# NCERT Exemplar Class 11 Maths Chapter 4 Solutions: Principle of Mathematical Induction

NCERT Exemplar Solutions, for Class 11 Maths Chapter 4, provided by Instasolv aim to provide you with detailed topic-wise solutions to understand the concepts easily. These NCERT Exemplar Class 11 Maths will not only help you score good marks in your academics exam as well in the entrance exams like JEE Main, JEE Advanced, and Engineering Entrance Exams.

The main topics of the NCERT Exemplar Class 11 Maths Solutions Chapter 4 are Mathematical Induction, Properties of Mathematical Induction, Inductive Hypothesis, and Principle of Mathematical Induction. Practising this chapter will make you understand about deductive reasoning. Also, you will learn how to verify a formula and how an equation is derived.

NCERT Exemplar Chapter 4 – Principle of Mathematical Induction has 4 main exercises with a total of 45 questions. All the exercises consist of different types of questions including the short answer type, long answer type, fill in the blanks, true or false statement, and objective type questions. All the questions are explained well with the help of examples to help you clear your concepts.

**Important Topics for NCERT Maths Exemplar Solutions Class 11 Chapter 4 – Principle of Mathematical Induction**

**What is Mathematical Induction?**

Mathematical induction is a dedicated form of working on various cases and coming up with ideas. Induction is the collection from a specific set of facts. This technique is used to define a wide range of reports in which we examine the accuracy of the case. The set should be numerable in order for mathematical induction to work with an infinite set, signifying that it should be having a one-to-one correspondence between the characters of the set in question and the set of positive integers.

In general, this procedure states that the set in the form an implied list of distinct elements such as {1, 2, 3, 4, 5, 6, 7,…}.

**Properties of Mathematical Induction**

It is essential for the Mathematical induction to follow the statements with respect to the properties they act upon:

- When for the value of n is true for statements such as x ≥ 7, we should start with sustaining the value x=7 i.e. Q(7)
- If the statement is given is true for x=k, and if it sustains the value x=k then it will also fulfil x=k+1. In order to prove the reality of the statement, we have to prove x=k+1.

**Inductive Hypothesis**

The step that is provided above is the statement of the reality of the statement x=k and is mentioned as the inductive step or inductive hypothesis. For example, we follow the pattern:

\(1=1^{2}=1\) \(4=2^{2}=1+3\) \(9=3^{2}=1+3+5\) \(16=4^{2}=1+3+5+7\)

Here, we can observe that the sum of the first two odd natural numbers is square of the second number which in the result is also a natural number and the pattern keeps going on.

P(x)=\(1+3+5+7+11..+(2n-1)=n^{2}\)

And P(1) is fulfilled, then, it is the first step and the value will fulfil for all natural numbers.

### Exercise-wise Discussion of NCERT Maths Exemplar Solutions Class 11 Chapter 4 – Principle of Mathematical Induction

Chapter 4 – Principle of Mathematical Induction is an important chapter that deals with the concepts of deductive reasoning and inductive reasoning which are a key aspect of scientific reasoning. It has 4 main exercises with a total of 45 questions. The detailed description of these exercises is provided below:

**Exercise 4.1 – Solved Examples**

This is the first exercise of NCERT Exemplar Class 11 Maths Solutions for Chapter 4. It contains 15 solved examples for your reference. Practising these questions will help you clear your concepts to solve further exercises.

**Exercise 4.2 – Short Answer Type Questions**

Exercise 4.2 of Chapter 4 – Principle of Mathematical Induction consists of a total of 16 short answer type questions. These questions are based on the topics deduction, induction, and principle of induction.

**Exercise 4.3 – Long Answer Type Questions**

This exercise consists of a total of 9 questions based on the properties of mathematical induction. They are long answer type questions where you will find detailed solutions to the problems.

**Exercise 4.4 – Objective Type Questions**

It is the last exercise of the chapter and is considered a very important exercise from the entrance exam point of view. It contains a total of 5 questions including fill in the blanks, state true or false, or multiple-choice questions. Solving the questions of this exercise will help you understand how clear your concepts are.

## Why Use NCERT Maths Exemplar Solutions Class 11 Chapter 4 – Principle of Mathematical Induction by Instasolv?

- NCERT Exemplar Class 11 for Maths Solutions for Chapter 4 will not only help you resolve your doubts but also make you understand even the most difficult topics of the chapter.
- The solutions are developed in sync with the understanding level of the students, which assists them to practice the topics regularly.
- Each topic within this Chapter 4 – Principle of Mathematical Induction is described step-by-step to make learning simpler and quicker.

These solutions are prepared by the subject matter experts at Instasolv who are well-versed with class 11 maths CBSE and NCERT syllabus.