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

## Mathematics Class 11

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## Chapter 31: Mathematical reasoning

Ex. 31.10Ex. 31.20Ex. 31.30Ex. 31.40Ex. 31.50Ex. 31.60

#### Chapter 31: Mathematical reasoning Exercise 31.10 solutions [Page 3]

Ex. 31.10 | Q 1.01 | Page 3

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

Listen to me, Ravi !

Ex. 31.10 | Q 1.02 | Page 3

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

Every set is a finite set.

Ex. 31.10 | 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.

Ex. 31.10 | Q 1.04 | Page 3

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

The cat pussy is black.

Ex. 31.10 | Q 1.05 | Page 3

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

Are all circles round?

Ex. 31.10 | Q 1.06 | Page 3

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

All triangles have three sides.

Ex. 31.10 | Q 1.07 | Page 3

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

Every rhombus is a square.

Ex. 31.10 | 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.

Ex. 31.10 | Q 1.09 | Page 3

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

This sentence is a statement.

Ex. 31.10 | Q 1.1 | Page 3

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

Is the earth round?

Ex. 31.10 | Q 1.11 | Page 3

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

Go !

Ex. 31.10 | 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.

Ex. 31.10 | Q 1.13 | Page 3

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

There are 35 days in a month.

Ex. 31.10 | Q 1.14 | Page 3

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

Mathematics is difficult.

Ex. 31.10 | Q 1.15 | Page 3

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

All real numbers are complex numbers.

Ex. 31.10 | 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.

Ex. 31.10 | Q 2 | Page 3

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

#### Chapter 31: Mathematical reasoning Exercise 31.20 solutions [Pages 6 - 7]

Ex. 31.20 | Q 1.1 | Page 6

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

Ex. 31.20 | Q 1.2 | Page 6

Write the negation of the statement:

It rained on July 4, 2005.

Ex. 31.20 | Q 1.3 | Page 6

Write the negation of the statement:

Ravish is honest.

Ex. 31.20 | Q 1.4 | Page 6

Write the negation of the statement:

The earth is round.

Ex. 31.20 | Q 1.5 | Page 6

Write the negation of the statement:

The sun is cold.

Ex. 31.20 | Q 2.1 | Page 7

All birds sing.

Ex. 31.20 | Q 2.2 | Page 7

Some even integers are prime.

Ex. 31.20 | Q 2.3 | Page 7

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

Ex. 31.20 | Q 2.4 | Page 7

I will not go to school.

Ex. 31.20 | Q 2.5 | Page 7

Both the diagonals of a rectangle have the same length.

Ex. 31.20 | Q 2.6 | Page 7

All policemen are thieves.

Ex. 31.20 | 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.

Ex. 31.20 | 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.

Ex. 31.20 | 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.

Ex. 31.20 | Q 4.2 | Page 7

Write the negation of the statement:

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

Ex. 31.20 | Q 4.3 | Page 7

Write the negation of the statement:

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

Ex. 31.20 | 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.

#### Chapter 31: Mathematical reasoning Exercise 31.30 solutions [Page 14]

Ex. 31.30 | Q 1.1 | Page 14

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

Ex. 31.30 | Q 1.2 | Page 14

Find the component statement of the compound statement:

The earth is round or the sun is cold.

Ex. 31.30 | Q 1.3 | Page 14

Find the component statement of the compound statement:

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

Ex. 31.30 | Q 1.4 | Page 14

Find the component statement of the compound statement:

25 is a multiple of 5 and 8.

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | 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

Ex. 31.30 | 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.

Ex. 31.30 | 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.

Ex. 31.30 | Q 4.1 | Page 14

Determine whether the compound statement are true or false:

Delhi is in India and 2 + 2 = 4.

Ex. 31.30 | Q 4.2 | Page 14

Determine whether the compound statement are true or false:

Delhi is in England and 2 + 2 = 4.

Ex. 31.30 | Q 4.3 | Page 14

Determine whether the compound statement are true or false:

Delhi is in India and 2 + 2 = 5.

Ex. 31.30 | Q 4.4 | Page 14

Determine whether the compound statement are true or false:

Delhi is in England and 2 + 2 =5.

