed manner. Thus, students can understand complex problems much easily.
- Students can use these solutions for reference in case of doubt. This will help students in learning the correct method to solve a problem.
- The concepts in Class 11 Maths lay the foundation for Class 12 as well as graduate and postgraduate levels.

At Embibe, some of the top subject matter experts have prepared the class 11 Maths NCERT book solutions. You may download their PDF versions for free, from Embibe website. You may also attend live classes on the platform taught by some of the country’s leading academicians.

These NCERT Class 11 math answers have been organised chapter-wise. Experts have destructured even the most difficult issues in a rational manner to facilitate easy learning. In addition, all of the solutions in Class 11 are 100 percent accurate and follow the NCERT (CBSE Board) rules and syllabus.

With these NCERT Solutions, achieving excellent math scores is no more a pipe dream. These NCERT Class 11 Maths Solutions contain all of the tasks in the Class 11 Maths textbook as well as the current curriculum. Students’ approaches to competitive such as JEE will be refined after they start referring to the 11th Maths NCERT Solutions.

In the following sections, you will find a brief description of each chapter along with answers PDF for each. At the end of each section, you will find a link to the NCERT Solutions PDF. Simply click on the link to view and download the NCERT Solutions PDF for each chapter.

Problems and solutions in this chapter will help us know how to ascertain whether a collection is set, different types of sets, algebra of sets and power sets, how sets are represented using symbols, and Venn diagrams. You can download Sets Class 11 PDF solutions from below.

Chapter 1 of Class 11 Maths consists of the following exercises:

- 1.1 Introduction
- 1.2 Sets and their Representations
- 1.3 The Empty Set
- 1.4 Finite and Infinite Sets
- 1.5 Equal Sets
- 1.6 Subsets
- 1.7 Power Set
- 1.8 Universal Set
- 1.9 Venn Diagrams
- 1.10 Operations on Sets
- 1.11 Complement of a Set
- 1.12 Practical Problems on Union and Intersection of Two Sets

In this chapter, you will learn how to link pairs of objects from two sets and introduce relations between the two objects in the pair. You will also learn about special relations which will qualify to be functions. The concept of function is very important in mathematics since it captures the idea of a mathematically precise correspondence between one quantity with the other.

Chapter 2 of Class 11 Maths consists of the following exercises:

- 2.1 Introduction
- 2.2 Cartesian Product of Sets
- 2.3 Relations
- 2.4 Functions

In this chapter, the concepts of trigonometric ratios to trigonometric functions have been generalized and you will study their properties in a comprehensive way. Chapter 3 of Class 11 Maths consists of the following exercises:

- 3.1 Introduction
- 3.2 Angles
- 3.3 Trigonometric Functions
- 3.4 Trigonometric Functions of Sum and Difference of Two Angles

The concept pf Mathematical Induction is frequently used in mathematics and is a key aspect of scientific reasoning, where collecting and analyzing data is the norm. Thus, in simple language, you can say the word *induction *means generalization from particular cases or facts.

In algebra or in other disciplines of mathematics, there are certain results or statements that are formulated in terms of n, where n is a positive integer. To prove such statements, the well-suited principle that is used based on the specific technique is known as the principle of mathematical induction.

Chapter 4 of Class 11 Maths consists of the following exercises:

- 4.1 Introduction
- 4.2 Motivation
- 4.3 The Principle of Mathematical Induction

**Practice 11th CBSE Exam Questions**

In earlier classes, you must have studied linear equations in one and two variables and quadratic equations in one variable. You have seen that the equation x^{2} + 1 = 0 has no real solution as x^{2} + 1 = 0 gives x^{2} = – 1 and square of every real number is non-negative. So, you need to extend the real number system to a larger system so that you can find the solution of the equation x^{2} = – 1.

