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 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 is: p: “Each radius of a circle is a chord of the circle.” The chord of a circle is defined as the straight line segment whose both end points lies on the circle while the radius touches the circle with its one end point because other end point of a radius is at the center of the circle. Thus, the given statement is false. (ii) The given statement is: q: “The center of the circle bisects each chord of the circle.” If the chord is a diameter, then it can be said that the centre of the circle bisects the chord of the circle, but if the chord is not the diameter, then it does not bisect the center of the circle. Thus, the given statement is false. (iii) The given statement is: r: “Circle is a particular case of an ellipse.” The equation of an ellipse is, x 2 a 2 + y 2 b 2 =1 Consider a=1 and b=1 , then the equation becomes, x 2 + y 2 =1 This is the required equation of a circle. Thus, the given statement is true. (iv) The given statement is: s “If x and y are integers such that x>y , then −x<−y .” If x>y , then by the rule of inequality, when the sign of the numbers changes, then the sign of inequality also changes i.e. −x<−y . Thus, the given statement is true. (v) The given statement is: t: “ 11 is a rational number.” The integer 11 is a prime number and it is known that the square root of a prime number is an irrational number. Thus, the given statement is false.

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