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

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.