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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 $n$ is a real number with $n < 3$, then $n^2 < 9$ (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 $x$ such that $0 < x < 1$.

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 $x$, either $x >1$ or $x < 1$.

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 $x$, the number $x -1$ 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 : $\sqrt{11}$ 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 $x$ and $y$ are integers such that $x > y$, then $-x < -y$.

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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