# RD Sharma solutions for Class 11 Mathematics Textbook chapter 31 - Mathematical reasoning [Latest edition]

## Chapter 31: Mathematical reasoning

Exercise 31.1Exercise 31.2Exercise 31.3Exercise 31.4Exercise 31.5Exercise 31.6
Exercise 31.1 [Page 3]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.1 [Page 3]

Exercise 31.1 | Q 1.01 | Page 3

Find out the following  sentence is a statement and is not. Justify your answer.

Listen to me, Ravi !

Exercise 31.1 | Q 1.02 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Every set is a finite set.

Exercise 31.1 | Q 1.03 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Two non-empty sets have always a non-empty intersection.

Exercise 31.1 | Q 1.04 | Page 3

Find out the sentence are statement and are not. Justify your answer.

The cat pussy is black.

Exercise 31.1 | Q 1.05 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Are all circles round?

Exercise 31.1 | Q 1.06 | Page 3

Find out the sentence are statement and are not. Justify your answer.

All triangles have three sides.

Exercise 31.1 | Q 1.07 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Every rhombus is a square.

Exercise 31.1 | Q 1.08 | Page 3

Find out the sentence are statement and are not. Justify your answer.

x2 + 5 | x | + 6 = 0 has no real roots.

Exercise 31.1 | Q 1.09 | Page 3

Find out the sentence are statement and are not. Justify your answer.

This sentence is a statement.

Exercise 31.1 | Q 1.1 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Is the earth round?

Exercise 31.1 | Q 1.11 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Go !

Exercise 31.1 | Q 1.12 | Page 3

Find out the sentence are statement and are not. Justify your answer.

The real number x is less than 2.

Exercise 31.1 | Q 1.13 | Page 3

Find out the sentence are statement and are not. Justify your answer.

There are 35 days in a month.

Exercise 31.1 | Q 1.14 | Page 3

Find out the sentence are statement and are not. Justify your answer.

Mathematics is difficult.

Exercise 31.1 | Q 1.15 | Page 3

Find out the sentence are statement and are not. Justify your answer.

All real numbers are complex numbers.

Exercise 31.1 | Q 1.16 | Page 3

Find out the sentence are statement and are not. Justify your answer.

The product of (−1) and 8 is 8.

Exercise 31.1 | Q 2 | Page 3

Give three examples of sentences which are not statements. Give reasons for the answers.

Exercise 31.2 [Pages 6 - 7]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.2 [Pages 6 - 7]

Exercise 31.2 | Q 1.1 | Page 6

Write the negation of the statement:
Banglore is the capital of Karnataka.

Exercise 31.2 | Q 1.2 | Page 6

Write the negation of the statement:

It rained on July 4, 2005.

Exercise 31.2 | Q 1.3 | Page 6

Write the negation of the statement:

Ravish is honest.

Exercise 31.2 | Q 1.4 | Page 6

Write the negation of the statement:

The earth is round.

Exercise 31.2 | Q 1.5 | Page 6

Write the negation of the statement:

The sun is cold.

Exercise 31.2 | Q 2.1 | Page 7

All birds sing.

Exercise 31.2 | Q 2.2 | Page 7

Some even integers are prime.

Exercise 31.2 | Q 2.3 | Page 7

There is a complex number which is not a real number.

Exercise 31.2 | Q 2.4 | Page 7

I will not go to school.

Exercise 31.2 | Q 2.5 | Page 7

Both the diagonals of a rectangle have the same length.

Exercise 31.2 | Q 2.6 | Page 7

All policemen are thieves.

Exercise 31.2 | Q 3.1 | Page 7

Are the pair of statement are negation of each other:
The number is not a rational number.
The number x is not an irrational number.

Exercise 31.2 | Q 3.2 | Page 7

Are the pair of statement are negation of each other:

The number x is not a rational number.
The number is an irrational number.

Exercise 31.2 | Q 4.1 | Page 7

Write the negation of the statement:

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

Exercise 31.2 | Q 4.2 | Page 7

Write the negation of the statement:

q : For every real number x, either x > 1 or x < 1.

Exercise 31.2 | Q 4.3 | Page 7

Write the negation of the statement:

r : There exists a number x such that 0 < x < 1.

Exercise 31.2 | Q 5 | Page 7

Check whether the following pair of statements are negation of each other. Give reasons for your answer.
a + b = b + a is true for every real number a and b.
There exist real numbers a and b for which a + b = b + a.

Exercise 31.3 [Page 14]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.3 [Page 14]

Exercise 31.3 | Q 1.1 | Page 14

Find the component statement of the compound statement:
The sky is blue and the grass is green.

Exercise 31.3 | Q 1.2 | Page 14

Find the component statement of the compound statement:

The earth is round or the sun is cold.

Exercise 31.3 | Q 1.3 | Page 14

Find the component statement of the compound statement:

All rational numbers are real and all real numbers are complex.

Exercise 31.3 | Q 1.4 | Page 14

Find the component statement of the compound statement:

25 is a multiple of 5 and 8.

