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 then
(v) t :  is a rational number

Medium

Solution

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(i) The given statement is false as chord of a circle is the line segment joining  any two distinct points on a circle. 

(ii) The given statement 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

If we put then we get  which is an equation of a circle.
Therefore circle is a particular case of an ellipse.
Thus statement is true

(iv)
(By a rule of inequality)
Thus the given statement is true

(v) is a prime number and we know that the square root of any prime number is an irrational number.
Therefore  is an irrational number.
Thus the given statement is false

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