Question

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

(i) p: $\sqrt{11}$ is an irrational number.

(ii) q: Circle is a particular case of an ellipse.

(iii) r: Each radius of a circle is a chord of the circle.

(iv) s: The centre of a circle bisects each chord of the circle.

(v) t: If a nd b are integers such that $a<b$, then $-a<-b.$

(vi) u: The quadratic equation $x^2+x+1=0$ has no real roots.

Answer 1

(i) True, as 11 being prime $\sqrt{11}$ is irrational.

(ii) True. The equation of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$

Putting $b=a,$ it becomes $x^2+y^2=a^2,$ which is the equation of a circle.

(iii) False, since the radius of a circle is not a chord.

(iv) False, since there are chords which do not pass through the centre

(v) True, since $a<b\Rightarrow -a>-b$ is true.

(vi) True. Since $D=(1^2-4\times 1\times 1)=-3<0.$