NCERT Solutions for Miscellaneous Exercise Chapter 14 Class 11 - Mathematical Reasoning

Question:1.(ii) Write the negation of the following statement:

q: All cats scratch.

Answer:

The negation of the statement is:

It is false that all cats scratch.

Or

There exists a cat which does not scratch.

Question:1.(iii) Write the negation of the following statement:

r: For every real number $x$, either $x >1$ or $x < 1$.

Answer:

The negation of the statement is:

There exists a real number x such that neither x > 1 nor x < 1.

Question:1.(iv) Write the negation of the following statement:

s: There exists a number $x$ such that $0 < x < 1$.

Answer:

The negation of the statement is:

There does not exist a number x such that 0 < x < 1.

Question:2.(i) State the converse and contrapositive of the following statement:

p: A positive integer is prime only if it has no divisors other than 1 and itself.

Answer:

The given statement as "if-then" statement is: If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is:

If a positive integer has no divisors other than 1 and itself, then it is a prime.

The contrapositive of the statement is:

If positive integer has divisors other than 1 and itself then it is not prime.

Question:2.(ii) State the converse and contrapositive of the following statement:

q: I go to a beach whenever it is a sunny day.

Answer:

The given statement as "if-then" statement is: If it is a sunny day, then I go to a beach.

The converse of the statement is:

If I go to the beach, then it is a sunny day.

The contrapositive of the statement is:

If I don't go to the beach, then it is not a sunny day.

Question:2.(iii) State the converse and contrapositive of the following statement:

r: If it is hot outside, then you feel thirsty.

Answer:

The given statement is in the form "if p then q".

The converse of the statement is:

If you feel thirsty, then it is hot outside.

The contrapositive of the statement is:

If you don't feel thirsty, then it is not hot outside.

Question:3.(i) Write the statement in the form “if p, then q”

p: It is necessary to have a password to log on to the server.

Answer:

The statement in the form “if p, then q” is :

If you log on to the server, then you have a password.

Question:3.(ii) Write the statement in the form “if p, then q”

q: There is traffic jam whenever it rains.

Answer:

The statement in the form “if p, then q” is :

If it rains, then there is a traffic jam.

Question:3.(iii) Write the statement in the form “if p, then q”

r: You can access the website only if you pay a subsciption fee.

Answer:

The statement in the form “if p, then q” is :

If you can access the website, then you pay a subscription fee.

Question:4.(i) Rewrite the following statement in the form “p if and only if q”

p: If you watch television, then your mind is free and if your mind is free, then you watch television.

Answer:

The statement in the form “p if and only if q” is :

You watch television if and only if your mind is free.

Question:4.(ii) Rewrite the following statement in the form “p if and only if q”

q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

Answer:

The statement in the form “p if and only if q” is :

You get an A grade if and only if you do all the homework regularly.

Question:4.(iii) Rewrite the following statement in the form “p if and only if q”

r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Answer:

The statement in the form “p if and only if q” is :

A quadrilateral is equiangular if and only if it is a rectangle.

Question:5 Given below are two statements
p : 25 is a multiple of 5.
q : 25 is a multiple of 8.
Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

Answer:

Given,

p: 25 is a multiple of 5.
q: 25 is a multiple of 8.

p is true while q is false.

The compound statement with 'And' is: 25 is a multiple of 5 and 8.

This is a false statement.

The compound statement with 'Or' is: 25 is a multiple of 5 or 8.

This is a true statement.

Question:6.(i) Check the validity of the statement given below by the method given against it

p: The sum of an irrational number and a rational number is irrational (by contradiction method).

Answer:

Assume that the given statement p is false.

The statement becomes: The sum of an irrational number and a rational number is rational.

Let $\sqrt p + \frac{s}{t} = \frac{q}{r}$

Where $\sqrt p$ is irrational number and $\frac{q}{r}$ and $\frac{s}{t}$ are rational numbers.

$\therefore \frac{q}{r} - \frac{s}{t}$ is a rational number and $\sqrt p$ is an irrational number, which is not possible.

This is a contradiction.

Hence our assumption is wrong.

Thus, the given statement p is true.

Question:6.(ii) Check the validity of the statements given below by the method given against it.

q: If $n$ is a real number with $n < 3$, then $n^2 < 9$ (by contradiction method).

Answer:

Assume that the given statement q is false.

The statement becomes: If n is a real number with n > 3, then $n^2 < 9$.

Therefore n>3 and n is a real number.

$\\ \therefore n^2 > 3^2 \\ \implies n^2 > 9$

This is a contradiction.

Therefore our assumption is wrong.

Thus, the given statement q is true.

Question:7 Write the following statement in five different ways, conveying the same meaning.
p: If a triangle is equiangular, then it is an obtuse angled triangle.

Answer:

a.) A triangle is equiangular implies it is an obtuse angled triangle.

b.) Knowing that a triangle is equiangular is sufficient to conclude that it is an obtuse angled triangle.

c.) A triangle is equiangular only if it is an obtuse angled triangle.

d.) When a triangle is equiangular, it is necessarily an obtuse angled triangle.

e.) If a triangle is not an obtuse-angled triangle, it is not equiangular.