GSEB Solutions Class 11 Maths Chapter 14 Mathematical Reasoning Miscellaneous Exercise

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:

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

Solution:

1. ~ p : There exists at least one positive real number x for which x – 1 is not positive.
2. ~ q : All cats do not scratch or we may say that there is at least one cat which does not scratch.
3. ~ r : There exists at least one number x such that neither x > 1, nor x < 1.
4. ~ 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:

1. p : A positive integer is prime only, if it has no divisor other than 1 and itself.
2. q : 1 go to a beach, whenever it is a sunny day.
3. 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”:

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

Solution:

1. If you log on to server, then you have a password.
2. If it rains, then there is a traffic jam.
3. 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”.

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

Solution:

1. You watch a television if and only if your mind is free.
2. You will get grade A if and only if you do all the home work regularly.
3. 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.

1. p : The sum of an irrational number and a rational number is irrational (by contradiction method).
2. q : If n is a real number with n > 3, then n² > 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 n² ≤ 9
Put n = 3 + a
⇒ n² = 9 + 6a + a²
= 9 + a(6 + a)
∴ n² > 9, which is contradiction
⇒ If n > 3, then n² > 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:

1. A triangle is equiangular, implies that it is an obtuse angled triangle.
2. A triangle is equiangular only if it is an obtuse angled triangle.
3. For a triangle to be equiangular, it is necessary that it is an obtuse angled triangle.
4. For a triangle to be obtuse angled triangle, it is sufficient that it is equiangular.
5. If a triangle is not obtuse angled triangle, then it is not an equiangular triangle.