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: is a rational number.

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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,

If we put a = b = 1, then we obtain

x² + y² = 1, which is an equation of a circle

Therefore, circle is a particular case of an ellipse.

Thus, statement r is true.

(iv) x > 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, is an irrational number.

Thus, the given statement t is false.

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\sqrt{11}

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 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 am ellipse.

(iv) s If x and y are integers such that x > y, then -x < -y .

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

x > y

- x < - y

\sqrt{11}

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.

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