NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning free in PDF file format updated for new academic session 2022-23 following latest CBSE Syllabus 2022-2023. Offline Apps of NCERT Solutions 2022-2023 for all subjects are available all to download free from App Store or Play Store. These solutions are based on updated NCERT Books for 2022-23 for intermediate students of CBSE, MP Board as well as UP Board who are following NCERT Books for session 2022-2023.

## NCERT Solutions for Class 11 Maths Chapter 14

• ### NCERT Solutions for Class 11 Maths Chapter 14 in PDF

 Class: 11 Subject: Maths Chapter 14: Mathematical Reasoning

### 11th Maths Chapter 14 Solutions

#### Important Terms on Mathematical Reasoning

1. A sentence is called a statement if it is either true or false but not both simultaneously.
2. The denial of a statement p is called its negation and is written as ~p and read as not p.
3. Compound statement is made up of two or more simple statements. These simple statements are called component statements.

4. ‘And’, ‘or’, ‘If–then, ‘only if’, ‘If and only if’ etc. are connecting words, which are used to form a compound statement.
5. Two simple statements p and q connected by the word ‘and’ namely ‘p and q’ is called a conjunction of p and q and is written as p^q.

1. Two simple statements p and q connected by the word ‘or’ the resulting compound statement ‘p or q’ is called disjunction of p and q and is written as pvq.
2. If in a compound statement containing the connective ‘or’ all the alternatives cannot occur simultaneously, then the connecting word ‘or’ is called as exclusive ‘or’.
3. If, in a compound statement containing the connective ‘or’, all the alternative can occur simultaneously, then the connecting word ‘or’ is called as inclusive ‘or’.

###### Questions from Exam Papers

1. Verify by the method of contradiction that 7 is irrational.
2. By giving counter example, show that the following statement is false: ‘If n is an odd integer, then n is prime’.
3. Show that the following statement is true by method of contra positive: ‘If x is an integer and x² is even, then x is also even’.
4. Prove by direct method that for any integer ‘n’, n³- n is always even’.

### Give three examples of sentences which are not statements. Give reasons for the answers.

The three examples of sentences, which are not statements, are as follows.
(i) He is a doctor. It is not evident from the sentence as to whom ‘he’ is referred to. Therefore, it is not a statement.
(ii) Geometry is difficult. This is not a statement because for some people, geometry can be easy and for some others, it can be difficult.
(iii) Where is she going? This is a question, which also contains ‘she’, and it is not evident as to who ‘she’ is. Hence, it is not a statement.

### Find the component statements of the following compound statements and check whether they are true or false: Number 3 is prime or it is odd.

The component statements are as follows. p: Number 3 is prime. q: Number 3 is odd. Both the statements are true.

### Identify the quantifier in the following statement and write the negation of the statements. There exists a number which is equal to its square.

The quantifier is “There exists”. The negation of this statement is as follows. There does not exist a number which is equal to its square.

### Show that the statement “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.

The given statement can be written in the form of “if-then” as follows.
If a and b are real numbers such that a² = b², then a = b.
Let p: a and b are real numbers such that a² = b². q: a = b
The given statement has to be proved false.
For this purpose, it has to be proved that if p, then ∼q.
To show this, two real numbers, a and b, with a² = b² are required such that a ≠ b.
Let a = 1 and b = –1
a² = (1)² = 1 and b² = (– 1)² = 1
∴ a² = b²
However, a ≠ b
Thus, it can be concluded that the given statement is false.

### What are the main topics to study in chapter 14 of class 11th Maths?

In chapter 14 of class 11th Maths, students will discuss about some basic ideas of Mathematical reasoning and some fundamental of deductive reasoning. Topics under chapter 14 of grade 11th Maths are:

1. Statements
2. New statements from old
3. Negation of a statement
4. Compound statements
5. Special words/phrases
6. The word “And”
7. The word “Or”
8. Quantifiers
9. Implications
10. Contrapositive and converse
11. Validating statements

### Which problems of chapter 14 of class 11th Maths are compulsory to do?

Chapter 14 of class 11th Maths is important. This chapter is useful in higher mathematics studies also. All questions and examples of this chapter are important. But some problems of this chapter are most important and are compulsory to do. These problems are question 1 of exercise 14.1, questions 1, 2, 3 of exercise 14.2, questions 1, 2 of exercise 14.3, questions 2, 3, 4 of exercise 14.4, questions 1, 3, 4, 5 of exercise 14.5, questions 2, 3, 4, 5, 6, 7 of miscellaneous exercise on chapter 14 and examples 1, 5, 7, 8, 13, 14, 18, 19, 20.

### Is there any need to practice extra questions of chapter 14 of class 11th Maths?

There is no need to practice extra questions of chapter 14 (Mathematical Reasoning) of class 11th Maths. NCERT’s questions and examples are sufficient. But, if someone wants to practice extra questions of chapter 14, they can practice extra questions from R.L. Arora, R.S. Aggarwal, R.D. Sharma, Previous year’s question paper, NCERT Exemplar, etc.

### Is chapter 14 of class 11th Math easy or complicated?

Chapter 14 of class 11th mathematics is not easy and not complicated. It is just a normal chapter because some problems of this chapter are easy whereas some are little bit difficult. But difficulty level of any chapter varies from student to student. So, Chapter 14 of class 11th mathematics is easy or not depends on students also. Some students find it hard, some find it simple, and some find it in the middle of simple and hard.

### Does the chapter 14 of class 11 Maths take more time to learn properly?

No, Chapter 14 of class 11th Maths is not lengthy. Chapter 14 of grade 11th Maths has 25 questions and 20 illustrations. Students need a maximum of 8-10 days to finish chapter 14 of grade 11th Maths if they give 1-2 hours per day to this chapter with complete concentration. This time can vary because no students have the same working speed.