In fact, the main objective is to solve the equation ax^{2} + bx + c = 0, where D = b2 – 4ac < 0, which is not possible in the system of real numbers. Chapter 5 of Class 11 Maths consists of the following exercises:

**Check Out Class 11 Maths Important Formulas From Here**

- 5.1 Introduction
- 5.2 Complex Numbers
- 5.3 Algebra of Complex Numbers
- 5.4 The Modulus and the Conjugate of a Complex Number
- 5.5 Argand Plane and Polar Representation

In previous classes, you have studied equations in one variable and two variables and also solved some statement-based problems by translating them in the form of equations. Here you 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, you 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, optimization problems, economics, psychology, etc. Chapter 6 of Class 11 Maths consists of the following exercises:

- 6.1 Introduction
- 6.2 Inequalities
- 6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
- 6.4 Graphical Solution of Linear Inequalities in Two Variables
- 6.5 Solution of System of Linear Inequalities in Two Variables

Permutation and Combination are two separate ways to represent a group of elements. When the order of arrangement doesn’t matter then we call it a combination. If the order does matter then we have a permutation.

In this chapter, you shall learn some basic counting techniques which will enable you to answer questions without actually listing the arrangements. Chapter 7 of Class 11 Maths consists of the following exercises:

- 7.1 Introduction
- 7.2 Fundamental Principle of Counting
- 7.3 Permutations
- 7.3.1 Permutations when all the objects are distinct
- 7.3.2 Factorial notation
- 7.3.3 Derivation of the formula for nPr
- 7.3.4 Permutations when all the objects are not distinct objects

- 7.4 Combinations

In prior classes, you learned how to find the squares and cubes of binomials like (a + b) and (a – b). However, for higher powers like 98^{5}, 98^{6}, etc., the calculations become difficult due to repeated multiplication. This difficulty was overcome by a theorem known as the Binomial Theorem. It gives an easier way to expand (a + b)^{n}, where n is an integer or a rational number. In this chapter, you will study the Binomial Theorem for positive integral indices only. Chapter 8 of Class 11 Maths consists of the following exercises:

- 8.1 Introduction to Binomial Theorem
- 8.2 Binomial Theorem for Positive Integral Indices
- 8.2.1 Binomial theorem for any positive integer n,
- 8.2.2 Some special cases

- 8.3 General and Middle Terms

In the previous class, you have studied Arithmetic Progression (A.P). In this chapter, besides discussing A.P, we will also study Arithmetic mean, Geometric mean, the Relationship between A.M. and G.M. Special series in forms of a sum to n terms of consecutive natural numbers, sum to n terms of squares of natural numbers, and sum to n terms of cubes of natural numbers.

Chapter 9 of Class 11 Maths consists of the following exercises:

- 9.1 Introduction
- 9.2 Sequences
- 9.3 Series
- 9.4 Arithmetic Progression (A.P.)
- 9.4.1 Arithmetic mean

- 9.5 Geometric Progression (G. P.)
- 9.5.1 General term of a G.P
- 9.5.2. Sum to n terms of a G.P
- 9.5.3 Geometric Mean (G.M.)

- 9.6 Relationship Between A.M. and G.M.
- 9.7 Sum to n Terms of Special Series

You must now be familiar with two-dimensional coordinate geometry from earlier classes. Mainly, it is a combination of algebra and geometry. A systematic study of geometry by the use of algebra was first carried out by celebrated French philosopher and mathematician René Descartes, in his book ‘La Géométry, published in 1637. We will study straight lines in this chapter.

Chapter 10 of Class 11 Maths consists of the following exercises:

- 10.1 Introduction
- 10.2 Slope of a Line
- 10.2.1 Slope of a line when coordinates of any two points on the line are given
- 10.2.2 Conditions for parallelism and perpendicularity of lines in terms of their slopes
- 10.2.3 Angle between two lines
- 10.2.4 Collinearity of three points

- 10.3 Various Forms of the Equation of a Line
- 10.3.1 Horizontal and vertical lines
- 10.3.2 Point-slope form
- 10.3.3 Two-point form
- 10.3.4 Slope-intercept form
- 10.3.5 Intercept – form
- 10.3.6 Normal form

- 10.4 General Equation of a Line
- 10.4.1 Different forms of Ax + By + C = 0

- 10.5 Distance of a Point From a Line
- 10.5.1 Distance between two parallel lines

In Chapter 10, you studied various forms of the equations of a line. In this chapter, you will study some other curves, viz., circles, ellipses, parabolas, and hyperbolas. The names parabola and hyperbola are given by Apollonius. These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double-napped right circular cone.

