# NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning

## NCERT Solutions for Class 11 Maths Chapter 14 Free PDF Download

The NCERT Solutions for Class 11 Maths Chapter 14, "Mathematical Reasoning," are available here in PDF format. These solutions include all the questions from the NCERT textbook, carefully solved and clearly explained. You will find comprehensive solutions for all exercises of Chapter 14 Exercise 14.1, 14.2, 14.3, 14.4, 14.5, and the miscellaneous exercise all in one place. Prepared by subject experts, these solutions adhere to the latest NCERT syllabus and guidelines.

### NCERT Solutions for Class 11Maths chapter 14 Exercise 14.1

### EX 14.1 Class 11 Maths Question 1:

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

(i) There are 35 days in a month.

(ii) Mathematics is defficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal lenght.

(vi) Answer this quetion.

(vii) The product of (-1) and 8 is 8.

(viii) The sum of all interior angles of tringle is 180^{0}.

(ix)Today is a windy day.

(x) All real numbers are complex numbers.

**Ans:**

(i) This sentence in incorrect because the maximum of days in a month is 31.Hence, it is a statemen.

(ii) This sentence is subjective in the sense that some people, mathematics can be easy and for some others, it can be difficult. Hence, is not a statement.

(iii) The, sum of 5 and 7 is 12, which is greater than 10. Therefore, this sentence is always correct. Hence, it is a statement.

(iv) This sentence is sometimes correct and incorrect. For example, the aquare of 2 is an even number. However, the square of 3 is an odd number. Hence, it is not a statement.

(v) This sentence is sometimes correct and sometimes incorrect. For example, squares and rhombus have sides of equal lenghts, However,trapezium and rectangles have sides of unequal lenghts. Hence, it is not a statement.

(vi) It is an order. Therefore, it not a statement,

(vii) The product of (-1) and 8 is (-8) Therefore, the given sentence is incorrect. Hence, it is statement.

(viii) The sentence is correct and hence, it is a statement.

(ix) The day that is being referred to is not evident from the sentence, Hence, it is not a statement.

(x) All real numbers can be written as a x 1 + 0 x I, Therefore, the given sentence is always correct, Hence, it is a statement.

### EX 14.1 Class 11 Maths Question 2:

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

**Ans:**

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

Solution-

The three examples of sentences, Ehich are not statement, are as follows.

(i) He is a doctor.

It is not evident from the sentence as to whon "he" is referred to. Therefore, it i not a statement.

(ii) Geometry is difficult.

This is not a statement bcause for some people, geometry can be easy and for some others, it con be difficult.

This is a question, which also contains 'she' and it is not evident as to who 'she' is Hence, it is not a statement.

## NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.2

### EX 14.2 Class 11 Maths Question 1:

Write the negation of the following statements:

(i) Chennai is the capital of Tamil Nadu.

### EX 14.2 Class 11 Maths Question 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.

(ii) The number x is a rational number.

The number x is an irrational number.

**Ans:**

(i) The negation of the first statement is "the number x is rational number".

This is same as the second statement. This is because if a number is not an irrational number, then it is a rational number.

Therefore, the given statements are negations of each other.

(ii) The negation of the first statement is "the number x is not a rational number" This means that the number x is an irrtional number, which is the same as the second statement.

Therefore, the given statement are nagations of each other.

## NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.3

### EX 14.3 Class 11 Maths Question 1:

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

(i) All rational numbers are real and all real numbers are not complex.

(ii) S quare of an integer is positive or negative.

(iii) The sand heats up quickly in the Sunand dies not coll down fast at night.

(iv) x = 2 and x = 3 are the roots of the equation 3x^{2} - x - 10 = 0.

**Ans:**

(i) Here, the connectiong word is 'and'

The Component statements are as follows.

*p*:All rational numbers are real.

*q*:All real numbers are not complex.

(ii) Here, the connectiong word is 'or'

The Component statements are as follows.

*p*: Square of an integer is positive

*q*: Square of an integer is negative

(iii) Here, the connectiong word is 'and'

The Component statements are as follows.

*p*: The sand heats up quickly in the sun.

*q*: The sand does not coll down fast at night.

(iv) Here, the connectiong word is 'and'

The Component statements are as follows.

*p*: x = 2 is a root of the equation 3x^{2} - x - 10 = 0

*q*: x = 2 is a root of the equation 3x^{2} - x - 10 = 0

### EX 14.3 Class 11 Maths Question 2:

Identify the quantifer in the following statements and write the nagation of the statements.

(i) There exists a number which is equal to its square.

(ii) For every real nu,ber x, x is less than x + 1.

(iii) There exists a capital for every state in India.

**Ans:**

(i) The quantifier is "There exists"

The negation of this statement is as follws.

There does not exist a number which is equal to its square.

(ii) The quantifier is "For every"

The negation of this statement is as follws.

There exist a real number x such that x is not than x + 1.

(iii) The quantifier is "There exists"

The negation of this statement is as follws.

There exists a state in India which does not have a capital.

## NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.4

### EX 14.4 Class 11 Maths Question 1:

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 add.*

**Ans:**

The given statement can be written in five defferent ways as follows.

(i) A natural number is odd implies that its square is odd

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd, it is necessary that its square is odd

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

(v) If the square of a natural number is not odd, then th natural number is not odd.

### EX 14.4 Class 11 Maths Question 2:

Write the contrapositive and converse of the following statements.

(i) If x is a prime number, then x is odd.

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

(iii) Something is cold implies that it has low temperature.

(iv)You connot comperhend geomentry if you do not know how to reason deductively.

(v) x is an even number implies that x is divisible by 4

**Ans:**

(i) The cotrapositive is as follows.

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

The converse is as follows.

IF a number x is odd, then is a prime number.

(ii) The contrapositive is as follows.

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

The converse is as follows.

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

(iii) The contrapositive is as follows.

If something does not have low temperature, then it is not cold.

The converse is as follows.

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

(iv) The contrapositive is as follows.

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

The converse is as follows.

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

(v) The given statement can be written as follows.

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

The contrapositive is as follows.

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

The converse is as follows.

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

## NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.4

### EX 14.4 Class 11 Maths Question 1:

Show that the statement.

p: "Ifx is a real number such that x^{2} + 4x = 0, then x is 0" is true by

(i) direct method

(ii) method of contradiction

(iii) method of contrapositive

**Ans:**

*p*: "If x is a real number such that x^{3} + 4x = 0, then x is 0"

Let *q*: x is a real number such that x^{3} + 4x = 0

*r*: x is 0.

(i) To show that statement *p* is true, we assume that *q* is true and then that *r* is true. Therefore, let statement *q* be true.

Show that the statement "For any real numbers a and b, a^{2} = ^{2} implies that a = b" is not true by giving a counter-example

**Ans:**

## Class 11 Maths NCERT Solutions - Mathematical Resoning Miscellaneous Questions

### Miscellaneos exercise Class 11 Maths Questions-1:

Write the negation of the following statements:

### Miscellaneos exercise Class 11 Maths Questions-2:

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