RBSE Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Ex 14.3

Rajasthan Board RBSE Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Ex 14.3 Textbook Exercise Questions and Answers.

RBSE Class 11 Maths Solutions Chapter 14 Mathematical Reasoning Ex 14.3

Question 1.
For each of the following compound statements first identify the connecting words and then break it into component statements.
(i) All rational numbers are real and all real numbers are not complex.
(ii) Square of an integer is positive or negative.
(iii) The sand heats up quickly in the Sun and does not cool down fast at night.
(iv) x = 2 and x = 3 are the roots of the equation 3x² - x - 10 = 0.
Answer:
(i) In the given statement:
Connective word ‘and’
Component statements are:
p: All rational number are real.
q : All real numbers are not complex.

(ii) In the given compound statement:
Connective word ‘or’
Component statements are:
p: square of an interger is positive.
q: square of an integer is negative.

(iii) In the given compound statement Here, connective word ‘and’
Component statements are:
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.

(iv) In the given compound statements :
Here, connective word ‘and’
Component statements are:
p: x = 2 is a root of equation 3x² - x - 10 = 0.
q : x = 3 is a root of equation 3x² - x - 10 = 0.

Question 2.
Identify the quantifier in the following statements and write the negation of the statements :
(i) There exists a number which is equal to its square.
(ii) For every real number x, x is less than x +1.
(iii) There exists a capital for every state in India.
Answer:
(i) Given statement-There exists a number which is equal to its square. quantifier: ‘There exists’
Negation of the statement: There exist no number which is equal to its square.

(ii) Given statement: For every real number x, x is less than x + 1.
Quantifier: ‘For all’
Negation of the statement: For every real number x, x is not less than x + 1.

(iii) Given statement: ‘There exists a capital for every state in India’.
Quantifier : “For all” or “For each”
Negation of the statement: There does not exist a capital for every state in India.

Question 3.
Check whether the following pair of statements are negation of each other. Give reason for your answer.
(i) x + y - y + x is true for every real number x and y.
(ii) There exists real numbers x and y for which
Answer:
First statement p : x + y = y + x is true for every real number x and y.
Thus, statement p = x + y - y + x is true for all real numebr x and y.
Negation of statement p: x + y = y + x is not true for all real number x and y.

Second statement q : The real numbers exists for which x + y = y + x is true since q is same statement as p.
Thus, p = q
Any statement cannot be negation of itself. Thus, pair of statement is not the negation of each other.

Question 4.
State whether the “or” used in the following statements is “exclusive” of “inclusive” Give reasons for your answer.
(i) Sun rises or Moon sets.
(ii) To apply for a driving licence, you should have a ration card or a passport.
(iii) All integers are positive or negative.
Answer:
(i) Given statement : Sim rises or Moon sets, here component statement are:
Coomponent statement p : Sun rises
Component statement q : Moon sets
Here, ‘or ’ is exclusive, since component p and q cannot occur at same time.

(ii) Given statement : to apply for a driving licence, you should have a ration card or a passport.
Component statement are:
p : To apply for a driving lincense, you should have a ration card.
q : To apply for a driving licence, you should have a passport.
Here, to apply for a driving licence, either ration card or passport is required or may be both (Ration card, Passport). Thus, here ‘or’ is inclusive.

(iii) Given statemt : All integers are positive or negative here, component statement are :
p : All integers are positive. q : All integers are negative.
Here, if first component statement is true then second component statement is false or vice-versa.
But both component statement cannot be together true or false. Since, any integer cannot be positive and negative both.
Thus, ‘or’ is exclusive.