# Miscellaneous Exercise Mathmeatical Reasoning - NCERT Class 11

## Chapter 14 Ex.14.ME Question 1

Write the negation of the following statements:

(i) p: For every positive real number $$x$$ , the number $$x - 1$$ is also positive.

(ii) q: All cats scratch.

(iii) r: For every real number $$x$$, either $$x > 1$$ or $$x < 1$$.

(iv) s: There exists a number $$x$$ such that $$0 < x < 1$$.

### Solution

(i) The negation of statement p is as follows

There exists a positive real number $$x$$ such that $$x - 1$$ is not positive.

(ii) The negation of statement q is as follows

There exists a cat that does not scratch.

(iii) The negation of statement r is as follows

There exists a real number $$x$$ such that neither $$x > 1$$ nor $$x < 1$$.

(iv) The negation of statement s is as follows

There does not exist a number $$x$$ such that $$0 < x < 1$$.

## Chapter 14 Ex.14.ME Question 2

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

(i) p: A positive integer is prime only if it has no divisors other than 1 and itself.

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

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

### Solution

(i) Statement p can be written as follows

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is as follows

If a positive integer has no divisors other than $${\rm{1}}$$ and itself, then it is prime.

The contrapositive of the statement is as follows

If positive integer has divisors other than 1 and itself, then it is not prime.

(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 go to a beach, then it is not a sunny day.

(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.

## Chapter 14 Ex.14.ME Question 3

Write each of the statements in the form ‘if p, then q’.

(i) p: It is necessary to have a password to log on to the server.

(ii) q: There is traffic jam whenever it rains.

(iii) r: You can access the website only if you pay a subscription fee.

### Solution

(i) Statement p can be written as follows.

If you log on to the server, then you have a password.

(ii) Statement q can be written as follows.

If it rains, then there is a traffic jam.

(iii) Statement r can be written as follows.

If you can access the website, then you pay a subscription fee.

## Chapter 14 Ex.14.ME Question 4

Rewrite each of the following statements in the form ‘p if and only if q’.

(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

### Solution

(i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

## Chapter 14 Ex.14.ME Question 5

Given below are two statements

p: 25 is a multiple of 5.

q: 25 is a multiple of 8.

Write the compound statements connecting these two statements with ‘And’ and ‘Or’. In both cases check the validity of the compound statement.

### Solution

The compound statement with ‘And’ is:

‘25 is a multiple of 5 and 8’

This is a false statement, since 25 is not a multiple of 8.

The compound statement with ‘Or’ is:

‘25 is a multiple of 5 or 8’

This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.

## Chapter 14 Ex.14.ME Question 6

Check the validity of the statements given below by the method given against it.

(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) q: If n is a real number with $$n > 3$$, then $${n^2} > 9$$ (by contradiction method).

### Solution

(i) The given statement is as follows

p: the sum of an irrational number and a rational number is irrational.

Let us assume that the given statement, p, is false.

i.e., we assume that the sum of an irrational number and a rational number is rational.

Therefore, $$\sqrt a + \frac{b}{c} = \frac{d}{e}$$, when $$\sqrt a$$ is irrational and $$b,c,d,e$$ are integers.

$$\frac{d}{e} - \frac{b}{c}$$ is a rational number and is an irrational number.

This is a contradiction.

Therefore, our assumption is wrong.

(ii) The given statement, q is as follows.

If $$n$$ is a real number with $$n > 3$$, then $${n^2} > 9$$.

Let us assume that $$n$$ is a real number with $$n > 3$$, but $${n^2} > 9$$ is not true i.e., $${n^2} \le 9$$

Then, $$n > 3$$ and $$n$$ is a real number. Squaring both the sides, we obtain

\begin{align}{n^2} &> {\left( 3 \right)^2}\\{n^2} &> 9\end{align}

which is a contradiction, since we have assumed that $${n^2} \le 9$$.

Thus, the given statement is true. That is, if $$n$$ is a real number with $$n > 3$$, then $${n^2} > 9$$.

## Chapter 14 Ex.14.ME Question 7

Write the following statement in five different ways, conveying the same meaning.

p: If a triangle is equiangular, then it is an obtuse angled triangle.

### Solution

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

(i) A triangle is equiangular implies that is an obtuse-angled triangle.

(ii) A triangle is equilateral only if it an obtuse-angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle.

(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.

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