Exercise 31.3 | Q 2.1 | Page 14

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

Students can take Hindi or Sanskrit as their third language.

Exercise 31.3 | Q 2.2 | Page 14

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

To entry a country, you need a passport or a voter registration card.

Exercise 31.3 | Q 2.3 | Page 14

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

A lady gives birth to a baby boy or a baby girl.

Exercise 31.3 | Q 2.4 | Page 14

For statement, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer.

To apply for a driving licence, you should have a ration card or a passport.

Exercise 31.3 | Q 3.1 | Page 14

Write the component statement of the compound statement and check whether the compound statement is true or false:

To enter into a public library children need an identity card from the school or a letter from the school authorities.

Exercise 31.3 | Q 3.2 | Page 14

Write the component statement of the compound statement and check whether the compound statement is true or false:

All rational numbers are real and all real numbers are not complex.

Exercise 31.3 | Q 3.3 | Page 14

Write the component statement of the compound statement and check whether the compound statement is true or false:

Square of an integer is positive or negative.i

Exercise 31.3 | Q 3.4 | Page 14

Write the component statement of the compound statement and check whether the compound statement is true or false:

x = 2 and x = 3 are the roots or the equation 3x2 − x − 10 = 0.

Exercise 31.3 | Q 3.5 | Page 14

Write the component statement of the compound statement and check whether the compound statement is true or false:

The sand heats up quickly in the sun and does not cool down fast at night.

Exercise 31.3 | Q 4.1 | Page 14

Determine whether the compound statement are true or false:

Delhi is in India and 2 + 2 = 4.

Exercise 31.3 | Q 4.2 | Page 14

Determine whether the compound statement are true or false:

Delhi is in England and 2 + 2 = 4.

Exercise 31.3 | Q 4.3 | Page 14

Determine whether the compound statement are true or false:

Delhi is in India and 2 + 2 = 5.

Exercise 31.3 | Q 4.4 | Page 14

Determine whether the compound statement are true or false:

Delhi is in England and 2 + 2 =5.

Exercise 31.4 [Page 16]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.4 [Page 16]

Exercise 31.4 | Q 1.1 | Page 16

Write the negation of  statement:

For every x ϵ Nx + 3 < 10

Exercise 31.4 | Q 1.2 | Page 16

Write the negation of  statement:

There exists x ϵ Nx + 3 = 10

Exercise 31.4 | Q 2.1 | Page 16

Negate  statement:
All the students completed their homework.

Exercise 31.4 | Q 2.2 | Page 16

Negate of the  statement :

There exists a number which is equal to its square.

Exercise 31.5 [Page 21]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.5 [Page 21]

Exercise 31.5 | Q 1.1 | Page 21

Write of the statement in the form "if p, then q".

You can access the website only if you pay a subscription fee.

Exercise 31.5 | Q 1.2 | Page 21

Write of the statement in the form "if p, then q".

There is traffic jam whenever it rains.

Exercise 31.5 | Q 1.3 | Page 21

Write of the statement in the form "if p, then q".

It is necessary to have a passport to log on to the server.

Exercise 31.5 | Q 1.4 | Page 21

Write of the statement in the form "if p, then q".

It is necessary to be rich in order to be happy.

Exercise 31.5 | Q 1.5 | Page 21

Write of the statement in the form "if p, then q".

The game is cancelled only if it is raining.

Exercise 31.5 | Q 1.6 | Page 21

Write of the statement in the form "if p, then q".

It rains only if it is cold.

Exercise 31.5 | Q 1.7 | Page 21

Write of the statement in the form "if p, then q".

Whenever it rains it is cold.

Exercise 31.5 | Q 1.8 | Page 21

Write of the statement in the form "if p, then q".

It never rains when it is cold.

Exercise 31.5 | Q 2.1 | Page 21

State the converse and contrapositive of  statement:

If it is hot outside, then you feel thirsty.

Exercise 31.5 | Q 2.2 | Page 21

State the converse and contrapositive of  statement:

I go to a beach whenever it is a sunny day.

Exercise 31.5 | Q 2.3 | Page 21

State the converse and contrapositive of  statement:

A positive integer is prime only if it has no divisors other than 1 and itself.

Exercise 31.5 | Q 2.4 | Page 21

State the converse and contrapositive of  statement:

If you live in Delhi, then you have winter clothes.

Exercise 31.5 | Q 2.5 | Page 21

State the converse and contrapositive of  statement:

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Exercise 31.5 | Q 3.1 | Page 21

Rewrite of the  statement in the form "p if and only if q".

p : If you watch television, then your mind is free and if your mind is free, then you watch television.

Exercise 31.5 | Q 3.2 | Page 21

Rewrite of the  statement in the form "p if and only if q".

q : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Exercise 31.5 | Q 3.3 | Page 21

Rewrite of the  statement in the form "p if and only if q".

r : For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

Exercise 31.5 | Q 3.4 | Page 21

Rewrite of the  statement in the form "p if and only if q".

s : If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

Exercise 31.5 | Q 4.01 | Page 21

Determine the contrapositive of the statement:

If Mohan is a poet, then he is poor.

