NCERT Solutions for Exercise 14.3 Class 11 Maths Chapter 14 - Mathematical Reasoning

Mathematical Reasoning Class 11 Chapter 14-Exercise: 14.3

Question:1.(i) For the following compound statements first identify the connecting words and then break it into component statements.

All rational numbers are real and all real numbers are not complex.

Answer:

The connecting word here is 'and'.

The component statements are:

p: All rational numbers are real.

q: All real numbers are not complex.

Question:1.(ii) For the following compound statement first identify the connecting words and then break it into component statements.

Square of an integer is positive or negative.

Answer:

The connecting word here is 'Or'.

The component statements are:

p: Square of an integer is positive.

q: Square of an integer is negative.

Question:1.(iii) For the following compound statement first identify the connecting words and then break it into component statements.

The sand heats up quickly in the Sun and does not cool down fast at night.

Answer:

The connecting word here is 'and'.

The component statements are:

p: The sand heats up quickly in the Sun.

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

Question:1.(iv) For the following compound statement first identify the connecting words and then break it into component statements.

and are the roots of the equation .

Answer:

The connecting word here is 'and'.

The component statements are:

p: x = 2 is a root of the equation .

q: x = 3 is a root of the equation .

Question:2.(i) Identify the quantifier in the following statement and write the negation of the statement.

There exists a number which is equal to its square.

Answer:

Given, p: There exists a number which is equal to its square.

Quantifier is "There exists".

Negation is, p': There does not exist a number which is equal to its square.

Question:2.(ii) Identify the quantifier in the following statement and write the negation of the statement.

For every real number , is less than .

Answer:

Given, p: For every real number , is less than .

Quantifier is "For Every".

Negation is, p': There exists a real number x such that x is not less than x + 1.

Question:2.(iii) Identify the quantifier in the following statement and write the negation of the statement.

There exists a capital for every state in India.

Answer:

Given, p: There exists a capital for every state in India.

Quantifier is "There exists".

Negation is, p': There does not exist a capital for every state in India. Or, There exists a state in India which does not have a capital.

Question:3. Check whether the following pair of statements are negation of each other. Give reasons for your answer.

(i) is true for every real numbers and .
(ii) There exists real numbers and for which .

Answer:

p: is true for every real numbers and .
q: There exists real numbers and for which .

The negation of p is:

There exists no real numbers x and y for which

which is not equal to q.

Hence the given pair of statements are not negation of each other.

Question:4.(i) State whether the “Or” used in the following statement is “exclusive “or” inclusive. Give reasons for your answer.

Sun rises or Moon sets.

Answer:

It is not possible for the Sun to rise and the moon to set simultaneously.

Here 'Or' is exclusive

Question:4.(ii) State whether the “Or” used in the following statement is “exclusive “or” inclusive. Give reasons for your answer.

To apply for a driving licence, you should have a ration card or a passport.

Answer:

A person can have both ration card or a passport to apply for a driving license.

Here 'Or' is inclusive.

Question:4.(iii) State whether the “Or” used in the following statement is “exclusive “or” inclusive. Give reasons for your answer.

All integers are positive or negative.

Answer:

All integers are either positive or negative but cannot be both.

Here 'Or' is exclusive.