**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.*

a.) A triangle is equiangular implies it is an obtuse angled triangle.
b.) Knowing that a triangle is equiangular is sufficient to conclude that it is an obtuse angled triangle.
c.) A triangle is equiangular only if it is an obtuse angled triangle.
d.) When a triangle is equiangular, it is necessarily an obtuse angled triangle.
e.) If a triangle is not an obtuse-angled triangle, it is not equiangular.

**6.(ii)** Check the validity of the statements given below by the method given against it.

(ii) q: If is a real number with , then (by contradiction method).

Assume that the given statement q is false.
The statement becomes: If n is a real number with n > 3, then .
Therefore n>3 and n is a real number.
This is a contradiction.
Therefore our assumption is wrong.
Thus, the given statement q is true.

**6.(i) **Check the validity of the statement 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).

Assume that the given statement p is false.
The statement becomes: The sum of an irrational number and a rational number is rational.
Let
Where is irrational number and and are rational numbers.
is a rational number and is an irrational number, which is not possible.
This is a contradiction.
Hence our assumption is wrong.
Thus, the given statement p is true.

**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.

Given,
p: 25 is a multiple of 5.
q: 25 is a multiple of 8.
p is true while q is false.
The compound statement with 'And' is: 25 is a multiple of 5 and 8.
This is a false statement.
The compound statement with 'Or' is: 25 is a multiple of 5 or 8.
This is a true statement.

**4.(iii) **Rewrite the following statement in the form “p if and only if q”

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

The statement in the form “p if and only if q” is :
A quadrilateral is equiangular if and only if it is a rectangle.

**4.(ii)** Rewrite the following statement in the form “p if and only if q”

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

The statement in the form “p if and only if q” is :
You get an A grade if and only if you do all the homework regularly.

**4.(i) **Rewrite the following statement 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.

The statement in the form “p if and only if q” is :
You watch television if and only if your mind is free.

**3.(iii) **Write the statement in the form “if p, then q”

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

The statement in the form “if p, then q” is :
If you can access the website, then you pay a subscription fee.

**3.(ii) **Write the statement in the form “if p, then q”

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

The statement in the form “if p, then q” is :
If it rains, then there is a traffic jam.

**3.(i) **Write the statement in the form “if p, then q”

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

The statement in the form “if p, then q” is :
If you log on to the server, then you have a password.

**2.(iii) **State the converse and contrapositive of the following statement:

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

The given statement is in the form "if p then q".
The converse of the statement is:
If you feel thirsty, then it is hot outside.
The contrapositive of the statement is:
If you don't feel thirsty, then it is not hot outside.

**2.(ii) **State the converse and contrapositive of the following statement:

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

The given statement as "if-then" statement is: If it is a sunny day, then I go to a beach.
The converse of the statement is:
If I go to the beach, then it is a sunny day.
The contrapositive of the statement is:
If I don't go to the beach, then it is not a sunny day.

**2.(i) **State the converse and contrapositive of the following statement:

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

The given statement as "if-then" statement is: If a positive integer is prime, then it has no divisors other than 1 and itself.
The converse of the statement is:
If a positive integer has no divisors other than 1 and itself, then it is a prime.
The contrapositive of the statement is:
If positive integer has divisors other than 1 and itself then it is not prime.

**1.(iv) **Write the negation of the following statement:

(iv) *s*: There exists a number such that .

The negation of the statement is:
There does not exist a number x such that 0 < x < 1.

**1.(iii) **Write the negation of the following statement:

(iii) r: For every real number , either or .

The negation of the statement is:
There exists a real number x such that neither x > 1 nor x < 1.

**1.(ii) **Write the negation of the following statement:

(ii) *q*: All cats scratch.

The negation of the statement is:
It is false that all cats scratch.
Or
There exists a cat which does not scratch.

**1.(i) **Write the negation of the following statement:

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

The negation of the statement is:
There exists a positive real number x such that x–1 is not positive.

**5.(v) **Is the following statement true or false? Give a valid reason for saying so.

(v) *t* : is a rational number.

The statement is False.
Since 11 is a prime number, therefore is irrational.

**5.(iv) **Is the following statement true or false? Give a valid reason for saying so.

(iv) *s*: If and are integers such that , then .

The statement is True.
Give, x>y
Multiplying -1 both sides
(-1)x<(-1)y -x < -y
(When -1 is multiplied to both L.H.S & R.H.S, sign of inequality changes)
By the rule of inequality.

**5.(iii) **Is the following statement true or false? Give a valid reason for saying so.

(iii) *r*: Circle is a particular case of an ellipse.

The statement is True.
In the equation of an ellipse if we put a = b, then it is a circle.

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