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

Answer 1

(i) The given statement $p$ is false as chord of a circle is the line segment joining any two distinct points on a circle.
(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 get $x^2+y^2=1$ which is an equation of a circle.
Therefore circle is a particular case of an ellipse.
Thus statement $r$ is true
(iv) $x>y$
$\Rightarrow -x<-y$ (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