#### Chapter 31: Mathematical reasoning Exercise 31.40 solutions [Page 16]

Ex. 31.40 | Q 1.1 | Page 16

Write the negation of  statement:

For every x ϵ Nx + 3 < 10

Ex. 31.40 | Q 1.2 | Page 16

Write the negation of  statement:

There exists x ϵ Nx + 3 = 10

Ex. 31.40 | Q 2.1 | Page 16

Negate  statement:
All the students completed their homework.

Ex. 31.40 | Q 2.2 | Page 16

Negate of the  statement :

There exists a number which is equal to its square.

#### Chapter 31: Mathematical reasoning Exercise 31.50 solutions [Page 21]

Ex. 31.50 | 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.

Ex. 31.50 | Q 1.2 | Page 21

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

There is traffic jam whenever it rains.

Ex. 31.50 | 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.

Ex. 31.50 | 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.

Ex. 31.50 | 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.

Ex. 31.50 | Q 1.6 | Page 21

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

It rains only if it is cold.

Ex. 31.50 | Q 1.7 | Page 21

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

Whenever it rains it is cold.

Ex. 31.50 | Q 1.8 | Page 21

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

It never rains when it is cold.

Ex. 31.50 | Q 2.1 | Page 21

State the converse and contrapositive of  statement:

If it is hot outside, then you feel thirsty.

Ex. 31.50 | Q 2.2 | Page 21

State the converse and contrapositive of  statement:

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

Ex. 31.50 | 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.

Ex. 31.50 | Q 2.4 | Page 21

State the converse and contrapositive of  statement:

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

Ex. 31.50 | Q 2.5 | Page 21

State the converse and contrapositive of  statement:

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

Ex. 31.50 | 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.

Ex. 31.50 | 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.

Ex. 31.50 | 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.

Ex. 31.50 | 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.

Ex. 31.50 | Q 4.01 | Page 21

Determine the contrapositive of the statement:

If Mohan is a poet, then he is poor.

Ex. 31.50 | Q 4.02 | Page 21

Determine the contrapositive of the statement:

Only if Max studies will he pass the test.

Ex. 31.50 | Q 4.03 | Page 21

Determine the contrapositive of the statement:

If she works, she will earn money.

Ex. 31.50 | Q 4.04 | Page 21

Determine the contrapositive of the statement:

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

Ex. 31.50 | Q 4.05 | Page 21

Determine the contrapositive of the statement:

It never rains when it is cold.

Ex. 31.50 | Q 4.06 | Page 21

Determine the contrapositive of the statement:

If Ravish skis, then it snowed.

Ex. 31.50 | Q 4.07 | Page 21

Determine the contrapositive of the statement:

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

Ex. 31.50 | Q 4.08 | Page 21

Determine the contrapositive of the statement:

If he has courage he will win.

Ex. 31.50 | Q 4.09 | Page 21

Determine the contrapositive of the statement:

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

Ex. 31.50 | Q 4.1 | Page 21

Determine the contrapositive of the statement:

Only if he does not tire will he win.

Ex. 31.50 | Q 4.11 | Page 21

Determine the contrapositive of the statement:

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

#### Chapter 31: Mathematical reasoning Exercise 31.60 solutions [Pages 28 - 29]

Ex. 31.60 | Q 1.1 | Page 28

Check the validity of the statement:

p : 100 is a multiple of 4 and 5.

Ex. 31.60 | Q 1.2 | Page 28

Check the validity of the statement:

q : 125 is a multiple of 5 and 7.

Ex. 31.60 | Q 1.3 | Page 28

Check the validity of the statement:

r : 60 is a multiple of 3 or 5.

Ex. 31.60 | 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.

Ex. 31.60 | 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.

Ex. 31.60 | 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
(iii) method of contradition.

Ex. 31.60 | 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"

Ex. 31.60 | Q 5 | Page 29

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

Ex. 31.60 | 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".

Ex. 31.60 | 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.

Ex. 31.60 | 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.

Ex. 31.60 | 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.

Ex. 31.60 | 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.

Ex. 31.60 | 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.

Ex. 31.60 | 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

Ex. 31.10Ex. 31.20Ex. 31.30Ex. 31.40Ex. 31.50Ex. 31.60

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

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing 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 chapter 31 Mathematical reasoning are Consolidating the Understanding, Difference Between Contradiction, Converse and Contrapositive, Validation by Contradiction, Introduction of Validating Statements, Contrapositive and Converse, Special Words Or Phrases, New Statements from Old, Mathematically Acceptable Statements.

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.

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