NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

# NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

Edited By Ramraj Saini | Updated on Sep 25, 2023 08:48 PM IST

## Mathematical Reasoning Class 11 Questions And Answers

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning are provided here. The quality that made humans “superior” to other species was the ability to reason. In mathematics, there are mainly two kinds of reasoning i.e. Inductive reasoning and Deductive reasoning. Here students can find class 11 maths chapter 14 NCERT solutions. These solutions are prepared by expert teachers considering the latest syllabus of CBSE 2023. These NCERT solutions are comprehensive, easy to understand, with step by step solution of each problem.

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This NCERT Syllabus for this chapter includes different types of statements like simple statements, compound statements, negation, conditional statements, biconditional statements, conjunction, disjunction, negation, etc. There are a total of 18 questions in 5 exercises of NCERT textbook. First, try to solve all NCERT problems on your own. If you are not able to do so you can take help from NCERT solutions for class 11 .

## Mathematical Reasoning Class 11 Solutions - Important Formulae

Statements: A statement is a sentence that is either true or false, but not both simultaneously. For example, "A triangle has four sides" and "New Delhi is the capital of India" are statements.

Negation of a Statement: The negation of a statement p, denoted as ∼p, is a statement that is true when p is false and false when p is true.

Compound Statement: A compound statement is made up of two or more smaller statements, which are called component statements. Compound statements can be formed using connectives like "AND" and "OR," quantifiers like "there exists" and "for every," and implications like "If," "only if," and "if and only if."

Implications:

"p ⇒ q" represents that p implies q, where p is a sufficient condition for q, and q is a necessary condition for p. The converse of "p ⇒ q" is "q ⇒ p."

"p ⇔ q" signifies that p implies q (p ⇒ q), and q implies p (q ⇒ p). It states that p is a sufficient condition for q, and q is a sufficient condition for p.

Contrapositive:

The contrapositive of "p ⇒ q" is "∼ q ⇒ ∼ p."

Contradiction: To check whether p is true, we assume the negation of p (∼p) is true.

Free download NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning for CBSE Exam.

## Mathematical Reasoning Class 11 NCERT Solutions (Intext Questions and Exercise)

Class 11 maths chapter 14 NCERT solutions - Exercise: 14.1

Question:1.(i) Which of the following sentences are statements? Give reasons for your answer.
There are 35 days in a month.

There are 30 or 31 days in a month (And 28 or 29 in some cases). The given sentence is false. Hence it is a statement.

Question:1.(ii) Which of the following sentences are statements? Give reasons for your answer.

Mathematics can be difficult for some and easy for others. So it is neither true nor false. Hence it is not a statement.

Question:1.(iii) Which of the following sentences are statements? Give reasons for your answer.

Sum of 5 and 7 = 5 + 7 = 12 > 10. Hence the sentence is true. So it is a statement.

Question:1.(iv) Which of the following sentences are statements? Give reasons for your answer.

$2^2 = 4$ , which is even, and $3^2 = 9$ , which is odd. So the square of a number may be even or may be odd. Hence it is not a statement.

Question:1.(v) Which of the following sentences are statements? Give reasons for your answer.

The sentence is true for square and rhombus but not true for the rectangle. Hence it is not a statement.

The give sentence is an order. Hence it is not a statement.

Question:1.(vii) Which of the following sentences are statements? Give reasons for your answer.

The product of (-1) and 8 = -1 x 8 = -8. Hence the given sentence is false. So it is a statement.

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

The sum of all interior angles of a triangle is 180 °. This sentence is true always. Hence this is a statement.

Today is a windy day.

This may be true or false. Hence this is not a statement.

All real numbers are complex numbers.

All real numbers can be written in the form of a + i(0) (when a is a real number). This shows that all real numbers are complex numbers. Hence the sentence is always true. So it is a statement.

Following are three examples of sentences which are not statements.

How beautiful!

- This is an exclamation. Hence not a statement.

Open the door.

- This is an order. Hence not a statement.

Where are you going?

- This is a question. Hence not a statement.

Class 11 maths chapter 14 question answer - Exercise: 14.2

Question:1(i) Write the negation of the following statements:

Chennai is not the capital of Tamil Nadu.

Or

It is false to say that Chennai is the capital of Tamil Nadu.

Or

It is not the case that Chennai is the capital of Tamil Nadu.

Question:1.(ii) Write the negation of the following statements:

$\sqrt{2}$ is a complex number.

Or

It is false to say that $\sqrt{2}$ is not a complex number.

