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 : square root of 11 is a rational number.
Answers
SOLUTION :-
(i) Given: Each radius of a circle is a chord of the circle
The radius meets the circle at only one point whereas a chord meets the circle at two different points.
Any radius of the circle cannot be the chord of the circle.
∴ The given statement is false.
(ii) Given: The centre of a circle bisects each chord of the circle
only the diameter of a circle is a chord at which the centre of the circle exists.
The centre does not bisect all chords.
∴ The given statement is false
(iii) Given: a circle is a particular case of an ellipse.
Let us consider the equation of an eclipse
x2/a2 + y2/b2 = 1
If a=b then
x2 + y2 = 1
This is an equation of circle.
So the circle is a particular case of an eclipse.
∴ The given statement is true.
(iv) Given: if x and y are integers such that x > y then -x < -y
Where x > y then by the equation of inequality
⇒ -x < -y
∴ The given statement is true.
(v) Given: √11 is a rational number.
Every rational number can be expressed in the form P/P where p and q are integers and q≠0.
But √11cannot be expressed in the form of p/q.
∴ The given statement is false.
p is false because chord connects two points which lie on the circumference of the circle.
q is false because it is not necessary that all the chords should pass from the center of the circle.
r is true because circle has all distances from the center to the circumference equidistant.
s is true only if x and y are different integers - for example => let x =3 and y =1 hence x>y holds true here . making both negative we get -3<-1
t is false because 11 is not a perfect square and hence its square root can not be a rational number.