Chapter 11 of Class 11 Maths consists of the following exercises:

**CBSE Class 11 Preparation Tips**

- 11.1 Introduction
- 11.2 Sections of a Cone
- 11.2.1 Circle, ellipse, parabola and hyperbola
- 11.2.2 Degenerated conic sections

- 11.3 Circle
- 11.4 Parabola
- 11.4.1 Standard equations of parabola
- 11.4.2 Latus rectum

- 11. 5 Ellipse
- 11.5.1 Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse
- 11.5.2 Special cases of an ellipse
- 11.5.3 Eccentricity
- 11.5.4 Standard equations of an ellipse
- 11.5.5 Latus rectum

- 11.6 Hyperbola
- 11.6.1 Eccentricity
- 11.6.2 Standard equation of Hyperbola
- 11.6.3 Latus rectum

You may recall that to locate the position of a point in a plane, you need two intersecting mutually perpendicular lines in the plane. These lines are called the coordinate axes and the two numbers are called the coordinates of the point with respect to the axes. In actual life, you do not have to deal with points lying in a plane only. For example, consider the position of a ball thrown in space at different points of time or the position of an airplane as it flies from one place to another at different times during its flight. Chapter 12 of Class 11 Maths consists of the following exercises:

- 12.1 Introduction
- 12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space
- 12.3 Coordinates of a Point in Space
- 12.4 Distance between Two Points
- 12.5 Section Formula

Download – NCERT Solutions for Class 11 Maths Chapter 12 PDF |

This chapter introduced Calculus. Calculus mainly deals with the study of change in the value of a function as the points in the domain change. First, we give an intuitive idea of derivatives (without actually defining it). Then we give a naive definition of a limit and study some algebra of limits. Then we will bring you back to the definition of derivative and study some algebra of derivatives. You will also obtain derivatives of certain standard functions. In this chapter, Limits belongs to Term I examination and Derivatives belongs to Term II examinations.

Chapter 13 of Class 11 Maths consists of the following exercises:

- 13.1 Introduction
- 13.2 Intuitive Idea of Derivatives
- 13.3 Limits
- 13.4 Limits of Trigonometric Functions
- 13.5 Derivatives
- 13.5.1 Algebra of derivatives of functions
- 13.5.2 Derivative of polynomials and trigonometric functions

Download – NCERT Solutions for Class 11 Maths Chapter 13 PDF |

In this chapter, some basic ideas of Mathematical Reasoning will be discussed. Here, the process of reasoning, especially in the context of mathematics, will be talked about. In mathematical language, there are two kinds of reasoning – inductive and deductive. The inductive reasoning in the context of mathematical induction has already been discussed. In this chapter, you shall also learn about some fundamentals of deductive reasoning. Chapter 14 of Class 11 Maths consists of the following exercises:

- 14.1 Introduction
- 14.2 Statements
- 14.3 New Statements from Old
- 14.3.1 Negation of a statement
- 14.3.2 Compound Statements

- 14.4 Special Words / Phrases
- 14.4.1 The word “And”
- 14.4.2 The word “Or”

- 14.5 Implications
- 14.5.1 Contrapositive and converse

- 14.6 Validating Statements
- 14.6.1 By contradiction

Download – NCERT Solutions for Class 11 Maths Chapter 14 PDF |

You must be knowing that statistics deals with data collected for specific purposes. You can make decisions about the data by analyzing and interpreting it. In earlier classes, you have studied methods of representing data both graphically and in tabular form. This representation reveals certain salient features or characteristics of the data. You have also studied the methods of finding a representative value for the given data. This value is called the measure of central tendency. Chapter 15 of Class 11 Maths consists of the following exercises:

- 15.1 Introduction
- 15.2 Measures of Dispersion
- 15.3 Range
- 15.4 Mean Deviation
- 15.4.1Mean deviation for ungrouped data
- 15.4.2 Mean deviation for grouped data
- 15.4.3 Limitations of mean deviation

