# NCERT Class 11-Math՚S: Chapter – 14 Mathematical Reasoning Part 6 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 8**:

Rewrite each of the following statements in the form of conditional statements:

(i) Mohan will be a good student if he studies hard.

(ii) Ramesh will get dessert only if he eats his dinner.

(iii) When you sing, my ears hurt.

(iv) A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder.

(v) A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan.

**Answer**:

(i) The given statement is of the form “*q* if *p*” , where

Mohan studies hard.

He will be a good student.

It is an equivalent form (Remark (b) 14.1. 12) of the statement “if *p* then *q*” . So the equivalent formulation of the given statement is

“If Mohan studies hard, then he will be a good student” .

(Here, note that in *p* he is replaced by Mohan and in *q* Mohan is replaced by he)

(ii) The given statement is of the form

“ only if ” which is an equivalent form (Remark (c) 14.1. 12) of the statement “if then ” . So, the equivalent formulation of the given statement is:

“If Ramesh eats his dinner, then he will get dessert”

(iii) Here ‘when’ means the same as ‘if’ and so the equivalent formulation of the given statements is:

“If you sing, then my ears hurt”

(iv) The given statement is of the form “ is necessary for ” where

: Indian team wins a cricket match

: The selection committee selects an all-rounder

Which is an equivalent form (Remark (e) 14.1. 12) of “if *p* then *q*” . So the equivalent formulation of the given statement is

″ If the teams-wins a cricket match then selection committee selects an all-rounder.

(v) The given statement is of the form “*p* is sufficient for *q*” where

: Tara goes to Rashtrapati Bhawan

: She visits New Delhi

Which is an equivalent form (Remark (d) 14.1. 12) of “if , then ” , so the equivalent formulation of the given statement is

“If Tara goes to Rashtrapati Bhawan, then she visits New Delhi” .

**Question 9**:

Express in English, the statement , where

It is raining today

**Answer**:

The conditional statement is

“If it is raining today, then ” .

**Question 10**:

Translate the following statements in symbolic form:

If and ″ then .

**Answer**:

Let and and

Then the given statement is symbolic form is

**Question 11**:

Form the biconditional of the following statements:

Today is 14^{th} of August

Tomorrow is Independence day

**Answer**:

The biconditional is given by

“Today is 14^{th} of August if and only if tomorrow is Independence Day” .

**Question 12**:

Translate the following biconditional into symbolic form:

“ABC is an equilateral triangle if and only if its each interior angle is ”

**Answer**:

Let ABC is an equilateral triangle

And Each interior angle of triangle ABC is

Then the given statement in symbolic form is .

**Question 13**:

Identify the quantifiers and write the negation of the following statements

(i) There exists a number which is equal to its square.

(ii) For all even integers , is also even.

(iii) There exists a number which is a multiple of and .

**Answer**:

(i) The quantifier is “there exists” and the negation is

“There does not exist a number which is equal to its square”

(ii) The quantifier is “for all” and the negation is

“There exists an even integer *x* such that is not even”

(iii) The quantifier is “there exists” and the negation is

“There does not exist a number which is a multiple of both and ” .