NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 12

Q: Which concepts are covered in class 12 maths chapter 1 miscellaneous exercises

This exercise discusses concepts related to inverse, injective, onto functions.

Q: What are the important topics in chapter relations and functions ?

Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this chapter.

Q: What is the weightage held by the chapter relations and functions for CBSE board exam ?

The weightage of chapter relation and function is more than 5% in the CBSE board examination.

Q: How are the NCERT solutions helpful in the board exam ?

As students can assess from previous year questions that many questions are asked directly from NCERT exercise. Hence to score well in the examination, it is required to practice NCERT exercise for class 12 maths before the exam.

Q: What are relations in Class 12 Maths?

In maths, relation defines the relationship between sets of values of ordered pairs

Q: Mention the number of questions in class 12 maths chapter 1 miscellaneous exercise?

Total 19 questions are the in miscellaneous exercise Class 12 Maths chapter 1

Q: How many exercises are there in the NCERT class 12 maths chapter 1 ?

There are a total of 5 exercises including a miscellaneous exercise in the NCERT class 12 maths chapter 1.

NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 12 - Relations and Functions

Edited By Komal Miglani | Updated on Apr 21, 2025 07:45 PM IST | #CBSE Class 12th

Relations and functions are two essential concepts in class 12. The first thing that comes to a student's mind when checking these solutions is what relations and functions are. Suppose you went to a biriyani shop and ordered only chicken biriyani every time. Here, there is only one output for one input. So, it is a function. But whenever your brother goes to the biriyani shop, he brings 2-3 types of biriyani. Here, for one input, there are many outputs. So it is a relation. Careers360 experts create these solutions, and the latest NCERT guidelines have been followed.

Latest: JEE Main high scoring chapters | JEE Main 10 year's papers

NCERT Solutions for Class 12 Chapter 1 Miscellaneous Exercise is a set of questions that includes all the topics covered in this chapter. This exercise of the NCERT serves as a comprehensive revision and helps students apply various concepts together in one place.

Class 12 Maths Chapter 1 Miscellaneous Exercise Solutions: Download PDF

Download PDFNCERT solutions for Class 12 Maths Chapter 1 Relations and Functions: Miscellaneous Exercise

Question:1 Show that the function $f : R \to {x \in R : - 1 < x < 1}$ defined by $f (x) = \frac{x}{1 + | x |}$ $x \in R$ is one one and onto function.

Answer:

The function $f : R \to {x \in R : - 1 < x < 1}$ defined by

$f (x) = \frac{x}{1 + | x |}$ , $x \in R$

One- one:

Let $f (x) = f (y)$ , $x, y \in R$

$\frac{x}{1 + | x |} = \frac{y}{1 + | y |}$

It is observed that if x is positive and y is negative.

$\frac{x}{1 + x} = \frac{y}{1 + y}$

Since x is positive and y is negative.

$x > y \Rightarrow x - y > 0$ but 2xy is negative.

$x - y \neq 2 x y$

Thus, the case of x is positive and y is negative is removed.

Same happens in the case of y is positive and x is negative so this case is also removed.

When x and y both are positive:

$f (x) = f (y)$

$\frac{x}{1 + x} = \frac{y}{1 + y}$

$x (1 + y) = y (1 + x)$

$x + x y = y + x y$

$x = y$

When x and y both are negative : $f (x) = f (y)$

$\frac{x}{1 - x} = \frac{y}{1 - y}$

$x (1 - y) = y (1 - x)$

$x - x y = y - x y$

$x = y$

$∴$ f is one-one.

Onto:

Let $y \in R$ such that $- 1 < y < 1$

If y is negative, then $x = \frac{y}{y + 1} \in R$

$f (x) = f (\frac{y}{y + 1}) = \frac{\frac{y}{1 + y}}{1 + | \frac{y}{1 + y} |}$ $= \frac{\frac{y}{1 + y}}{1 + \frac{- y}{1 + y}} = \frac{y}{1 + y - y} = y$

If y is positive, then $x = \frac{y}{1 - y} \in R$

$f (x) = f (\frac{y}{1 - y}) = \frac{\frac{y}{1 - y}}{1 + | \frac{y}{1 - y} |}$ $= \frac{\frac{y}{1 - y}}{1 + \frac{- y}{1 - y}} = \frac{y}{1 - y + y} = y$

Thus, f is onto.

