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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.

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 and y are integers such that x > y, then –x < –y.

(v) t: $\sqrt{11}$  is a rational number.

Solution:

(i) The given statement p is false.

According to the definition of chord, it should intersect the circle at two distinct points.

(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 obtain

$x^{2}+y^{2}=1$, which is an equation of a circle

Therefore, circle is a particular case of an ellipse.

Thus, statement r is true.

(iv) y

⇒ –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.

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