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.

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