Hence, f is one-one and onto.

Question:2 Show that the function $f : R \to R$ given by $f (x) = x^{3}$ is injective.

Answer:

$f : R \to R$

$f (x) = x^{3}$

One-one:

Let $f (x) = f (y) x, y \in R$

$x^{3} = y^{3}$

We need to prove $x = y$ .So,

Let $x \neq y$ then there cubes will not be equal i.e. $x^{3} \neq y^{3}$ .
It will contradict given condition of cubes being equal.
Hence, $x = y$ and it is one -one which means it is injective

Question:3 Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if $A \subset B$ . Is R an equivalence relation on P(X)? Justify your answer.

Answer:

Given a non empty set X, consider P(X) which is the set of all subsets of X.

Since, every set is subset of itself , ARA for all $A \in P (x)$

$∴$ R is reflexive.

Let $A R B \Rightarrow A \subset B$

This is not same as $B \subset A$

If $A = {0, 1}$ and $B = {0, 1, 2}$

then we cannot say that B is related to A.

$∴$ R is not symmetric.

If $A R B a n d B R C, t h e n A \subset B a n d B \subset C$

this implies $A \subset C$ $= A R C$

$∴$ R is transitive.

Thus, R is not an equivalence relation because it is not symmetric.

Question:4 Find the number of all onto functions from the set ${1, 2, 3, . . ., n}$ to itself.

Answer:

The number of all onto functions from the set ${1, 2, 3, . . ., n}$ to itself is permutations on n symbols 1,2,3,4,5...............n.

Hence, permutations on n symbols 1,2,3,4,5...............n = n

Thus, total number of all onto maps from the set ${1, 2, 3, . . ., n}$ to itself is same as permutations on n symbols 1,2,3,4,5...............n which is n.

Question:5 Let $A = {- 1, 0, 1, 2}$ , $B = {- 4, - 2, 0, 2}$ and $f, g : A \to B$ be functions defined by $f (x) = x^{2} - x, x \in A$ and $g (x) = 2 | x - \frac{1}{2} | - 1, x \in A$ . Are $f$ and $g$ equal? Justify your answer. (Hint: One may note that two functions $f : A \to B$ and $g : A \to B$ such that $f (a) = g (a) \forall a \in A$ , are called equal functions).

Answer:

Given :

$A = {- 1, 0, 1, 2}$ , $B = {- 4, - 2, 0, 2}$

$f, g : A \to B$ are defined by $f (x) = x^{2} - x, x \in A$ and $g (x) = 2 | x - \frac{1}{2} | - 1, x \in A$ .

It can be observed that

$f (- 1) = (- 1)^{2} - (- 1) = 1 + 1 = 2$

$g (- 1) = 2 | - 1 - \frac{1}{2} | - 1 = 2 | \frac{- 3}{2} | - 1 = 3 - 1 = 2$

$f (- 1) = g (- 1)$

$f (0) = (0)^{2} - (0) = 0 + 0 = 0$

$g (0) = 2 | 0 - \frac{1}{2} | - 1 = 2 | \frac{- 1}{2} | - 1 = 1 - 1 = 0$

$f (0) = g (0)$

$f (1) = (1)^{2} - (1) = 1 - 1 = 0$

$g (1) = 2 | 1 - \frac{1}{2} | - 1 = 2 | \frac{1}{2} | - 1 = 1 - 1 = 0$

$f (1) = g (1)$

$f (2) = (2)^{2} - (2) = 4 - 2 = 2$

$g (2) = 2 | 2 - \frac{1}{2} | - 1 = 2 | \frac{3}{2} | - 1 = 3 - 1 = 2$

$f (2) = g (2)$

$∴ f (a) = g (a) \forall a \in A$

Hence, f and g are equal functions.

Question:6 Let $A = {1, 2, 3}$ . Then number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is

(A) 1
(B) 2
(C) 3
(D) 4

Answer:

$A = {1, 2, 3}$

The smallest relations containing $(1, 2)$ and $(1, 3)$ which are
reflexive and symmetric but not transitive is given by

$R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (3, 1)}$

$(1, 1), (2, 2), (3, 3) \in R$ , so relation R is reflexive.

