# NCERT Class 11 Mathematics Solutions: Chapter 14 –Mathematical Reasoning Miscellaneous Exercise Part 1

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1.Write the negation of the following statements:

(i) For every positive real number , the number is also positive.

(ii) All cats scratch.

(iii) For every real number , either .

(iv) There exists a number such that .

Answer: (i)

The negation of statement is as follows.

There exists a positive real number , such that is not positive.

Answer: (ii)

The negation of statement is as follows.

There exists a cat that does not scratch.

Answer: (iii)

The negation of statement is as follows.

There exists a real number , such that neither .

Answer: (iv)

The negation of statement is as follows.

There does not exist a number , such that .

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

(i) A positive integer is prime only if it has no divisors other than and itself.

(ii) I go to a beach whenever it is a sunny day.

(iii) If it is hot outside, then you feel thirsty.

Answer: (i)

Statement p can be written as follows.

If a positive integer is prime, then it has no divisors other than and itself.

The converse of the statement is as follows.

If a positive integer has no divisors other than and itself, then it is prime.

The contrapositive of the statement is as follows.

If positive integer has divisors other than and itself, then it is not prime.

Answer: (ii)

The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is as follows.

If I go to a beach, then it is a sunny day.

The contrapositive of the statement is as follows.

If I do not go to a beach, then it is not a sunny day.

Answer: (iii)

The converse of statement r is as follows.

If you feel thirsty, then it is hot outside.

The contrapositive of statement r is as follows.

If you do not feel thirsty, then it is not hot outside.