## NCERT Solutions for Class 11 Maths Chapter 14 – Mathematical Reasoning Miscellaneous Exercise

#### Page No 345:

#### Question 1:

Write the negation of the following statements:

(i)

*p*: For every positive real number*x*, the number*x*– 1 is also positive.
(ii)

*q*: All cats scratch.
(iii)

*r*: For every real number*x*, either*x*> 1 or*x*< 1.
(iv)

*s*: There exists a number*x*such that 0 <*x*< 1.#### Answer:

(i) The negation of statement

*p*is as follows.
There exists a positive real number

*x*, such that*x*– 1 is not positive.
(ii) The negation of statement

*q*is as follows.
There exists a cat that does not scratch.

(iii) The negation of statement

*r*is as follows.
There exists a real number

*x*, such that neither*x*> 1 nor*x*< 1.
(iv) The negation of statement

*s*is as follows.
There does not exist a number

*x*, such that 0 <*x*< 1.#### Question 2:

State the converse and contrapositive of each of the following statements:

(i)

*p*: A positive integer is prime only if it has no divisors other than 1 and itself.
(ii)

*q*: I go to a beach whenever it is a sunny day.
(iii)

*r*: If it is hot outside, then you feel thirsty.#### Answer:

(i) Statement

*p*can be written as follows.
If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is as follows.

If a positive integer has no divisors other than 1 and itself, then it is prime.

The contrapositive of the statement is as follows.

If positive integer has divisors other than 1 and itself, then it is not prime.

(ii) The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is as follows.

If I go to a beach, then it is a sunny day.

The contrapositive of the statement is as follows.

If I do not go to a beach, then it is not a sunny day.

(iii) The converse of statement

*r*is as follows.
If you feel thirsty, then it is hot outside.

The contrapositive of statement

*r*is as follows.
If you do not feel thirsty, then it is not hot outside.

#### Question 3:

Write each of the statements in the form “if

*p*, then*q*”.
(i)

*p*: It is necessary to have a password to log on to the server.
(ii)

*q*: There is traffic jam whenever it rains.
(iii)

*r*: You can access the website only if you pay a subscription fee.#### Answer:

(i) Statement

*p*can be written as follows.
If you log on to the server, then you have a password.

(ii) Statement

*q*can be written as follows.
If it rains, then there is a traffic jam.

(iii) Statement

*r*can be written as follows.
If you can access the website, then you pay a subscription fee.

#### Question 4:

Re write each of the following statements in the form “

*p*if and only if*q*”.
(i)

*p*: If you watch television, then your mind is free and if your mind is free, then you watch television.
(ii)

*q*: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.
(iii)

*r*: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.#### Answer:

(i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

#### Question 5:

Given below are two statements

*p*:

*25 is a multiple of 5.*

*q: 25 is a multiple of 8.*

Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

#### Answer:

The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.

This is a false statement, since 25 is not a multiple of 8.

The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”.

This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.

#### Question 6:

Check the validity of the statements given below by the method given against it.

(i)

*p*: The sum of an irrational number and a rational number is irrational (by contradiction method).
(ii)

*q*: If*n*is a real number with*n*> 3, then*n*^{2}> 9 (by contradiction method).#### Answer:

(i) The given statement is as follows.

*p*: the sum of an irrational number and a rational number is irrational.
Let us assume that the given statement,

*p*, is false. That is, we assume that the sum of an irrational number and a rational number is rational.
Therefore, , whereis irrational and

*b*,*c*,*d*,*e*are integers.
⇒ de – bc = aBut here, is a rational number andis an irrational number.

This is a contradiction. Therefore, our assumption is wrong.

Therefore, the sum of an irrational number and a rational number is rational.

Thus, the given statement is true.

(ii) The given statement,

*q*, is as follows.
If

*n*is a real number with*n*> 3, then*n*^{2}> 9.
Let us assume that

*n*is a real number with*n*> 3, but*n*^{2}> 9 is not true.
That is,

*n*^{2}< 9
Then,

*n*> 3 and*n*is a real number.
Squaring both the sides, we obtain

*n*

^{2}> (3)

^{2}

⇒

*n*^{2}> 9, which is a contradiction, since we have assumed that*n*^{2}< 9.
Thus, the given statement is true. That is, if

*n*is a real number with*n*> 3, then*n*^{2}> 9.#### Question 7:

Write the following statement in five different ways, conveying the same meaning.

*p: If triangle is equiangular, then it is an obtuse angled triangle.*

#### Answer:

The given statement can be written in five different ways as follows.

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.

(ii) A triangle is equiangular only if it is an obtuse-angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle.

(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.

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