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

Mathematical Reasoning Class 11 Chapter 14-Exercise: 14.4

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 odd.

Answer:

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.

Question:2.(i) Write the contrapositive and converse of the following statement.

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

Answer:

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.

Question:2.(ii) Write the contrapositive and converse of the following statement.

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

Answer:

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.

Question:2.(iii) Write the contrapositive and converse of the following statement.

Something is cold implies that it has low temperature.

Answer:

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 .

Question:2.(iv) Write the contrapositive and converse of the following statement.

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

Answer:

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.

Question:2.(v) Write the contrapositive and converse of the following statement.

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

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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.

Question:4.(a) Given statements in (a). Identify the statements given below as contrapositive or converse of each other.

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.

Answer:

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.

Question:4.(b) Given statements in (b). Identify the statements given below as contrapositive or converse of each other.

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.

Answer:

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.