$(1, 2), (2, 1) \in R$ and $(1, 3), (3, 1) \in R$ , so relation R is symmetric.

$(2, 1), (1, 3) \in R$ but $(2, 3) \notin R$ , so realation R is not transitive.

Now, if we add any two pairs $(2, 3)$ and $(3, 2)$ to relation R, then relation R will become transitive.

Hence, the total number of the desired relation is one.

Thus, option A is correct.

Question:7 Let $A = {1, 2, 3}$ . Then number of equivalence relations containing $(1, 2)$ is
(A) 1
(B) 2
(C) 3
(D) 4

Answer:

$A = {1, 2, 3}$

The number of equivalence relations containing $(1, 2)$ is given by

$R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}$

We are left with four pairs $(2, 3)$ , $(3, 2)$ , $(1, 3), (3, 1)$ .

$(1, 1), (2, 2), (3, 3) \in R$ , so relation R is reflexive.

$(1, 2), (2, 1) \in R$ and $(2, 3), (3, 2) \notin R$ , so relation R is not symmetric.

$(1, 3), (3, 1) \notin R$ , so realation R is not transitive.

Hence , equivalence relation is bigger than R is universal relation.

Thus the total number of equivalence relations cotaining $(1, 2)$ is two.

Thus, option B is correct.

Also, Read

Topics covered in Chapter 1 Number System: Miscellaneous Exercise

Concept

Definition

Composition of Functions

If there are two functions, $f : A \to B$ and $g : B \to C$ ,

then the composition of functions $g \circ f$ is defined as:

$(g \circ f) (x) = g (f (x))$

Invertible Function

A function is invertible if there exists an inverse function that reverses the effect of the original function. The function must be bijective.
$f^{- 1} (f (x)) = x$

Also, read,

NEET/JEE Coaching Scholarship

Get up to 90% Scholarship on Offline NEET/JEE coaching from top Institutes

Apply

JEE Main high scoring chapters and topics

As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE

Get now

NCERT Solutions Subject Wise

Unlock the NCERT textbook solutions of other subjects by clicking the following links.

JEE Main Highest Scoring Chapters & Topics

Just Study 40% Syllabus and Score upto 100%

Download EBook

Subject-wise NCERT Exemplar solutions

These resources cover the NCERT exemplar solutions of other subjects, prepared for students of class 12 to help them learn more systematically.

JEE Main Important Mathematics Formulas

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

Get now

Frequently Asked Questions (FAQs)

1. Which concepts are covered in class 12 maths chapter 1 miscellaneous exercises

This exercise discusses concepts related to inverse, injective, onto functions.

2. What are the important topics in chapter relations and functions ?

Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this chapter.

3. What is the weightage held by the chapter relations and functions for CBSE board exam ?

The weightage of chapter relation and function is more than 5% in the CBSE board examination.

4. How are the NCERT solutions helpful in the board exam ?

As students can assess from previous year questions that many questions are asked directly from NCERT exercise. Hence to score well in the examination, it is required to practice NCERT exercise for class 12 maths before the exam.

5. What are relations in Class 12 Maths?

In maths, relation defines the relationship between sets of values of ordered pairs

6. Mention the number of questions in class 12 maths chapter 1 miscellaneous exercise?

Total 19 questions are the in miscellaneous exercise Class 12 Maths chapter 1

7. How many exercises are there in the NCERT class 12 maths chapter 1 ?

There are a total of 5 exercises including a miscellaneous exercise in the NCERT class 12 maths chapter 1.

