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

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

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

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

**(iv)** s: If x and y are integers such that x > y, then –x < –y.

**(v)** t:\( \sqrt{11}\) is a rational number.

Asked by Pragya Singh | 1 year ago | 91

**(i)** Right answer is** False.**

**Explanation:-**

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

**(ii)** Right answer is** False.**

**Explanation:-**

If the chord is not the diameter of the circle, then the centre will not bisect that chord.

**(iii) **Right answer is** True.**

**Explanation:-**

The equation of an ellipse is,

If we put a = b = 1, then we get

x^{2} + y^{2} = 1, which is an equation of a circle

Hence, circle is a particular case of an ellipse.

Therefore, statement r is true

**(iv) **Right answer is** True.**

**Explanation:-**

x > y

By a rule of inequality

-x < – y

Hence, the given statement s is true

**(v)** Right answer is** False.**

**Explanation:-**

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

Therefore, \( \sqrt{11}\) is an irrational number. Thus, the given statement t is false.

Determine whether the argument used to check the validity of the following statement is correct: p: “If x^{2} is irrational, then x is rational.” The statement is true because the number x^{2} = π^{2} is irrational, therefore x = π is irrational.

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

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

**(ii) **q: The centre of a circle bisect each chord of the circle.

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

**(iv)** s: If x and y are integers such that x > y, then – x < – y.

**(v)** t: \( \sqrt{11}\) is a rational number.

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.”

Show that the following statement is true “The integer n is even if and only if n^{2} is even”

Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x^{2} is odd, then x is also odd.”