Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise

Question 1.

Write the negations of the following statements:

- p : For every positive real number x, the number x – 1 is also positive.
- q : All cats scratch.
- r : For every real number x, either x > 1 or x < 1.
- s : There exists a number x such that 0 < x < 1.

Solution:

- ~ p : There exists at least one positive real number x for which x – 1 is not positive.
- ~ q : All cats do not scratch or we may say that there is at least one cat which does not scratch.
- ~ r : There exists at least one number x such that neither x > 1, nor x < 1.
- ~ s : There does not exist a number such that 0 < x < 1.

Question 2.

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

- p : A positive integer is prime only, if it has no divisor other than 1 and itself.
- q : 1 go to a beach, whenever it is a sunny day.
- r : If it is hot outside, then you feel thirsty.

Solution:

1. Converse:

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

Contrapositive:

If a positive integer has no divisor other than 1 and itself, then it is not a prime.

2. Converse:

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

Contrapositive:

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

3. Converse:

If you feel thirsty, then it is hot outside.

Contrapositive:

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

Question 3.

Write each of the following statements in the form “if p then q”:

- p : It is necessary to have a password to log on to server.
- q : There is a traffic jam whenever it rains.
- r : You can access the website if you pay a subscription fee.

Solution:

- If you log on to server, then you have a password.
- If it rains, then there is a traffic jam.
- If you pay a subscription fee, then you can access the website.

Question 4.

Rewrite each of the following statements in the form “p” if and only if “q”.

- p : If you watch television, then your mind is free and if your mind is free, then you watch a television.
- q : For you to get an A grade, it is necessary and sufficient that you do all the home work regularly.
- r : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle then it is equiangular.

Solution:

- You watch a television if and only if your mind is free.
- You will get grade A if and only if you do all the home work regularly.
- A quadrilateral is equiangular if and only if it is a rectangle.

Question 5.

Given below are two statements:

p : 25 is a multiple of 5.

q : 25 is a multiple of 8.

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

Solution:

(i) Compound statement with ‘AND’

25 is a multiple of 5 and 8.

This is a false statement since p and q both are not true at the same time.

(ii) Compound statement with ‘OR’

25 is a multiple of 5 or it is a multiple of 8

This is a true statement.

Question 6.

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

- p : The sum of an irrational number and a rational number is irrational (by contradiction method).
- q : If n is a real number with n > 3, then n
^{2}> 9 (by contradiction method).

Solution:

1. Let \(\sqrt{a}\) be an irrational number and b be a rational number.

Their sum = b + \(\sqrt{a}\).

Let b + \(\sqrt{a}\) is not irrational. Therefore, it is a rational number. …………….. (1)

b + \(\sqrt{a}\) = \(\frac{p}{q}\), where p, q are co-prime.

\(\sqrt{a}\) = \(\frac{p}{q}\) – b ……………………. (2)

L.H.S. = \(\sqrt{a}\) = An irrational number

R.H.S. = \(\frac{p}{q}\) – b = A rational number

It is a contradiction.

Therefore, the sum irrational.

2. Let n > 3 and n^{2} ≤ 9

Put n = 3 + a

⇒ n^{2} = 9 + 6a + a^{2}

= 9 + a(6 + a)

∴ n^{2} > 9, which is contradiction

⇒ If n > 3, then n^{2} > 9.

Question 7.

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

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

Solution:

- A triangle is equiangular, implies that it is an obtuse angled triangle.
- A triangle is equiangular only if it is an obtuse angled triangle.
- For a triangle to be equiangular, it is necessary that it is an obtuse angled triangle.
- For a triangle to be obtuse angled triangle, it is sufficient that it is equiangular.
- If a triangle is not obtuse angled triangle, then it is not an equiangular triangle.