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

**Question:**

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.

**Solution:**

(i) The given statement *p* is false.

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

(ii) The given statement *q* is false.

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

In other words, the centre of a circle only bisects the diameter, which is the chord of the circle.

(iii) The equation of an ellipse is,

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

If we put *a* = *b* = 1, then we obtain

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

Therefore, circle is a particular case of an ellipse.

Thus, statement *r* is true.

(iv) *x *> *y*

⇒ –*x* < –*y* (By a rule of inequality)

Thus, the given statement *s* is true.

(v) 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.

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