Question

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

(i) p: √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 center of a circle bisects each chord of the circle.

(v) t: If a and b are integers such that a < b, then –a > -b.

(vi) y: The quadratic equation x² + x + 1 = 0 has no real roots.

Answer 1

(i) p: √11 is an irrational number.

True statement.

Reason:

An irrational number is any number which cannot be expressed as a fraction of two integers.

√11 cannot be expressed as a fraction of two integers, so √11 is an irrational number.

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

True statement.

Reason:

The equation of an ellipse is x²/a² + y²/b² = 1

Special case: When a = b

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

So, we can say that, a circle is a particular case of an ellipse with the same radius in all points.

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

False statement.

Reason:

A chord intersects the circle at two points, but radius intersects the circle only at one point.

So the radius is not a chord of the circle.

(iv) S: The center of a circle bisects each chord of the circle.

False statement.

Reason:

The only diameter of a circle is bisected by the center of the circle. Except for diameter, no other chords are passes through the center of a circle.

(v) t: If a and b are integers such that a < b, then –a > -b.

True statement.

Reason:

a < b, then –a > - b [By rule of inequality]

(vi) y: The quadratic equation x² + x + 1 = 0 has no real roots.

True statement.

Reason:

General form of a quadratic equation, ax² + bx + c = 0, has no real roots if discriminant, D < 0.

Where D= b² – 4ac < 0.

Given equation; x² + x + 1 = 0

Here, a= 1, b = 1 and c = 1

Now, b² – 4ac = 1 – 4 x 1 x 1 = -3 < 0

So, there is no real root.