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# NCERT solutions for Class 11 Mathematics chapter 14 - Mathematical Reasoning

## Chapter 14 - Mathematical Reasoning

#### Page 324

Q 1.1 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

All real numbers are complex numbers.

Q 1.1 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

There are 35 days in a month.

Q 1.2 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

Mathematics is difficult.

Q 1.3 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

The sum of 5 and 7 is greater than 10.

Q 1.4 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

The square of a number is an even number.

Q 1.5 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

The sides of a quadrilateral have equal length.

Q 1.6 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

Q 1.7 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

The product of (–1) and 8 is 8.

Q 1.8 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

The sum of all interior angles of a triangle is 180°.

Q 1.9 | Page 324

Which of the following sentences are statements? Give reasons for your answer.

Today is a windy day.

Q 2 | Page 324

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

#### Page 329

Q 1.1 | Page 329

Write the negation of the following statements:

Chennai is the capital of Tamil Nadu.

Q 1.2 | Page 329

Write the negation of the following statements:

sqrt2 is not a complex number.

Q 1.3 | Page 329

Write the negation of the following statements:

All triangles are not equilateral triangle.

Q 1.4 | Page 329

Write the negation of the following statements:

The number 2 is greater than 7.

Q 1.5 | Page 329

Write the negation of the following statements:

Every natural number is an integer.

Q 2.1 | Page 329

Are the following pairs of statements negations of each other?

The number is not a rational number.

The number x is not an irrational number.

Q 2.2 | Page 329

Are the following pairs of statements negations of each other?

The number x is a rational number.

The number x is an irrational number.\

Q 3.1 | Page 329

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.

Q 3.2 | Page 329

Find the component statements of the following compound statements and check whether they are true or false.

All integers are positive or negative.

Q 3.3 | Page 329

Find the component statements of the following compound statements and check whether they are true or false.

100 is divisible by 3, 11 and 5.

#### Pages 334 - 335

Q 1.1 | Page 334

For each of the following compound statements first identify the connecting words and then break it into component statements.

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

Q 1.2 | Page 334

For each of the following compound statements first identify the connecting words and then break it into component statements.

Square of an integer is positive or negative.

Q 1.3 | Page 334

For each of the following compound statements first identify the connecting words and then break it into component statements.

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

Q 1.4 | Page 334

For each of the following compound statements first identify the connecting words and then break it into component statements.

= 2 and x = 3 are the roots of the equation 3x2 – x – 10 = 0.

Q 2.1 | Page 335

Identify the quantifier in the following statements and write the negation of the statements.

There exists a number which is equal to its square.

Q 2.2 | Page 335

Identify the quantifier in the following statements and write the negation of the statements.

For every real number xx is less than x + 1.

Q 2.3 | Page 335

Identify the quantifier in the following statements and write the negation of the statements.

There exists a capital for every state in India.

Q 3 | Page 335

Check whether the following pair of statements is negation of each other. Give reasons for the answer.

(i) x + y y + x is true for every real numbers x and y.

(ii) There exists real number x and y for which x + y = y + x

Q 4.1 | Page 335

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

Sun rises or Moon sets.

Q 4.2 | Page 335

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

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

Q 4.3 | Page 335

State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.

All integers are positive or negative.

#### Pages 338 - 339

Q 1 | Page 338

Rewrite the following statement with “if-then” in five different ways conveying the same meaning.

If a natural number is odd, then its square is also odd.

Q 2.1 | Page 338

Write the contrapositive and converse of the following statements.

If x is a prime number, then x is odd

Q 2.2 | Page 338

Write the contrapositive and converse of the following statements.

It the two lines are parallel, then they do not intersect in the same plane.

Q 2.3 | Page 338

Write the contrapositive and converse of the following statements.

Something is cold implies that it has low temperature.

Q 2.4 | Page 338

Write the contrapositive and converse of the following statements.

You cannot comprehend geometry if you do not know how to reason deductively.

Q 2.5 | Page 338

Write the contrapositive and converse of the following statements.

x is an even number implies that x is divisible by 4

Q 3.1 | Page 338

Write each of the following statement in the form “if-then”.

You get a job implies that your credentials are good.

Q 3.2 | Page 338

Write each of the following statement in the form “if-then”.

The Banana trees will bloom if it stays warm for a month.

Q 3.3 | Page 338

Write each of the following statement in the form “if-then”.

A quadrilateral is a parallelogram if its diagonals bisect each other.

Q 3.4 | Page 338

Write each of the following statement in the form “if-then”.

To get A+ in the class, it is necessary that you do the exercises of the book.

Q 4 | Page 339

Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other.

(a) If you live in Delhi, then you have winter clothes.

(i) If you do not have winter clothes, then you do not live in Delhi.

(ii) If you have winter clothes, then you live in Delhi.

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

(i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.

(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

#### Pages 342 - 345

Q 1 | Page 342

Show that the statement

p: “If x is a real number such that x3 + 4= 0, then x is 0” is true by

(i) direct method

(iii) method of contrapositive

Q 2 | Page 342

Show that the statement “For any real numbers a and ba2 = b2 implies that a = b” is not true by giving a counter-example.

Q 3 | Page 342

Show that the following statement is true by the method of contrapositive.

pIf x is an integer and x2 is even, then x is also even.

Q 4.1 | Page 342

By giving a counter example, show that the following statements are not true.

p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.

Q 4.2 | Page 342

By giving a counter example, show that the following statements are not true.

q: The equation x2 – 1 = 0 does not have a root lying between 0 and 2.

Q 5 | Page 345

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.

Q 5.1 | Page 343

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

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

Q 5.2 | Page 343

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

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

Q 5.3 | Page 343

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

r: Circle is a particular case of an ellipse.

Q 5.4 | Page 343

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

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

Q 5.5 | Page 343

Which of the following statements are true and which are false? In each case give a valid reason for saying so.

tsqrt11 is a rational number.

#### Page 345

Q 1.1 | Page 345

Write the negation of the following statements:

p: For every positive real number x, the number x – 1 is also positive.

Q 1.2 | Page 345

Write the negation of the following statements:

q: All cats scratch.

Q 1.3 | Page 345

Write the negation of the following statements:

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

Q 1.6 | Page 345

Write the negation of the following statements:

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

Q 2.1 | Page 345

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

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

Q 2.2 | Page 345

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

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

Q 2.3 | Page 345

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

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

Q 3 | Page 345

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

Q 4.1 | Page 345

Re write each of the following statements 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.

Q 4.2 | Page 345

Re write each of the following statements in the form “p if and only if q”.

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

Q 4.3 | Page 345

Re write each of the following statements in the form “p if and only if q”.

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

Q 6.1 | Page 345

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

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

Q 6.2 | Page 345

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

q: If n is a real number with n > 3, then n2 > 9 (by contradiction method).

Q 7 | Page 345

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

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

## NCERT solutions for Class 11 Mathematics chapter 14 - Mathematical Reasoning

NCERT solutions for Class 11 Mathematics chapter 14 (Mathematical Reasoning) include all questions with solution and detail explanation from Mathematics Textbook for Class 11. 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 created the CBSE Mathematics Textbook for 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. These NCERT 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 14 Mathematical Reasoning are Mathematically Acceptable Statements, Consolidating the Understanding, Introduction of Validating Statements, Difference Between Contradiction, Converse and Contrapositive, New Statements from Old, Special Words Or Phrases, Validation by Contradiction, Contrapositive and Converse.

Using NCERT solutions for Class 11 Mathematics 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 NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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