Exercise 31.5 | Q 4.02 | Page 21

Determine the contrapositive of the statement:

Only if Max studies will he pass the test.

Exercise 31.5 | Q 4.03 | Page 21

Determine the contrapositive of the statement:

If she works, she will earn money.

Exercise 31.5 | Q 4.04 | Page 21

Determine the contrapositive of the statement:

If it snows, then they do not drive the car.

Exercise 31.5 | Q 4.05 | Page 21

Determine the contrapositive of the statement:

It never rains when it is cold.

Exercise 31.5 | Q 4.06 | Page 21

Determine the contrapositive of the statement:

If Ravish skis, then it snowed.

Exercise 31.5 | Q 4.07 | Page 21

Determine the contrapositive of the statement:

If x is less than zero, then x is not positive.

Exercise 31.5 | Q 4.08 | Page 21

Determine the contrapositive of the statement:

If he has courage he will win.

Exercise 31.5 | Q 4.09 | Page 21

Determine the contrapositive of the statement:

It is necessary to be strong in order to be a sailor.

Exercise 31.5 | Q 4.1 | Page 21

Determine the contrapositive of the statement:

Only if he does not tire will he win.

Exercise 31.5 | Q 4.11 | Page 21

Determine the contrapositive of the statement:

If x is an integer and x2 is odd, then x is odd.

Exercise 31.6 [Pages 28 - 29]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 31 Mathematical reasoning Exercise 31.6 [Pages 28 - 29]

Exercise 31.6 | Q 1.1 | Page 28

Check the validity of the statement:

p : 100 is a multiple of 4 and 5.

Exercise 31.6 | Q 1.2 | Page 28

Check the validity of the statement:

q : 125 is a multiple of 5 and 7.

Exercise 31.6 | Q 1.3 | Page 28

Check the validity of the statement:

r : 60 is a multiple of 3 or 5.

Exercise 31.6 | Q 2.1 | Page 28

Check whether the statement are true or not:

p : If x and y are odd integers, then x + y is an even integer.

Exercise 31.6 | Q 2.2 | Page 28

Check whether the statement are true or not:

q : If xy are integers such that xy is even, then at least one of x and y is an even integer.

Exercise 31.6 | Q 3 | Page 29

Show that the statement
p : "If x is a real number such that x3 + x = 0, then x is 0"
is true by
(i) direct method
(ii) method of contrapositive

Exercise 31.6 | Q 4 | Page 29

Show that the following statement is true by the method of contrapositive
p : "If x is an integer and x2 is odd, then x is also odd"

Exercise 31.6 | Q 5 | Page 29

Show that the following statement is true
"The integer n is even if an only if n2 is even"

Exercise 31.6 | Q 6 | Page 29

By giving a counter example, show that the following statement is not true.
p : "If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle".

Exercise 31.6 | Q 7.1 | Page 29

statement are true and false? In each case give a valid reason for saying so

p : Each radius of a circle is a chord of the circle.

Exercise 31.6 | Q 7.2 | Page 29

statement are true and false? In each case give a valid reason for saying so

q : The centre of a circle bisects each chord of the circle.

Exercise 31.6 | Q 7.3 | Page 29

statement are true and false? In each case give a valid reason for saying so

r : Circle is a particular case of an ellipse.

Exercise 31.6 | Q 7.4 | Page 29

statement are true and false? In each case give a valid reason for saying so

s : If x and y are integers such that x > y, then − x < − y.

Exercise 31.6 | Q 7.5 | Page 29

statement are true and false? In each case give a valid reason for saying so

t :  $\sqrt{11}$  is a rational number.

Exercise 31.6 | Q 8 | Page 29

Determine whether the argument used to check the validity of the following statement is correct:
p : "If x2 is irrational, then x is rational"
The statement is true because the number x2 = π2 is irrational, therefore x = π is irrational.

## Chapter 31: Mathematical reasoning

Exercise 31.1Exercise 31.2Exercise 31.3Exercise 31.4Exercise 31.5Exercise 31.6

## RD Sharma solutions for Class 11 Mathematics Textbook chapter 31 - Mathematical reasoning

RD Sharma solutions for Class 11 Mathematics Textbook chapter 31 (Mathematical reasoning) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics Textbook chapter 31 Mathematical reasoning are Mathematically Acceptable Statements, New Statements from Old, Special Words Or Phrases, Contrapositive and Converse, Introduction of Validating Statements, Validation by Contradiction, Difference Between Contradiction, Converse and Contrapositive, Consolidating the Understanding.

Using RD Sharma Class 11 solutions Mathematical reasoning exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 31 Mathematical reasoning Class 11 extra questions for Class 11 Mathematics Textbook and can use Shaalaa.com to keep it handy for your exam preparation