NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning (Ex 14.2) Exercise 14.2

1. Write the negation of the following statements.

i. Chennai is the capital of Tamil Nadu.

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

The given statement is that, ‘Chennai is the capital of Tamil Nadu’.

The given statement does not carry the word ‘not’. 

To form the required negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation formed of the given statement is, ‘Chennai is not the capital of Tamil Nadu’.

ii. $\sqrt 2 $ is not a complex number.

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

The given statement is that, ‘$\sqrt 2 $ is not a complex number’.

The given statement carries the word ‘not’. 

To form the required negation of the given statement, we will remove the word ‘not’ from it. 

Therefore, the negation formed of the given statement is, ‘$\sqrt 2 $ is a complex number’.

iii. All triangles are not equilateral triangles. 

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

The given statement is that, ‘All triangles are not equilateral triangles’.

The given statement carries the word ‘not’. 

To form the required negation of the given statement, we will remove the word ‘not’ from it. 

Therefore, the negation formed of the given statement is, ‘All triangles are equilateral triangles’.

iv. The number 2 is greater than 7.

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

The given statement is that, ‘The number 2 is greater than 7’.

The given statement does not carry the word ‘not’. 

To form the required negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation formed of the given statement is, ‘The number 2 is not greater than 7’.

v. Every natural number is not an integer. 

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

The given statement is that, ‘Every natural number is not an integer’.

The given statement carries the word ‘not’. 

To form the required negation of the given statement, we will remove the word ‘not’ from it. 

Therefore, the negation formed of the given statement is, ‘Every natural number is an integer’.

2. Are the following pairs of statements negations of each other?

i. The number $x$ is not a rational number.

    The number $x$ is not an irrational number.

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

Consider the first statement. 

The given statement is that, ‘The number $x$ is not a rational number’.

The given statement carries the word ‘not’. 

To form the required negation of the given statement, we will remove the word ‘not’ from it. 

Therefore, the negation formed of the given statement is, ‘The number $x$ isa rational number’.

Now this statement is similar to the second statement. This is because if a number is not an irrational number, then it will be rational. And this is what the negation of our first statement says. 

Therefore, the given pair of statements are the negations of each other. 

ii. The number $x$ is a rational number.

     The number $x$ is an irrational number.

Ans: The negation of a statement means to negate the statement. In other words, it is writing the opposite of the given statement. Negation reverses the meaning of the statement. If a given statement is false then its negation will be true and vice versa. To write the negation of any statement, we usually add or remove the word ‘not’. 

Consider the first statement. 

The given statement is that, ‘The number $x$ is a rational number’.

The given statement does not carry the word ‘not’. 

To form the required negation of the given statement, we will add the word ‘not’ to it. 

Therefore, the negation formed of the given statement is, ‘The number $x$ is not a rational number’.

Now this statement is similar to the second statement. This is because if a number is not a rational number, then it will be irrational. And this is what the negation of our first statement says. 

Therefore, the given pair of statements are the negations of each other. 

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

i. Number 3 is prime or it is odd.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Consider the given statement, ‘Number 3 is prime or it is odd’. 

We will determine the first component statement and check whether it is true or false. 

Let this first component statement be ${\text{p}}$.

${\text{p}}$: Number 3 is prime. 

This statement is true because 3 is a prime number as it has only two factors, 1 and itself. 

We will now determine the second component statement and check whether it is true or false. 

Let this second component statement be ${\text{q}}$.

${\text{q}}$: Number 3 is odd. 

This statement is true because 3 is an odd number as it is not completely divisible by 2. 

Therefore, the component statements for the given compound statement are,

${\text{p}}$: Number 3 is prime. 

${\text{q}}$: Number 3 is odd. 

Both these above component statements are true. 

ii. All integers are positive or negative.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Consider the given statement, ‘All integers are positive or negative’. 

We will determine the first component statement and check whether it is true or false. 

Let this first component statement be ${\text{p}}$.

${\text{p}}$: All integers are positive. 

This statement is false because integers are both positive and negative numbers. 

We will now determine the second component statement and check whether it is true or false. 

Let this second component statement be ${\text{q}}$.

${\text{q}}$:All integers are positive. 

This statement is false because integers are both positive and negative numbers. 

Therefore, the component statements for the given compound statement are,

${\text{p}}$: All integers are positive. 

${\text{q}}$: All integers are negative. 

Both these above component statements are false. 

iii. 100 is divisible by 3, 11, and 5.

Ans: Compound statements are those statements that are made up of two or more simpler or smaller statements. These smaller statements are complete in themselves and have their own independent meanings. 

Consider the given statement, ‘100 is divisible by 3, 11, and 5’. 

We will determine the first component statement and check whether it is true or false. 

Let this first component statement be ${\text{p}}$.

${\text{p}}$: 100 is divisible by 3. 

This statement is false because 100 is not divisible by 3. 

We will now determine the second component statement and check whether it is true or false. 

Let this second component statement be ${\text{q}}$.

${\text{q}}$: 100 is divisible by 11. 

This statement is false because 100 is not divisible by 11. 

We will now determine the third component statement and check whether it is true or false. 

Let this third component statement be ${\text{r}}$.

${\text{r}}$: 100 is divisible by 5. 

This statement is true because 100 is divisible by 5.

Therefore, the component statements for the given compound statement are,

${\text{p}}$: 100 is divisible by 3.

${\text{q}}$: 100 is divisible by 11.

${\text{r}}$: 100 is divisible by 5. 

The statements ${\text{p}}$ and ${\text{q}}$ are false, whereas the statement ${\text{r}}$ is true.

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

Opting for the NCERT solutions for Ex 14.2 Class 11 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 14.2 Class 11 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 11 students who are thorough with all the concepts from the Subject Mathematical Reasoning textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 11 Maths Chapter 14 Exercise 14.2 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.

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