- 15.5 Variance and standard deviation
- 15.5.1 Standard Deviation
- 15.5.2 Shortcut method to find Variance and standard deviation

- 15.6 Analysis of Frequency Distributions

Download – NCERT Solutions for Class 11 Maths Chapter 15 PDF |

In earlier classes, you studied the concept of probability as a measure of uncertainty of various phenomena. In general, to obtain the probability of an event, we find the ratio of the number of outcomes favourable to the event, to the total number of equally likely outcomes. This theory of probability is known as the classical theory of probability. Chapter 16 of Class 11 Maths consists of the following exercises:

- 16.1 Introduction
- 16.2 Random Experiments
- 16.2.1 Outcomes and sample space

- 16.3 Event
- 16.3.1 Occurrence of an event
- 16.3.2 Types of events
- 16.3.3 Algebra of events
- 16.3.4 Mutually exclusive events
- 16.3.5 Exhaustive events

- 16.4 -Axiomatic Approach to Probability
- 16.4.1- Probability of an event
- 16.4.2- Probabilities of equally likely outcomes
- 16.4.3 Probability of the event ‘A or B’
- 16.4.4 Probability of event ‘not A’

CBSE Class 11 Maths question paper will be prepared on the basis of unit-wise weightage prescribed by CBSE. As the Term I exams are approaching, students must be fully aware of the Term I unit-wise weightage. Check the below-mentioned table for CBSE Class 11 Maths Term I unit-wise weightage.

Units Name | Marks |

Unit 1: Sets and Functions | 11 |

Unit 2: Algebra | 13 |

Unit 3: Coordinate Geometry | 06 |

Unit 4: Calculus | 04 |

Unit 5: Statistics and Probability | 06 |

Internal Assessment | 10 |

Total | 50 |

** CBSE Class 11 Syllabus 2021-22**

As per the new assessment policy, the internal assessment of Class 11 Term I carries 10 marks. The details of internal assessment marks are listed as follows.

Internal Assessment | Marks |

Periodic Tests | 05 |

Mathematics Activities: Activity file record +Term end assessment of one activity & Viva | 05 |

Total Marks | 10 Marks |

Check the below-mentioned table for CBSE Class 11 Maths Term II unit-wise weightage.

Unit Name | Marks |

Unit 1: Sets and Functions (Cont.) | 08 |

Unit 2: Algebra (Cont.) | 11 |

Unit 3: Coordinate Geometry (Cont.) | 09 |

Unit 4: Calculus (Cont.) | 06 |

Unit 5: Statistics and Probability (Cont.) | 06 |

Internal Assessment | 10 |

Total | 50 |

As per the new assessment policy, the internal assessment of Class 11 Term II carries 10 marks. The details of internal assessment marks are listed as follows.

Internal Assessment | Marks |

Periodic Tests | 05 |

Mathematics Activities: Activity file record +Term end assessment of one activity & Viva | 05 |

Total Marks | 10 Marks |

The transition from Class 10 to Class 11 is huge and requires smart and hard work by the students. To get better at the subject and score well, we advise students to follow the below-mentioned tips:

- Students often avoid the theory part of Maths. Do not do that as the questions asked largely depend on your understanding. Once you get how a formula is derived you will also get the solutions for the question.
- Advance at your own pace. Competition with your friends is good but don’t skip any chapter or concept.
- Preparing a
*formula sheet*is a must. - Stick to the
**NCERT Book**and once you have completely solved the book then move on to reference books. - Use the NCERT solutions provided by us as reference material while solving questions.
- The last and most important point is
**Practice**. Practice makes a man perfect and maths requires lots and lots of practice.

