NCERT · Class 11 · Maths

Chapter 14: Mathematical Reasoning Exercise 14.4

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Questions

4 questions

Q1Written answer

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.

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Worked solution

The five different ways of the given statement can be written as follows:

(i) A natural number is odd, indicating 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) It is sufficient that the number is odd for the square of a natural number to be odd.

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

Q2Written answer

Write the contrapositive and converse of the following statements.

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

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

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

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

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

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Worked solution

(i) The contrapositive of the given statement is as follows:

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

The converse of the given statement is as follows:

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

(ii) The contrapositive of the given statement is as follows:

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

The converse of the given statement is as follows:

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

(iii) The contrapositive of the given statement is as follows:

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

The converse of the given statement is as follows:

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

(iv) The contrapositive of the given statement is as follows:

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

The converse of the given statement is as follows:

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

(v) The given statement can be written as ‘if x is an even number, then x is divisible by 4’.

The contrapositive of the given statement is as follows:

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

The converse of the given statement is as follows:

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

Q3Written answer

Write each of the following statements in the form “if-then”.

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

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

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

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

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Worked solution

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

(ii) If the Banana trees stay warm for a month, then the trees will bloom.

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

(iv) If you want to score an A⁺ in the class, then you do all the exercises in the book.

Q4Written answer

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.

(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.

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Worked solution

(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 [Contrapositive of statement (a)].

(ii) If you have winter clothes, then you live in Delhi [Converse of statement (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 [Contrapositive of statement (b)].

(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram [Converse of statement (b)].