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Class: | 11 |

Subject: | Maths |

Chapter 14: | Mathematical Reasoning |

Table of Contents

- 1 NCERT Solutions for Class 11 Maths Chapter 14
- 1.1 Class 11 Maths Chapter 14 Mathematical Reasoning Sols
- 1.1.1 Important Terms on Mathematical Reasoning
- 1.1.2 Give three examples of sentences which are not statements. Give reasons for the answers.
- 1.1.3 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.
- 1.1.4 Identify the quantifier in the following statement and write the negation of the statements. There exists a number which is equal to its square.
- 1.1.5 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.

- 1.1 Class 11 Maths Chapter 14 Mathematical Reasoning Sols

## NCERT Solutions for Class 11 Maths Chapter 14

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### Class 11 Maths Chapter 14 Mathematical Reasoning Sols

**Download Exercise 14.1****Download Exercise 14.2****Download Exercise 14.3****Download Exercise 14.4****Download Exercise 14.5****Download Miscellaneous Exercise 14****NCERT Books for Class 11****Revision Books for Class 11**- Hindi Medium Solutions will be uploaded very soon.

#### Important Terms on Mathematical Reasoning

- A sentence is called a
*statement*if it is either true or false but not both simultaneously. - The denial of a statement p is called its negation and is written as ~p and read as not p.
- Compound statement is made up of two or more simple statements. These simple statements are called
*component statements*. - ‘And’, ‘or’, ‘If–then, ‘only if’, ‘If and only if’ etc. are connecting words, which are used to form a compound statement.
- 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.

- 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. - 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’.
- 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’.

##### Important Extra Questions on Mathematical Reasoning

- Verify by the method of contradiction that 7 is irrational.
- By giving counter example, show that the following statement is false: ‘If n is an odd integer, then n is prime’.
- 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’.
- Prove by direct method that for any integer ‘n’, n³- n is always even’.

Table of Contents

- 1 Give three examples of sentences which are not statements. Give reasons for the answers.
- 2 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.
- 3 Identify the quantifier in the following statement and write the negation of the statements. There exists a number which is equal to its square.
- 4 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.

#### 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.

(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.

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.

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.

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.