Also Read

NCERT Book Class 12 English 2025-26 Free PDF Download

NCERT Book Class 12 Physics 2025-26 FREE PDF Download

NCERT Book Class 12 Maths 2025-26 PDF Free Download

NCERT Books for Class 12 Biology 2025-26: Download Free PDF

NCERT Books for Class 12 Hindi 2025-26; Download Chapter Wise PDF here

NCERT Books for Class 12 - Download All Subjects PDFs Here

NCERT Syllabus 2025-26 for Class 6-12 - Download All Subjects PDF Free

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Solutions for Class 12 Physics Chapter 7 - Alternating Current

NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

Dual nature of radiation and matter Class 12th Notes - Free NCERT Class 12 Physics Chapter 11 Notes - Download PDF

Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF

NCERT Class 12th Maths Chapter 5 Continuity and Differentiability Notes

NCERT Class 12 Physics Chapter 4 Notes, Moving Charges and Magnetism Class 12 Chapter 4 Notes

NCERT Class 12 Physics Chapter 3 Notes, Current Electricity Class 12 Chapter 3 Notes

Atoms Class 12th Notes - Free NCERT Class 12 Physics Chapter 12 Notes - Download PDF

NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves

Articles

CBSE Class 12 Preparation Tips for Computer Science 2024-25

May 17, 2025

Himachal Pradesh 12th Class Result 2025 Declared, Check hpbose.org 12th Results Here

May 17, 2025

HP Board 12th Result 2025 Out; Download Link, Check HPBOSE Class 12th Results Here

May 17, 2025

HPBOSE HP Board Result 2025 Out: Check Class 10th, 12th Results @hpbose.org

May 17, 2025

HP Board 12th Revaluation & Rechecking 2025, Check Dates, Fees, Apply Online at hpbose.org

May 17, 2025

CBSE Class 12 English Preparation Tips 2025 - Tips to Score Maximum Marks

May 17, 2025

CBSE Sports Calendar 2025-26: Cluster, Zonal, and National level Games Dates

May 16, 2025

CBSE Sports Registration 2025-26, Games Login, Portal, Schedule, Venue

May 16, 2025

Upcoming School Exams

National Institute of Open Schooling 10th examination

Admit Card Date:02 April,2025 - 19 May,2025

Result

Application Process

Exam Pattern

National Institute of Open Schooling 12th Examination

Admit Card Date:02 April,2025 - 19 May,2025

Application Process

Exam Pattern

Admit Card

Andhra Pradesh Board of Intermediate Education Examination

Admit Card Date:06 May,2025 - 20 May,2025

Exam Pattern

Mock Test

Admit Card

Andhra Pradesh Board of Intermediate Education (Year 1) Exam

Admit Card Date:06 May,2025 - 20 May,2025

Result

Admit Card

Maharashtra Higher Secondary School Certificate Examination

Application Date:07 May,2025 - 17 May,2025

Exam Pattern

Result

Mock Test

View All School Exams

Certifications By Top Providers

Full-Stack Web Development

Via Masai School

Digital Banking Business Model

Via State Bank of India

Google Artificial Intelligence for JavaScript Developers with TensorFlow JavaScript

Via Google

Introduction to Psychology

Via San Jose State University, San Jose

Banking 101 A Guide for Beginners in the Banking Sector

Via EduBridge

Forensic Science DNA Analysis

Via University of Cambridge, Cambridge

1114 courses

818 courses

394 courses

294 courses

Harvard University, Cambridge

98 courses

University of Michigan, Ann Arbor

84 courses

IT & Software

Teaching and Academics

Programming and Development

Health, Fitness and Medicine

Business and Management

Engineering and Architecture

General Management

Financial Management

Environmental Studies

History

Psychology

Artificial Intelligence

Explore Top Universities Across Globe

University of Essex, Colchester

Wivenhoe Park Colchester CO4 3SQ

University College London, London

Gower Street, London, WC1E 6BT

The University of Edinburgh, Edinburgh

Old College, South Bridge, Edinburgh, Post Code EH8 9YL

University of Bristol, Bristol

Beacon House, Queens Road, Bristol, BS8 1QU

University of Delaware, Newark

Newark, DE 19716

University of Nottingham, Nottingham

University Park, Nottingham NG7 2RD

ACT Vocabulary Words You Must Know in 2025

21 min ⋆ May 09, 2025 06:05 AM IST

Tips 5 ACT Preparation Tips 2025 - Section wise Strategies to Score High

7 min ⋆ May 09, 2025 05:05 AM IST

ACT Practice Test 2025 - Download PDF with Answers and Explanations

5 min ⋆ May 09, 2025 05:05 AM IST

MBBS Fees in Bangladesh for Indian Students 2025: Medical Colleges

5 min ⋆ Apr 15, 2025 05:04 AM IST

New German Student Visa Process: Online Portal at digital.diplo.de for Applications

5 min ⋆ May 09, 2025 03:05 AM IST

LSAT Abroad Sample Papers 2025, Question Paper - Download PDF Here!

