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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) $\mathrm{t}: \sqrt{11}$ is a rational number.
(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) $\mathrm{t}: \sqrt{11}$ is a rational number.
Solution:
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Verified Answer
(i) False : The end points of radius do not lie on the circle. Therefore, it is not a chord.
(ii) False : Only diameters are bisected at the centre. Other chords do not pass through the centre. Therefore centre cannot bisect them.
(iii) True : Equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
When $b=a$, the equation becomes $\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$ or $x^2+y^2=a^2$ which is the equation of the circle.
(iv) True : If $x$ and $y$ are integers and $x>y$ then $-x < -y$. By rule of inequality.
(v) False : 11 is prime number.
$\therefore \sqrt{11}$ is irrational.
(ii) False : Only diameters are bisected at the centre. Other chords do not pass through the centre. Therefore centre cannot bisect them.
(iii) True : Equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.
When $b=a$, the equation becomes $\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$ or $x^2+y^2=a^2$ which is the equation of the circle.
(iv) True : If $x$ and $y$ are integers and $x>y$ then $-x < -y$. By rule of inequality.
(v) False : 11 is prime number.
$\therefore \sqrt{11}$ is irrational.
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