# 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
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.