4 min ⋆ May 09, 2025 02:05 AM IST

Related E-books & Sample Papers

List of Competitive Exams for Class 9 to 12 Students

415+ Downloads

CBSE Class 11, 12 English (Core) Syllabus 2025-26

110+ Downloads

CBSE Class 11, 12 Biology Syllabus 2025-26

222+ Downloads

CBSE Class 11, 12 Physics Syllabus 2025-26

632+ Downloads

CBSE Class 11, 12 Maths Syllabus 2025-26

318+ Downloads

CBSE Class 11, 12 Applied Maths Syllabus 2025-26

35+ Downloads

Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

Can I change my board from CBSE to CHSE Odisha in Class 12th?

Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.

Read Complete Answer

Manasvini Gupta 30 January,2025

quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

Read Complete Answer

Jefferson 25 September,2024

i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

Re-evaluate Your Study Strategies:
- Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
- Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
- Practice Regularly: Consistent practice is key to mastering chemistry.
Consider Professional Help:
- Tutoring: A tutor can provide personalized guidance and support.
- Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.
Explore Alternative Options:
- Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
- Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.
Focus on NEET 2025 Preparation:
- Stay Dedicated: Continue your NEET preparation with renewed determination.
- Utilize Resources: Make use of study materials, online courses, and mock tests.
Seek Support:
- Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
- Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.

Read Complete Answer

Kanishka kaushikii 17 September,2024

i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

Type A: For candidates scoring 60% or above in the exam.

Type B: For candidates scoring between 50% and 60%.

Type C: For candidates scoring between 40% and 50%.

Cash Scholarship:

Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.

Hope you find this useful!

Read Complete Answer

Yuvan 30 August,2024

i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Read Complete Answer

Ashvadha 12 July,2024

View All

Stay up-to date with CBSE Class 12th News

Ask

Question

Student Community: Where Questions Find Answers

Ask and get expert answers on exams, counselling, admissions, careers, and study options.

Download Careers360 App

All this at the convenience of your phone

Regular Exam Updates
Best College Recommendations
College & Rank predictors
Detailed Books and Sample Papers
Question and Answers

Scan and download the app

Central Board of Secondary Education Class 12th Examination

JEE Test Series

Applications Open

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

Exams

Top Schools

Products & Resources

NCERT Study Material

Exams

Colleges

Predictors

Resources

Exams

Colleges & Courses

Predictors

Resources

Exams

Colleges

Predictors

Resources

Exams

Colleges

Predictors & E-Books

Resources

Exams

Predictors & Articles

Colleges

Resources

Exams

Colleges

Resources

Exams

Resources

Top Courses & Careers

Colleges

Exams

Colleges

Resources

Quick Links

Exams

Colleges

Resources

Exams

Colleges

Resources

Diploma Colleges

Exams

Resources

Upcoming Events

Other Exams

Exams

Colleges

Top Countries

Resources

Exams

Colleges

Upcoming Events

Resources

Engineering Preparation

Medical Preparation

Online Courses

Products

Top Streams

Specializations

Resources

Top Providers

NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 12 - Relations and Functions

Class 12 Maths Chapter 1 Miscellaneous Exercise Solutions: Download PDF

Download PDFNCERT solutions for Class 12 Maths Chapter 1 Relations and Functions: Miscellaneous Exercise

Topics covered in Chapter 1 Number System: Miscellaneous Exercise

NEET/JEE Coaching Scholarship

JEE Main high scoring chapters and topics

NCERT Solutions Subject Wise

Subject-wise NCERT Exemplar solutions

JEE Main Important Mathematics Formulas

Frequently Asked Questions (FAQs)

Also Read

Articles

Upcoming School Exams

Certifications By Top Providers

Explore Top Universities Across Globe

Related E-books & Sample Papers

Questions related to CBSE Class 12th

Popular CBSE Class 12th Questions