Or

It is not the case that $\sqrt{2}$ is not a complex number.

Question:1.(iii) Write the negation of the following statements:

All triangles are equilateral triangle.

Or

It is false to say that all triangles are not equilateral triangle.

Or

It is not the case that all triangles are not equilateral triangle.

Question:1.(iv) Write the negation of the following statements:

The number 2 is not greater than 7.

Or

It is false to say that the number 2 is greater than 7.

Or

It is not the case that the number 2 is greater than 7.

Question:1.(v) Write the negation of the following statements:

Every natural number is not an integer.

Or

It is false to say that every natural number is an integer.

Or

It is not the case that every natural number is an integer.

Question:2.(i) Are the following pairs of statements negations of each other:

p: The number x is not a rational number.
r: The number x is not an irrational number.

The negation of p is: The number x is a rational number, which is the same as statement r.

The negation of r is: The number x is an irrational number, which is the same as statement p.

Hence the pairs of statements are negations of each other.

Question:2(ii) Are the following pairs of statements negations of each other:

p: The number x is a rational number.
r: The number x is an irrational number.

The negation of p is: The number x is not a rational number, which is the same as statement r.

The negation of r is: The number x is not an irrational number, which is the same as statement p.

Hence the pairs of statements are negations of each other.

The component statements are

p: Number 3 is prime.

r: Number 3 is odd.

Both statements are true. Here the connecting word is ‘or’.

The component statements are:

p: All integers are positive.

r: All integers are negative.

Both the components statements are false. Here the connecting word is ‘or’.

The component statements are:

p: 100 is divisible by 3.

q: 100 is divisible by 11.

r: 100 is divisible by 5.

First two statements are false and the last statement is true. Here the connecting word is ‘and’.

Class 11 maths chapter 14 question answer - Exercise: 14.3

The connecting word here is 'and'.

The component statements are:

p: All rational numbers are real.

q: All real numbers are not complex.

The connecting word here is 'Or'.

The component statements are:

p: Square of an integer is positive.

q: Square of an integer is negative.

The connecting word here is 'and'.

The component statements are:

p: The sand heats up quickly in the Sun.

q: The sand does not cool down fast at night.

The connecting word here is 'and'.

The component statements are:

p: x = 2 is a root of the equation $3x^2 - x - 10 = 0$ .

q: x = 3 is a root of the equation $3x^2 - x - 10 = 0$ .

Given, p: There exists a number which is equal to its square.

Quantifier is "There exists".

Negation is, p': There does not exist a number which is equal to its square.

Given, p: For every real number $x$ , $x$ is less than $x + 1$ .

Quantifier is "For Every".

Negation is, p': There exists a real number x such that x is not less than x + 1.

Given, p: There exists a capital for every state in India.

Quantifier is "There exists".

Negation is, p': There does not exist a capital for every state in India. Or, There exists a state in India which does not have a capital.

(i) $x + y = y + x$ is true for every real numbers $x$ and $y$ .
(ii) There exists real numbers $x$ and $y$ for which $x + y = y + x$ .

p: $x + y = y + x$ is true for every real numbers $x$ and $y$ .
q: There exists real numbers $x$ and $y$ for which $x + y = y + x$ .

The negation of p is:

There exists no real numbers x and y for which $x + y = y + x$

which is not equal to q.

Hence the given pair of statements are not negation of each other.

It is not possible for the Sun to rise and the moon to set simultaneously.

Here 'Or' is exclusive

A person can have both ration card or a passport to apply for a driving license.

Here 'Or' is inclusive.

All integers are either positive or negative but cannot be both.

Here 'Or' is exclusive.

Mathematical reasoning NCERT solutions - Exercise: 14.4

a.) If the square of a natural number is odd, then the natural number is odd.

b.) A natural number is not odd only if its square is not odd.

c.) For a natural number to be odd it is necessary that its square is odd.

d.) For the square of a natural number to be odd, it is sufficient that the number is odd

e.) If the square of a natural number is not odd, then the natural number is not odd.

The contrapositive is :

If a number x is not odd, then x is not a prime number.

The converse is :

If a number x in odd, then it is a prime number.

The contrapositive is:

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

The converse is:

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

The contrapositive is:

If something is not at low temperature, then it is not cold.

The converse is:

If something is at low temperature, then it is cold .

The contrapositive is:

If you know how to reason deductively, then you can comprehend geometry.

The converse is:

If you do not know how to reason deductively, then you cannot comprehend geometry.

