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**Exercise 14.5**

**1. Show that the statement **

** “If is a real number such that then is 0” is true by (i) direct method (ii) method of contradiction (iii) method of contrapositive.**

**Ans. **The given compound statement is of the form “if then ”.

R such that

**(i) **Direct method: If we assume that is true then

R such that

R such that

R such that or

is true, i.e., when is true then is true.

Therefore, the given compound statement is true.

**(ii) **Method of contradiction: If we assume that is true and is false, then

R such that

R such that

R such that or

which is contradiction. SO our assumption that is false.

Therefore, the given compound statement is true.

**(iii) **Method of contrapositive: If we assume that is false, then

R such that

is false, i.e., if is false, is false.

Therefore, the given compound statement is true.

**2. Show that the statement “For any real numbers and implies that ” is not true by giving a counter example.**

**Ans. **The given compound statement is of the form “if then ”.

If we assume that is true then, R such that

Let and

Now but therefore, when is true, then is false.

Therefore, the given compound statement is not true.

**3. Show that the following statement is true by the method of contrapositive.**

** “If is an integer and is even, then is also even”**

**Ans. **The given compound statement is of the form “if then ”.

Z and is even.

is an even integer.

If we assume that is false then is not an even integer.

is an odd integer.

is an odd integer.

is false, i.e., when is false, then is false.

Therefore, the given compound statement is true.

**4. By giving a counter example, show that the following statements are not true:**

**(i) “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle”.**

**(ii) “The equation does not have a root lying between 0 and 2”.**

**Ans. (i) **Since the triangle is obtuse angles triangle then

Let

Also all the angles of the triangle are equal.

Sum of all angles of a triangle = which is not possible.

Therefore, the given compound statement is not true.

**(ii) **We see that is a root of the equation which lies between 0 and 2.

Therefore, the given compound statement is not true.

**5. Which of the following statements are true and which are false? In each case give a valid reason for saying so.**

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

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

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

**(iv) If and are integers such that then **

**(v) id a ration number.**

**Ans. (i) **A chord of a circle is a line whose two end points lie on the circles and all the points on the line lie inside the circle.

So the radius of the circle is not a chord of the circle.

Therefore, the given statement is false.

**(ii) **The centre of a circle bisects chord of circle when the chord is diameter of the circle.

When the chord is other than diameter then centre of the circle does not lie on the chord.

Therefore, the given statement is false.

**(iii) **In the equation of an ellipse if we put, then we get an equation of circle.

Therefore, the given statement is true.

**(iv) **It is given that Z such that

Multiplying both sides by negative sign, then Z such that

Therefore, the given statement is true.

**(v) **Since cannot be expressed in the form where and are integers and

Therefore, the given statement is false.