NCERT Solutions for Class 11 Maths Chapter 14 Mathematical Reasoning Exercise 14.3
Chapter 14 Mathematical Reasoning Exercise 14.3 NCERT Solutions is available here through which you can easily get solutions of every question. NCERT Solutions for Class 11 Maths is very helpful in knowing the concepts and important formula of the chapter. These NCERT Solutions are updated as per the latest pattern of CBSE.
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 3x 2 – x – 10 = 0.
Answer
(i) “AND”
The component statements are:
All rational numbers are real.
All real numbers are not complex.
(ii) “OR”
The component statements are:
Square of an integer is positive.
Square of an integer is negative.
(iii) “AND”
The component statements are:
The sand heats up quickly in the sun.
The sand does not cool down fast at night.
(iv) “AND”
The component statements are:
x = 2 is a root of the equation 3x 2 – x – 10 = 0
x = 3 is a root of the equation 3x 2 – x – 10 = 0
2. Identify the quantifier in the given 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) Quantifier is “there exists”.
Negation is: There does not exist a number which is equal to its square.
(ii) Quantifier: For every p: for every real number x, x is less than x + 1.
~ p: There exist a real number x such that x is not less than x + 1.
(iii) Quantifier is: “There exists”.
The negation statement is: There exists a state in India which does not have a capital.
3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.
(i) x + y = y + x is true for every real numbers x and y.
(ii) There exists real numbers x and y for which x + y = y + x.
Answer
Statement (i) and (ii) are not the negation of each other.
4. State whether the "OR" used in the following statements is "exclusive or 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) When sun rise the moon sets. One of the happenings will take place. Here 'OR' is exclusive
(ii) To apply for a driving licence either a ration card or a passport or both can be used.
∴ "OR" used here is inclusive.
(iii) All integers are positive or negative. An integer cannot be both + ve and – ve at time.
∴ Here "OR" is exclusive.
The component statements are:
All rational numbers are real.
All real numbers are not complex.
(ii) “OR”
The component statements are:
Square of an integer is positive.
Square of an integer is negative.
(iii) “AND”
The component statements are:
The sand heats up quickly in the sun.
The sand does not cool down fast at night.
(iv) “AND”
The component statements are:
x = 2 is a root of the equation 3x 2 – x – 10 = 0
x = 3 is a root of the equation 3x 2 – x – 10 = 0
2. Identify the quantifier in the given 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) Quantifier is “there exists”.
Negation is: There does not exist a number which is equal to its square.
(ii) Quantifier: For every p: for every real number x, x is less than x + 1.
~ p: There exist a real number x such that x is not less than x + 1.
(iii) Quantifier is: “There exists”.
The negation statement is: There exists a state in India which does not have a capital.
3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.
(i) x + y = y + x is true for every real numbers x and y.
(ii) There exists real numbers x and y for which x + y = y + x.
Answer
Statement (i) and (ii) are not the negation of each other.
4. State whether the "OR" used in the following statements is "exclusive or 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) When sun rise the moon sets. One of the happenings will take place. Here 'OR' is exclusive
(ii) To apply for a driving licence either a ration card or a passport or both can be used.
∴ "OR" used here is inclusive.
(iii) All integers are positive or negative. An integer cannot be both + ve and – ve at time.
∴ Here "OR" is exclusive.