First, we convert the given sentence into the "if-then" statement:

If x is an even number, then x is divisible by 4.

The contrapositive is:

If x is not divisible by 4, then x is not an even number.

The converse is:

If x is divisible by 4, then x is an even number.

Question:3.(i) Write the following statement in the form “if-then”

The given statement in the form “if-then” is :

If you get a job, then your credentials are good.

Question:3.(ii) Write the following statement in the form “if-then”

The given statement in the form “if-then” is :

If the Banana tree stays warm for a month, then it will bloom.

Question:3.(iii) Write the following statement in the form “if-then”

The given statement in the form “if-then” is :

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

Question:3.(iv) Write the following statement in the form “if-then”

The given statement in the form “if-then” is :

(iv) If you get A+ in the class, then you have done all the exercises in the book.

If you live in Delhi , then you have winter clothes . : (if p then q)

The Contrapositive is (~q, then ~p)

Hence (i) is the Contrapositive statement.

The Converse is (q, then p)

Hence (ii) is the Converse statement.

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.

If a quadrilateral is a parallelogram , then its diagonals bisect each other . (if p then q)

The Contrapositive is (~q, then ~p)

Hence (i) is the Contrapositive statement.

The Converse is (q, then p)

Hence (ii) is the Converse statement.

Mathematical reasoning class 11 NCERT solutions - Exercise: 14.5

If $x$ is a real number such that $x^3 + 4x = 0$ , then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$ .

q: x is 0.

In order to prove the statement “if p then q”

Direct Method: By assuming that p is true, prove that q must be true.

So,

p is true:There exists a real number x such that $\dpi{100} x^3 + 4x = 0 \implies x(x^2 + 4) = 0$

$\dpi{100} \implies x = 0\ or\ (x^2 + 4)= 0$

$\dpi{100} \implies x = 0\ or\ x^2 = -4\ (not\ possible)$

Hence, x = 0

Therefore q is true.

If $x$ is a real number such that $x^3 + 4x = 0$ , then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$ .

q: x is 0.

In order to prove the statement “if p then q”

Contradiction: By assuming that p is true and q is false.

So,

p is true: There exists a real number x such that $x^3 + 4x = 0$

q is false: $\dpi{100} x \neq 0$

Now, $\dpi{100} x^3 + 4x = 0 \implies x(x^2 + 4) = 0$

$\dpi{100} \implies x = 0\ or\ (x^2 + 4)= 0$

$\dpi{100} \implies x = 0\ or\ x^2 = -4\ (not\ possible)$

Hence, x = 0

But we assumed $\dpi{100} x \neq 0$ . This contradicts our assumption.

Therefore q is true.

If $x$ is a real number such that $x^3 + 4x = 0$ , then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$ .

q: x is 0.

In order to prove the statement “if p then q”

Contrapositive Method: By assuming that q is false, prove that p must be false.

So,

q is false: $\dpi{100} x \neq 0$

$\dpi{100} \implies$ x.(Positive number) $\dpi{100} \neq$ 0.(Positive number)

$\dpi{100} \implies x(x^2 + 4) \neq 0(x^2 + 4)$

$\dpi{100} \implies x(x^2 + 4) \neq 0 \implies x^3 + 4x \neq 0$

Therefore p is false.

Given,

For any real numbers a and b, $a^2 = b^2$ implies that $a = b$ .

Let a = 1 & b = -1

Now,

$\dpi{100} a^2 = (1)^2$ = 1

$\dpi{100} b^2 = (-1)^2$ = 1

$\implies a^2 =1= b^2$

But a $\dpi{80} \neq$ b

Hence $a^2 = b^2$ does not imply that $a = b$ .

Hence the given statement is not true.

Given, If x is an integer and $x^2$ is even, then $x$ is also even.

Let, p : x is an integer and $x^2$ is even

q: $x$ is even

In order to prove the statement “if p then q”

Contrapositive Method: By assuming that q is false, prove that p must be false.

So,

q is false: x is not even $\dpi{80} \implies$ x is odd $\dpi{80} \implies$ x = 2n+1 (n is a natural number)

$\\ \therefore x^2 = (2n+1)^2 \\ \implies x^2 = 4n^2 + 4n + 1 \\ \implies x^2 = 2.2(n^2 + n) + 1 = 2m + 1$

Hence $x^2$ is odd $\dpi{80} \implies$ $x^2$ is not even

Hence p is false.

Hence the given statement is true.

