Question 1: Check the validity of the following statements:

Answer:

(i)  p :100  is a multiple of 4 and 5. $

Since, 100 is a multiple of 4 and 5, the statement is true. Hence, it is a valid statement.

(ii)  q: 125  is a multiple of 5 and7. $

Since, 125 is a multiple of 5 but not a multiple of 7, the statement is not true. Hence, it is not a valid statement.

(iii)  r: 60  is a multiple of 3 or 5. $

Since, 60 is a multiple of 3 and 5, the statement is true. Hence, it is a valid statement.

Question 2: Check whether the following statement are true or not:

Answer: 

(i)      If and are odd integers, then is an even integer.

Let us assume that and be the statements given by

and are odd integers.

is an even integer

the given statement can be written as :

if then

Let be true. Then, and are odd integers

for some integers

is an integer

is true.

So, is true and is true.

Hence, “if p, then q “is a true statement.

(ii)      if are integer such that is even, then at least one of and is an even integer.

Let us assume that and be the statements given by

and are integers and is an even integer.

At least one of and is even.

Let be true, and then is an even integer.

So,

Now,

Let

Since, is an even integer, is also an even integer.

Now take

So, it is also true.

Hence, the statement is true.

Question 3: Show that the statement: is true by

(i) direct method      (ii) method of contrapositive      (iii) method of contradiction.

Answer:

(i) direct method

Let us assume that and be the statements given by

is a real Number such that

The given statement can be written as:

if then

Let be true. Then, is a real number such that

is a real number such that

is true

Thus, is true

Therefore, is true and is true.

Hence, is true.

(ii) Method of Contrapositive:

Let be false. Then,

is not true

is not true

Thus,

Hence, and is true

(iii) Method of Contradiction:

If possible, let be false.

Then, is not true

is true

is true

and is true is a real number such that

This is a contradiction.

Hence, is true.

Question 4: Show that the following statement is true by the method of contrapositive:

Answer:

Let us assume that and be the statements given

is an integer and is odd.

is an odd integer.

The given statement can be written as:

if then

Let be false.

Then, is not an odd integer, then is an even integer

for some integer n

is an even integer

Thus, is False

Therefore, is false and is false

Hence, “ if then ” is a true statement.

Question 5: Show that the following statement is true “The integer is even if and only if is even”

Answer:

Let the statements,

Integer is even

If is even Let be true. Then,

Let

Squaring both the sides, we get,

is an even number.

So, is true when is true.

Hence, the given statement is true.

Question 6: By giving a counter example, show that the following statement is not true.

p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle”

Answer:

Let be triangle in which then the is not an obtuse angled triangle.

Therefore given statement is not true. 

Question 7: Which of the following statements are true and which are false? In each case give a valid reason for saying so:

Answer:

(i) Each radius of a circle is a chord of the circle.

The given statement is false.

According the the definition of a chord, it should intersect the circumference of a circle at two distinct points.

(ii) The center of a circle bisects each chord of the circle.

The given statement is false.

If a chord is not a diameter of a circle, then the center does not bisect that chord. In other words, the center of a circle only bisects the diameter, which is the chord of the circle.

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

The statement is true.

If we put then we obtain which is the equation of a circle. Therefore, a circle is a particular case of an ellipse.

(iv) If and are integers such that then

The statement is true.

  (By the rule of inequality)

(v) is a rational number.

The given statement is false.

11 is a prime number and we know that the square root of any prime number is an irrational number. Therefore is an irrational number.

Question 8: Determine whether the argument used to check the validity of the following statement is correct:

The statement is true because the number is irrational, therefore is irrational.

Answer:

Argument Used: is irrational, therefore is irrational.

“If is irrational, then is rational.”

Let us take an irrational number given by where k is a rational number.

Squaring both sides, we get,

is a rational number and contradicts our statement.

Hence, the given argument is wrong.