Q3 of 18 Page 898

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


(i) p: √11 is an irrational number is a TRUE statement.


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 is a TRUE statement.


A circle is a particular case of an ellipse with the same radius in all points.


The equation of an ellipse is


When a = b, we will get the equation of the circle, x2 + y2 = 1


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


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 is a FALSE statement.


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


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


a < b, then –a > -b, is TRUE by the rule of inequality.


(vi) y: The quadratic equation x2 + x + 1 = 0 have no real roots is a TRUE statement.


General form of a quadratic equation is ax2 + bx + c = 0.


If b2 – 4ac < 0, there is no real solution.


In the given equation; x2 + x + 1 = 0


a= 1; b = 1; c = 1


b2 – 4ac = 1 – 4 x 1 x 1 = -3 < 0


So, there is no real root.


More from this chapter

All 18 →
1

Which of the following sentences are statements? In case of a statement mention whether it is true or false.

(i) The sun is a star.


(ii) √7 is an irrational number.


(iii) The sum of 5 and 6 is less than 10.


(iv) Go to your class.


(v) Ice is always cold.


(vi) Have you ever seen the Red Fort?


(vii) Every relation is a function.


(viii) The sum of any two sides of a triangle is always greater than the third side.


(ix) May God bless you!


 2

Which of the following sentences are statements? In case of a statement, mention whether it is true or false.

(i) Paris is in France.


(ii) Each prime number has exactly two factors.


(iii) The equation x2 + 5|x| + 6 = 0 has no real roots.


(iv) (2 + √3) is a complex number.


(v) Is 6 a positive integer?


(vi) The product of -3 and -2 is -6.


(vii) The angles opposite the equal sides of an isosceles triangle are equal.


(viii) Oh! It is too hot.


(ix) Monika is a beautiful girl.


(x) Every quadratic equation has at least one real root.


 4

Write the negation of each of the following statements:

(i) Every natural number is greater than 0.


(ii) Both the diagonals of a rectangle are equal.


(iii) The sum of 4 and 5 is 8.


(iv) The number 6 is greater than 4.


(v) Every natural number is an integer.


(vi) The number -5 is a rational number


(vii) All cats scratch.


(viii) There exists a rational number x such that x2 = 3.


(ix) All students study mathematics at the elementary level.


(x) Every student has paid the fees.


(xi) There is some integer k for which 2k = 6.


(xii) None of the students in this class has passed.


 1

Split each of the following into simple sentences and determine whether it is true or false.

(i) A line is straight and extends indefinitely in both the directions.


(ii) A point occupies a position, and its location can be determined.


(iii) The sand heats up quickly in the sun and does not cool down fast at night.


(iv) 32 is divisible by 8 and 12.


(v) x = 1 and x = 2 are the roots of the equation x2 – x – 2 = 0.


(vi) 3 is rational, and √3 is irrational.


(vii) All integers are rational numbers, and all rational numbers are not real numbers.


(viii) Lucknow is in Uttar Pradesh, and Kanpur is in Uttarakhand.