We know, Sum of all the angles of a triangle = $180^{\circ}$

If all the three angles are equal, then each angle is $60^{\circ}$

But $60^{\circ}$ is not an obtuse angle, and hence none of the angles of the triangle is obtuse.

Hence the triangle is not an obtuse-angled triangle.

Hence the given statement is not true.

Given,

The equation $x^2 - 1 = 0$ does not have a root lying between 0 and 2.

Let x = 1

$\therefore (1)^2 - 1 = 1 -1 =0$

Hence 1 is a root of the equation $x^2 - 1 = 0$ .

But 1 lies between 0 and 2.

Hence the given statement is not true.

The statement is False.

By definition, A chord is a line segment intersecting the circle in two points. But a radius is a line segment joining any point on circle to its centre.

The statement is False.

A chord is a line segment intersecting the circle in two points. But it is not necessary for a chord to pass through the centre.

The statement is True.

In the equation of an ellipse if we put a = b, then it is a circle.

The statement is True.

Give, x>y

Multiplying -1 both sides

(-1)x<(-1)y $\implies$ -x < -y

(When -1 is multiplied to both L.H.S & R.H.S, sign of inequality changes)

By the rule of inequality.

The statement is False.

Since 11 is a prime number, therefore $\sqrt{11}$ is irrational.

Mathematical Reasoning class 11 solutions - Miscellaneous Exercise

Question:1.(i) Write the negation of the following statement:

The negation of the statement is:

There exists a positive real number x such that x–1 is not positive.

Question: 1.(ii) Write the negation of the following statement:

The negation of the statement is:

It is false that all cats scratch.

Or

There exists a cat which does not scratch.

Question:1.(iii) Write the negation of the following statement:

The negation of the statement is:

There exists a real number x such that neither x > 1 nor x < 1.

Question:1.(iv) Write the negation of the following statement:

The negation of the statement is:

There does not exist a number x such that 0 < x < 1.

The given statement as "if-then" statement is: If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is:

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

The contrapositive of the statement is:

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

The given statement as "if-then" statement is: If it is a sunny day, then I go to a beach.

The converse of the statement is:

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

The contrapositive of the statement is:

If I don't go to the beach, then it is not a sunny day.

The given statement is in the form "if p then q".

The converse of the statement is:

If you feel thirsty, then it is hot outside.

The contrapositive of the statement is:

If you don't feel thirsty, then it is not hot outside.

Question:3.(i) Write the statement in the form “if p, then q”

The statement in the form “if p, then q” is :

If you log on to the server, then you have a password.

Question:3.(ii) Write the statement in the form “if p, then q”

The statement in the form “if p, then q” is :

If it rains, then there is a traffic jam.

Question:3.(iii) Write the statement in the form “if p, then q”

The statement in the form “if p, then q” is :

If you can access the website, then you pay a subscription fee.

The statement in the form “p if and only if q” is :

You watch television if and only if your mind is free.

The statement in the form “p if and only if q” is :

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

Question:4.(iii) Rewrite the following statement in the form “p if and only if q”

The statement in the form “p if and only if q” is :

A quadrilateral is equiangular if and only if it is a rectangle.

Given,

p: 25 is a multiple of 5.
q: 25 is a multiple of 8.

p is true while q is false.

The compound statement with 'And' is: 25 is a multiple of 5 and 8.

This is a false statement.

The compound statement with 'Or' is: 25 is a multiple of 5 or 8.

This is a true statement.

Assume that the given statement p is false.

The statement becomes: The sum of an irrational number and a rational number is rational.

Let $\sqrt p + \frac{s}{t} = \frac{q}{r}$

Where $\sqrt p$ is irrational number and $\frac{q}{r}$ and $\frac{s}{t}$ are rational numbers.

$\therefore \frac{q}{r} - \frac{s}{t}$ is a rational number and $\sqrt p$ is an irrational number, which is not possible.

Hence our assumption is wrong.

Thus, the given statement p is true.

Assume that the given statement q is false.

The statement becomes: If n is a real number with n > 3, then $n^2 < 9$ .

Therefore n>3 and n is a real number.

$\dpi{100} \\ \therefore n^2 > 3^2 \\ \implies n^2 > 9$

Therefore our assumption is wrong.

Thus, the given statement q is true.

a.) A triangle is equiangular implies it is an obtuse angled triangle.

b.) Knowing that a triangle is equiangular is sufficient to conclude that it is an obtuse angled triangle.

c.) A triangle is equiangular only if it is an obtuse angled triangle.

d.) When a triangle is equiangular, it is necessarily an obtuse angled triangle.

e.) If a triangle is not an obtuse-angled triangle, it is not equiangular.