**Attempt 11th CBSE Exam Mock Tests**

Below we have provided some important questions that will aid you in scoring better:

Q1. If P (A) = P (B), Show that A = B |

Q2. Draw the graph of the Constant function, f: R ϵ R; f(x) = 2 x ϵ R. Also find its domain and range. |

Q3. The minute hand of the watch is 1.5 cm long. How far does it tip to move in 40 minutes? |

Q4. Prove that n(n+1)(n+5) is a multiple of 3. |

Q5. Find real θ such that \(\frac{3+2iSin\theta}{1-2iSin\theta}\) is purely real. |

Q6. Find all pairs of consecutive odd natural no. both of which are larger than 10 such that their sum is less than 40. |

Q7. A bag contains 5 black and 6 red balls, determine the number of ways in which 2 black and 3 red balls can be selected. |

Q8. Using the binomial theorem, prove that \(6^{a}-5n\) always leaves remainder 1 when divided by 25 |

Q9. Show that the sum of (m+n)th and (m-n)th terms of an A. P. is equal to twice the mth term |

Q10. Find the equation to the straight line which passes through the point (3,4) and has intercept on the axes equal in magnitude but opposite in sign. |

Q11. Find the equation of hyperbola whose length of latus rectum is 36 & foci are \((0,\mp12)\) |

Q12. Find the coordinate of a point equidistant from the four points O(0,0,0), A (a,0,0), B(0,b,0), and C(0,0,c) |

Q13. Differentiate the function \(y=\frac{(x+2)(3x-1)}{(2x+5)}\) with respect to x |

Q14. Give three examples of sentences that are not statements. Give reasons for the answers. |

Q15. The mean of 2,7,4,6,8 and p is 7. Find the mean deviation about the median of these observations. |

Q16. Find the prob. that when a hand of 7 cards is drawn from a well-shuffled deck of 52 cards, it contains(i) all king (ii) 3 kings (iii) at least 3 kings |

Below are some FAQs by students on Class 11 Maths:

**Question 1: How to get ***NCERT Solutions for ***Class 11 Maths Trigonometry?**

Ans: In this article, we have provided the chapter-wise solutions for Class 11 Maths. You can download Trigonometry solutions by clicking on the link.

*Question 2: How can I learn Maths in Class 11?*

Ans: You can check out the NCERT solutions in this article. These will aid you in the subject. To study Maths, you must start from Chapter 1 and then proceed slowly as per your pace. We have provided the solutions to every chapter of the Class 11 Maths textbook so that students can use them as a guide as well as a study material.

**Question 3: Which concepts are important in Class 11 Maths? **

Ans: All concepts covered in the Class 11 NCERT Maths book are important. These concepts are not only important for Class 11 exams but for higher studies as well.

*Question 4: How many chapters are there in the Class 11 NCERT Maths Book?*

Ans: There are 16 total chapters in Maths Class 11. The first chapter is Sets and the last chapter is Probability. In this article, we have provided solutions to all the chapters

*Question 5*: *Is Class 11 Maths difficult to understand?*

Ans: Many students find Class 11 Maths to be difficult but that is not really the case. The most important thing to remember while practising Maths is to understand the concepts and practice by writing. If you are practising the sums and problems regularly, you will not find Class 11 Maths difficult.

*Question 6: Which are the best books for preparing Class 11 Maths?*

Ans: The textbooks prescribed by CBSE i.e NCERT Class 11 Maths are considered the best books. They cover all the chapters and clear all the concepts.

*Question 7: Is Class 11 Maths difficult? *

Ans: If we consider the increment in the syllabus from Class 10 to Class 11 then yes. But if you ask in general, it depends on the learning capability of the student. Students are also able to score full marks in the subject with constant practice. NCERT solutions surely help in the process.

**Question 8****:**** Which is the best textbook to study CBSE Class 11 Maths?**

Ans: According to experts, the NCERT textbook for Class Maths should be the book of choice for students. You should check other reference books only when you have completed the NCERT textbook thoroughly.

*Question 9: From where can I download the NCERT*

Ans: Students looking for Class 11 Maths Solutions as PDF can directly download them from this article.

**Download NCERT Solutions For Class 11**

Now that you have a detailed NCERT Guide for Class 11 Maths in PDF for all chapters, we hope you make the best use of it. All the questions have been solved by the best and top academicians at Embibe

*If you have any queries regarding this article, let us know about it in the comment section below and we will get back to you soon.* *Stay tuned to Embibe for more on CBSE NCERT Solutions For Class 11.*