# NCERT solutions for Mathematics Class 11 chapter 14 - Mathematical Reasoning [Latest edition]

## Solutions for Chapter 14: Mathematical Reasoning

Below listed, you can find solutions for Chapter 14 of CBSE, Karnataka Board PUC NCERT for Mathematics Class 11.

Exercise 14.1Exercise 14.2Exercise 14.3Exercise 14.4Exercise 14.5Miscellaneous Exercise
Exercise 14.1 [Page 324]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Exercise 14.1 [Page 324]

Exercise 14.1 | Q 1.1 | Page 324

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

All real numbers are complex numbers.

Exercise 14.1 | Q 1.1 | Page 324

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

There are 35 days in a month.

Exercise 14.1 | Q 1.2 | Page 324

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

Mathematics is difficult.

Exercise 14.1 | 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.

Exercise 14.1 | 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.

Exercise 14.1 | 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.

Exercise 14.1 | Q 1.6 | Page 324

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

Exercise 14.1 | 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.

Exercise 14.1 | 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°.

Exercise 14.1 | Q 1.9 | Page 324

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

Today is a windy day.

Exercise 14.1 | Q 2 | Page 324

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

Exercise 14.2 [Page 329]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Exercise 14.2 [Page 329]

Exercise 14.2 | Q 1.1 | Page 329

Write the negation of the following statements:

Chennai is the capital of Tamil Nadu.

Exercise 14.2 | Q 1.2 | Page 329

Write the negation of the following statements:

sqrt2 is not a complex number.

Exercise 14.2 | Q 1.3 | Page 329

Write the negation of the following statements:

All triangles are not equilateral triangle.

Exercise 14.2 | Q 1.4 | Page 329

Write the negation of the following statements:

The number 2 is greater than 7.

Exercise 14.2 | Q 1.5 | Page 329

Write the negation of the following statements:

Every natural number is an integer.

Exercise 14.2 | 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.

Exercise 14.2 | 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.\

Exercise 14.2 | 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.

Exercise 14.2 | 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.

Exercise 14.2 | 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.

Exercise 14.3 [Pages 334 - 335]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Exercise 14.3 [Pages 334 - 335]

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.3 | 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.

Exercise 14.4 [Pages 338 - 339]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Exercise 14.4 [Pages 338 - 339]

Exercise 14.4 | 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.

Exercise 14.4 | Q 2.1 | Page 338

Write the contrapositive and converse of the following statements.

If x is a prime number, then x is odd

Exercise 14.4 | 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.

Exercise 14.4 | Q 2.3 | Page 338

Write the contrapositive and converse of the following statements.

Something is cold implies that it has low temperature.

Exercise 14.4 | 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.

Exercise 14.4 | 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

Exercise 14.4 | 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.

Exercise 14.4 | 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.

Exercise 14.4 | 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.

Exercise 14.4 | 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.

Exercise 14.4 | 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.

Exercise 14.5 [Pages 342 - 343]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Exercise 14.5 [Pages 342 - 343]

Exercise 14.5 | 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

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Exercise 14.5 | 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.

Miscellaneous Exercise [Page 345]

### NCERT solutions for Mathematics Class 11 Chapter 14 Mathematical Reasoning Miscellaneous Exercise [Page 345]

Miscellaneous Exercise | 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.

Miscellaneous Exercise | Q 1.2 | Page 345

Write the negation of the following statements:

q: All cats scratch.

Miscellaneous Exercise | Q 1.3 | Page 345

Write the negation of the following statements:

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

Miscellaneous Exercise | Q 1.6 | Page 345

Write the negation of the following statements:

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

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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.

Miscellaneous Exercise | Q 3. (i) | Page 345

Write the statement in the form “if p, then q”.

p: It is necessary to have a password to log on to the server.

Miscellaneous Exercise | Q 3. (ii) | Page 345

Write the statement in the form “if p, then q”.

q: There is a traffic jam whenever it rains.

Miscellaneous Exercise | Q 3. (iii) | Page 345

Write the statement in the form “if p, then q”.

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

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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.

Miscellaneous Exercise | 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).

Miscellaneous Exercise | 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).

Miscellaneous Exercise | 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.

## Solutions for Chapter 14: Mathematical Reasoning

Exercise 14.1Exercise 14.2Exercise 14.3Exercise 14.4Exercise 14.5Miscellaneous Exercise

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

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC 14 (Mathematical Reasoning) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Mathematics Class 11 chapter 14 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 NCERT Mathematics Class 11 solutions Mathematical Reasoning exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Mathematics Class 11 students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 14, Mathematical Reasoning Mathematics Class 11 additional questions for Mathematics Mathematics Class 11 CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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