Q1 of 18 Page 904

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.


p: A line is straight.


q: A line extends indefinitely in both the directions.


Both the simple sentences are TRUE; therefore, the given sentence is TRUE.


(ii) p: A point occupies a position.


q: Its location can be determined.


Both the simple sentences are TRUE; therefore, the given sentence is TRUE.


(iii) p: The sand heats up quickly in the sun.


q: The sand does not cool down fast at night.


The first sentence is TRUE and the second sentence is FALSE.


Therefore the given sentence is FALSE.


(iv) p: 32 is divisible by 8.


q: 32 is divisible by 12.


First sentence is TRUE. Second sentence is FALSE.


Therefore the given sentence is FALSE.


(v) p: x = 1 is a root of the equation x2 – x – 2 = 0


q: x = 2 is a root of the equation x2 – x – 2 = 0


So, the first sentence is FALSE. Second sentence is TRUE.


Therefore, the given sentence is FALSE.


(vi) p: 3 is rational.


q: √3 is irrational.


First sentence is TRUE. Second sentence is TRUE.


Both the sentences are TRUE; therefore the given sentence is TRUE.


(vii) p: All integers are rational numbers.


q: All rational numbers are not real numbers.


First sentence is TRUE. Second sentence is FALSE.


Therefore, the given sentence is FALSE.


(viii) p: Lucknow is in Uttar Pradesh.


q: Kanpur is in Uttarakhand.


First sentence is TRUE. Second sentence is FALSE.


Therefore, the given sentence is FALSE.


More from this chapter

All 18 →
3

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


 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.


 2

Split each of the following into simple sentences and determine whether it is true or false. Also, determine whether an ‘inclusive or’ or ‘exclusive or’ is used.

(i) The sum of 3 and 7 is 10 or 11.


(ii) (1 + i) is a real or a complex number.


(iii) Every quadratic equation has one or two real roots.


(iv) You are wet when it rains, or you are in a river.


(v) 24 is a multiple of 5 or 8.


(vi) Every integer is rational or irrational.


(vii) For getting a driving license, you should have a ration card or a passport.


(viii) 100 is a multiple of 6 or 8.


(ix) Square of an integer is positive or negative.


(x) Sun rises or Moon sets.


 3

Find the truth set in case of each of the following open sentences defined on N:

(i) x + 2 < 10


(ii) x + 5 < 4


(iii) x + 3 > 2