NCERT Solutions for Class 11 Maths Chapter 14: Mathematical Reasoning - Exercise 14.4

Exercise 14.4

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 odd.

Ans: Consider the given statement, ‘If a natural number is odd, then its square is also odd’.

The given statement can be written in the ‘if-then’ in five different ways conveying the same meaning as follows,

1. A natural number is odd implies that its square is odd.

2. A natural number is odd only if its square is odd.

3. For a natural number to be odd, it is necessary that its square is odd.

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

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

2. Write the contrapositive and converse of the following statements.

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

Ans: Consider the given statement, ‘If $x$ is a prime number, then $x$ is odd’.

We will write the contrapositive of the given statement.

Contrapositive statements are a type of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’, where $A$ and $B$ are the component statements. 

The contrapositive of the given statement is as follows,

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

We will write the converse of the given statement.

Converse statements are a type of conditional statements. The converse of ‘If $A$, then $B$’, is ‘If $B$, then$A$’, where $A$ and $B$ are the component statements.

The converse of the given statement is as follows,

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

Therefore, the contrapositive and converse of the given statement is as follows,

Contrapositive: If a number $x$ is not odd, then $x$ is not a prime number. 

Converse: If a number $x$ is odd, then it is a prime number.

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

Ans: Consider the given statement, ‘If the two lines are parallel, then they do not intersect in the same plane’.

We will write the contrapositive of the given statement.

Contrapositive statements are a type of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’, where $A$ and $B$ are the component statements. 

The contrapositive of the given statement is as follows,

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

We will write the converse of the given statement.

Converse statements are a type of conditional statements. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’, where $A$ and $B$ are the component statements.

The converse of the given statement is as follows,

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

Therefore, the contrapositive and converse of the given statement is as follows,

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

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

iii. Something that is cold implies that it has a low temperature.

Ans: Consider the given statement, ‘Something is cold implies that it has low temperature’.

We will write the contrapositive of the given statement.

Contrapositive statements are a type of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’, where $A$ and $B$ are the component statements. 

The contrapositive of the given statement is as follows,

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

We will write the converse of the given statement.

Converse statements are a type of conditional statements. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’, where $A$ and $B$ are the component statements.

The converse of the given statement is as follows,

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

Therefore, the contrapositive and converse of the given statement is as follows,

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

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

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

Ans: Consider the given statement, ‘You cannot comprehend geometry if you do not know how to reason deductively’.

We will write the contrapositive of the given statement.

Contrapositive statements are a type of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’, where $A$ and $B$ are the component statements. 

The contrapositive of the given statement is as follows,

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

We will write the converse of the given statement.

Converse statements are a type of conditional statements. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’, where $A$ and $B$ are the component statements.

The converse of the given statement is as follows,

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

Therefore, the contrapositive and converse of the given statement is as follows,

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

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

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

Ans: Consider the given statement, ‘$x$ is an even number that implies that $x$ is divisible by 4’.

The given statement can be written in the ‘If-then’ form as follows,

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

We will write the contrapositive of the given statement.

Contrapositive statements are a type of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’, where $A$ and $B$ are the component statements. 

The contrapositive of the given statement is as follows,

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

We will write the converse of the given statement.

Converse statements are a type of conditional statements. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’, where $A$ and $B$ are the component statements.

The converse of the given statement is as follows,

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

Therefore, the contrapositive and converse of the given statement is as follows,

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

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

3. Write each of the following statements in the form ‘if-then’.

(i) Getting a job implies that your credentials are good.

Ans: The ‘If-then’ form are the conditional statements. They are written as, ‘If $A$, then $B$’, where $A$ and $B$ are the component statements.

Consider the given statement, ‘You get a job implies that your credentials are good’.

The ‘If-then’ form of the given statement is as follows,

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

(ii) The Banana trees will bloom if it stays warm for a month.

Ans: The ‘If-then’ form are the conditional statements. They are written as, ‘If $A$, then $B$’, where $A$ and $B$ are the component statements. 

Consider the given statement, ‘The Banana trees will bloom if it stays warm for a month’.

The ‘If-then’ form of the given statement is as follows,

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

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

Ans: The ‘If-then’ form are the conditional statements. They are written as, ‘If $A$, then $B$’, where $A$ and $B$ are the component statements. 

Consider the given statement, ‘A quadrilateral is a parallelogram if its diagonals bisect each other’.

The ‘If-then’ form of the given statement is as follows,

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

(iv). To get ${A^ + }$ in the class, it is necessary that you do the exercises in the book.

Ans: The ‘If-then’ form are the conditional statements. They are written as, ‘If $A$, then $B$’, where $A$ and $B$ are the component statements.

Consider the given statement, ‘To get ${A^ + }$ in the class, it is necessary that you do the exercises of the book’.

The ‘If-then’ form of the given statement is as follows,

If you want to get an ${A^ + }$ in the class, then you do all the exercises in the book.

4. 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.

Ans: Contrapositive statements and Converse statements are the types of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’. Here $A$ and $B$ are the component statements.

Consider the Given Statements.

(i) From the given statements it can be said that the second statement (i) is the contrapositive of the first statement (a). This is because it is of the form ‘If not $B$, then not $A$’.

(ii) Also, from the given statements it can be said that the third statement (ii) is the contrapositive of the first statement (a). This is because it is of the form ‘If $B$, then $A$’.

(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.

Ans: Contrapositive statements and Converse statements are the types of conditional statements. The contrapositive of ‘If $A$, then $B$’, is ‘If not $B$, then not $A$’. The converse of ‘If $A$, then $B$’, is ‘If $B$, then $A$’. Here $A$ and $B$ are the component statements. 

Consider the Given Statements.

i. From the given statements it can be said that the second statement (i) is the contrapositive of the first statement (a). This is because it is of the form ‘If not $B$, then not $A$’.

ii. Also, from the given statements it can be said that the third statement (ii) is the contrapositive of the first statement (a). This is because it is of the form ‘If $B$, then $A$’.

NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Exercise 14.4

Opting for the NCERT solutions for Ex 14.4 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 14.4 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.

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