# NCERT Solutions for Class 11 Maths Chapter 14

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 Class: 11 Subject: Maths Chapter 14: Mathematical Reasoning

## NCERT Solutions for Class 11 Maths Chapter 14

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### Class 11 Maths Chapter 14 Mathematical Reasoning Sols

#### 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
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.