## Mathematical Reasoning Class 11 Solutions - Topics

14.1 Introduction

14.2 Statements

14.3 New Statements from Old

14.4 Special Words/Phrases

14.5 Implications

14.6 Validating Statements

Interested students can study class 11 maths ch 14 question answer using the exercises given below.

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All these questions are prepared and explained in a step-by-step manner to understand the concepts very easily. There are 7 questions are given in the miscellaneous exercise. In solutions of NCERT for class 11 maths chapter 14 mathematical reasoning, you will get solutions of miscellaneous exercise too.

 Basis for comparison Deductive Reasoning Inductive Reasoning Approach Top-down approach Bottom-up approach Based on Truths, facts and rules Trend or Patterns Starting point Premises Conclusion Process Observation > Pattern > Hypothesis > Theory Theory > Hypothesis > Observation > Confirmation Structure Goes from specific statement to general statement Goes from general statement to specific statement

Example-

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

1. There are 35 days in a month.
2. Mathematics is difficult.
3. The sum of 5 and 7 is greater than 10.
4. The square of a number is an even number.

Solution-

1. There are 28 or 29 or 30 or 31 days in a month. The given sentence is false. Hence it is a statement.
2. Mathematics can be difficult for some and easy for others. So it is neither true nor false. Hence it is not a statement.
3. Sum of 5 and 7 = 5 + 7 = 12 > 10. Hence the sentence is true. So it is a statement.
4. $2^2 = 4$ , which is even, and $3^2 = 9$, which is odd. So the square of a number may be even or maybe odd. Hence it is not a statement.

## NCERT Solutions for Class 11 Mathematics - Chapter Wise

 chapter-1 Sets chapter-2 Relations and Functions chapter-3 Trigonometric Functions chapter-4 Principle of Mathematical Induction chapter-5 Complex Numbers and Quadratic equations chapter-6 Linear Inequalities chapter-7 Permutation and Combinations chapter-8 Binomial Theorem chapter-9 Sequences and Series chapter-10 Straight Lines chapter-11 Conic Section chapter-12 Introduction to Three Dimensional Geometry chapter-13 Limits and Derivatives chapter-14 Mathematical Reasoning chapter-15 Statistics chapter-16

## NCERT Solutions for Class 11- Subject Wise

 NCERT solutions for class 11 biology NCERT solutions for class 11 maths NCERT solutions for class 11 chemistry NCERT solutions for Class 11 physics

Benefits of NCERT Solutions

• NCERT solutions for class 11 maths chapter 14 mathematical reasoning will introduce you to different ways to approach the problems.
• This chapter is very easy and has three concepts. NCERT exercise questions are enough for the practice. If you are finding difficulties in solving the problems, you can take help from NCERT solutions for class 11 maths chapter 14 mathematical reasoning.
• All these exercise questions are solved in a very detailed manner. So it will be very easy for you to understand the concepts.
• You can solve 7 problems given in the miscellaneous exercise to get command on this chapter.

## NCERT Books and NCERT Syllabus

1. In Chapter 14 of NCERT Solutions for Class 11 Maths, why is the concept of Mathematical Reasoning considered significant?

The significance of the Mathematical Reasoning chapter in Class 11 syllabus lies in its ability to improve students' logical thinking and reasoning skills. The solutions developed by the experienced faculty at Careers360 are beneficial for students preparing for competitive exams such as JEE Main, JEE Advanced, in addition to the Class 11 yearly exams. The mathematical reasoning pdf enables students to comprehend fundamental concepts and their practical applications in everyday life, even offline.

2. Write down the main topics of the mathematical reasoning class 11 NCERT solutions.

Below are the primary subjects covered in NCERT Solutions for maths chapter 14 class 11, Mathematical Reasoning:

1. Introduction
2. Statements
3. Deriving New Statements from Old
4. Special Terminology/Phrases
5. Implications
6. Verification of Statements.
3. Where can I find the complete solutions of NCERT for class 11 maths ?

Students can find a detailed NCERT solutions for class 11 maths here. they can practice ch 14 maths class 11 mathematical reasoning to command the concepts that will raise their confidence during the exam and ultimately lead to score well in the competitive exams.

4. How many chapters are there in CBSE class 11 maths ?

There are 16 chapters starting from set to probability in CBSE class 11 maths.

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