# NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning: The asset that made humans “superior” to other species was the ability to reason. In mathematics, there are mainly two kinds of reasoning.

• Inductive reasoning (In the chapter 4 mathematical induction you have learnt about inductive reasoning)
• Deductive reasoning

In this article, you will get NCERT solutions for class 11 maths chapter 14 mathematical reasoning.  In this chapter, you will learn different types of statements like simple statements, compound statements, negation, conditional statement, 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 CBSE NCERT solutions for class 11 maths chapter 14 mathematical reasoning. All these questions are prepared and explained in a step-by-step manner to undersatnd 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. These solutions will help you in the preparation of CBSE class 11 final examination as well as in the various competitive exams like JEE Main, VITEEE, BITSAT, etc. This chapter is not included in the syllabus of Jee advanced. Check all NCERT solutions which will help you to understand the concepts in much easy way. There are five exercises and a miscellaneous exercise in this chapter.

Exercise:14.1

Exercise:14.2

Exercise:14.3

Exercise:14.4

Exercise:14.5

Miscellaneous Exercise

 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.

Topics of NCERT Grade 11 Maths Chapter-14 Mathematical  Reasoning

14.1 Introduction

14.2 Statements

14.3 New Statements from Old

14.4 Special  Words/Phrases

14.5 Implications

14.6 Validating Statements

The complete solutions of NCERT class 11 mathematics chapter 14 mathematical reasoning is provided below:

## NCERT solutions for class 11 maths chapter 14 mathematical reasoning-Exercise: 14.1

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.

Mathematics is difficult.

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

The sum of 5 and 7 is greater than 10.

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

The square of a number is an even number.

$2^2 = 4$, 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.

The sides of a quadrilateral have equal length.

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.

The product of (–1) and 8 is 8.

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.

Solutions of NCERT for class 11 maths chapter 14 mathematical reasoning-Exercise: 14.2

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

Chennai is the capital of Tamil Nadu.

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 not a complex number

$\sqrt{2}$ 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 not equilateral triangle.

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 greater than 7.

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 an integer.

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.

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

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.

The number x is a rational number.
The number x is an irrational number.

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’.

All integers are positive or negative.

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’.

100 is divisible by 3, 11 and 5.

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’.

## CBSE NCERT solutions for class 11 maths chapter 14 mathematical reasoning-Exercise: 14.3

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

The connecting word here is 'and'.

The component statements are:

p: All rational numbers are real.

q: All real numbers are not complex.

Square of an integer is positive or negative.

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

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.

$x = 2$ and $x = 3$ are the roots of the equation $3x^2 - x - 10 = 0$.

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 $3x^2 - x - 10 = 0$.

There exists a number which is equal to its square.

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.

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

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.

There exists a capital for every state in India.

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.

Sun rises or Moon sets.

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

Here 'Or' is exclusive

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

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

Here 'Or' is inclusive.

All integers are positive or negative.

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

Here 'Or' is exclusive.

NCERT solutions for class 11 maths chapter 14 mathematical reasoning-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.

If x is a prime number, then x is 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.

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

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.

Something is cold implies that it has low temperature.

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 .

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

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.

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

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”

You get a job implies that your credentials are good.

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 Banana tree will bloom if it stays warm for a month.

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”

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

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”

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

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.
(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.

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.

NCERT solutions for class 11 maths chapter 14 mathematical reasoning-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.

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

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 $\impies$$\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.

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

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.

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

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.

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

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.

r: Circle is a particular case of an ellipse.

The statement is True.

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

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

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.

t : $\sqrt{11}$ is a rational number.

The statement is False.

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

CBSE NCERT solutions for class 11 maths chapter 14 mathematical reasoning-Miscellaneous Exercise

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

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

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:

q: All cats scratch.

The negation of the statement is:

It is false that all cats scratch.

Or

There exists a cat which does not scratch.

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

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.

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

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.

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

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”

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

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”

q: There is traffic jam whenever it rains.

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”

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

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

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

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

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

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

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

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”

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

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.

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

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.

q: If $n$ is a real number with $n < 3$, then $n^2 < 9$ (by contradiction method).

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.

## NCERT solutions for class 11 mathematics

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

## NCERT solutions for class 11- Subject wise

 Solutions of NCERT for class 11 biology CBSE NCERT solutions for class 11 maths NCERT solutions for class 11 chemistry Solutions of NCERT 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 CBSE 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.