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

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

Edited By Ravindra Pindel | Updated on Jul 13, 2022 06:36 PM IST

You have already learned about the fundamentals of deductive reasoning in mathematical reasoning. In the NCERT solutions for Class 11 Maths chapter 14 miscellaneous exercise, you will get the questions from all the topics covered in this chapter. This exercise is considered to be tougher than the other exercises of this chapter. Class 11 Maths chapter 14 miscellaneous solutions are very important for competitive exams like JEE Main, SRMJEE, VITEEE as many problems in engineering entrance exams are directly asked from miscellaneous exercises.

Miscellaneous exercise chapter 14 Class 11 will check your understanding of this chapter. This exercise is not very for the CBSE Exam as only a few questions in the CBSE exams are asked from the miscellaneous exercises. If you have solved all the problems from previous exercises, you can start solving problems from this exercise. You can go through the Class 11 Maths chapter 14 miscellaneous exercise solutions to understand this exercise more effectively. You can check NCERT Solutions link for detailed NCERT solutions at one place.

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## Mathematical Reasoning Class 11 Chapter 14 Miscellaneous Exercise-

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

p: For every positive real number $x$, the number $x -1$ is also positive.

The negation of the statement is:

There exists a positive real number x such that x–1 is not positive.

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

q: All cats scratch.

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 $x < 1$.

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 $0 < x < 1$.

The negation of the statement is:

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

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

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.

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

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.

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

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.

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.

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.

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

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

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

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

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

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

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.

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

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

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.

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

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.

Hence our assumption is wrong.

Thus, the given statement p is true.

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

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.

$\dpi{100} \\ \therefore n^2 > 3^2 \\ \implies n^2 > 9$

Therefore our assumption is wrong.

Thus, the given statement q is true.

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.

## More About NCERT Solutions for Class 11 Maths Chapter 14 Miscellaneous Exercise:-

As the name suggests the miscellaneous exercise chapter 14 Class 11 consists of a mixture of questions from all the topics of mathematical reasoning. Questions related to finding negation of statements, converse and contrapositive of statements, compound statements and their validity, etc are also covered in the Class 11 Maths chapter 14 miscellaneous solutions.

Also Read| Mathematical Reasoning Class 11 Notes

## Benefits of NCERT Solutions for Class 11 Maths Chapter 14 Miscellaneous Exercise:-

• Class 11 Maths chapter 14 miscellaneous exercise solutions are designed by subject matter experts who are experienced in this field.
• You can use Class 11 Maths chapter 14 miscellaneous solutions for reference while the NCERT problems.

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## Subject Wise NCERT Exampler Solutions

Happy learning!!!

1. Does the sentence "Tomorrow is Friday" is a statement ?

No, sentences involving variable time such as today, tomorrow, yesterday are not mathematical acceptable statements.

2. Does the sentence "How beautiful " is a statement ?

No, because the given sentence is an exclamation which is not a mathematical acceptable statement.

3. Does the sentence "Open the door" is a statement ?

No, because the given sentence is an order which is not a mathematical acceptable statement.

4. Does the sentence "Where are you going" is a statement ?

No, because the given sentence is a question that is not a mathematical acceptable statement.

5. Does the sentence "She is a mathematics graduate" is a statement ?

No, In the given sentence the "she" pronoun is used which is not referred to a particular person. Hence it is not a mathematical acceptable statement.

6. Does the sentence "Kashmir is far from here" is a statement ?

No, In the given sentence the "here" pronoun is used which is not referred to a particular location. Hence it is not a mathematical acceptable statement.

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