(B) 2
(C) 3
(D) 0
(A) 20
(B) 23
(C) 26
(D) 28
(A) 22
(B) 23
(C) 55
(D) 61
(A) 3
(B) 2
(C) 5
(D) None of these
(A) 1 each
(B) 2 each
(C) 10 each
(D) 5 each
(A) 1000
(B) 00100
(C) 00001
(D) 00010
(A) 0
(B) 2
(C) 5
(D) 4
(A) 2
(B) 5
(C) 6
(D) 9
(A) 0
(B) 1
(C) 2
(D) 3
(A) 1
(B) 2
(C) 3
(D) None of these
1. What is 20 more than 347?2. What will be 100 less than 5680? 3. What is 29 plus 63? 4. 2017 3017 4017 ______ 5. 631 621 611 ______ |
In conclusion, these worksheets for class 2 maths can help the kids to understand the fundamental mathematical concepts and apply logical thinking in various scenarios. It is also beneficial to have a class 2 maths worksheet pdf to help the students practice additional questions related to the class 2 maths syllabus. For more assistance on maths preparation, stay tuned with BYJU’S and download the app from the play store or app store.
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Are you a parent worried about how to educate your child in Mathematics of Class 2? We understand your problem. The market is flooded with many resources, coaching institutes and their study materials for CBSE Class 2 students which might confuse parents and guardians. But don’t worry as Vedantu is here to your rescue. You need not go beyond CBSE Class 2 NCERT Maths books for your child. These books are published by NCERT keeping in mind the competencies of Class 2 students and are preferred by the CBSE as well.
Educating your kids about the concepts of Mathematics can be cumbersome at times. However, an interactive process can help you overcome this challenge. NCERT publishes textbooks with fun learning activities and interesting topics via NCERT books for Class 2 Maths.
NCERT has been the best educational guide for students of all Classes. Going through the Class 2 NCERT Maths book, you will find concepts presented interactively.
Due to its simple way of elaborations and lists of activities, these books are a mandatory buy. Also, they comply with the latest CBSE syllabus, so download the Class 2 Maths NCERT book PDF and help your child overcome the hurdles of Mathematics!
You can download NCERT Class 2 Maths Book PDF for free this page by going through the following steps:
Navigate through this webpage, locate the download button and click on it
A form will pop up on your screen asking for your contact details. Fill in the asked details.
Verify those details via an OTP that you will receive
Download NCERT Class 2 Maths Book PDF
To ensure that you are accessing the authentic materials only, download the study materials from Vedantu’s website or the mobile application for free.
The NCERT Book for Class 2 Maths provides clear explanations, solved examples, and practice exercises to enhance conceptual understanding and problem-solving skills. You can download the NCERT Book for Class 2 Maths in both English and Hindi Languages. Get the link to download Maths Class 2, NCERT Books from the table below.
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2 |
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Ncert solutions for class 2, ncert books for class 2 maths pdf - free download, an introduction to the class 2 ncert maths book.
Mathematics improves the problem-solving and reasoning abilities of a student. It is the discipline of science consisting of the means of counting and measuring values along with ascertaining shapes of objects. The NCERT books for Class 2 Maths explain these basic notions in detail.
If your child is preparing for Class 2 examinations, studying from maths magic book Class 2 can expand their knowledge. Also, try downloading the Class 2 Maths NCERT book PDF to extend the convenience of learning from any device.
The chapter-wise elaboration of Class 2 Mathematics gets elucidated below. Go through them to know what these chapters unfold!
The NCERT Class 2 Maths book PDF comprises a chapter explaining the shapes of various objects. It describes the physical properties of materials interactively. Students can learn about the dimensions of numerous objects such as coins, postcards, pencils, and circles through this lesson.
From this segment, pupils can also learn how an object differs from the other in shape and texture. They can also learn about the physical similarities between one or more objects. For instance, students can gather knowledge about the shape of a pencil or the strength of a postcard, how things roll and slide simultaneously, etc.
Further, this lesson contains fun exercises that keep the readers engaged like guessing the object’s name, trying this activity, etc.
Concepts learned in this chapter form a basic building block for many advanced concepts that will be covered in the latter Classes. For example, understanding various shapes like circles, spheres, etc. It will help in understanding concepts of area, circumference, etc in the higher standards.
The lesson of NCERT books for Class 2 Maths keeps the young minds engaged in learning about counting numbers. It educates them about the arrangements of objects in a particular order. Also, pupils can use the method of placing substances in proper order.
From this chapter, students can learn about the strategy of counting things by guessing the number of items present near them. It covers various exercises and fun learning activities for enhancing the knowledge of young minds.
For instance, interactive activities like joining the dots, puzzles and guessing the number allow increasing a student’s thinking abilities. You can also download the NCERT PDF formats to enjoy learning in the comfort of your place.
It is very important for pupils to understand this chapter as thoroughly as possible because, in Class 3 concepts of subtraction, addition, multiplication and division will be introduced and it will be expected from them to know the basic counting.
The next chapter of the Class 2 Maths NCERT book covers interactive facts on the measurement of objects. Through this lesson, one can understand the amount of weight one can carry.
It also explains these facts through pictures and interesting stories to keep the readers engaged. The lesson comprises exercises that enable the kids to compare the weights of various objects. There are activities in the lesson that allow children to hold various substances in their hands and evaluate the contrast in their weights.
From this lesson, the key takeaways will be interpreting the heavier and lighter objects and comparing their weights. NCERT publishes Mathematics textbooks for interactive content for Class 2 kids to help them grasp complex concepts quickly.
The Class 2 NCERT textbook has made such a complicated topic easier for the students to understand and this is the biggest advantage of referring to NCERT textbooks.
The fourth lesson of the NCERT Maths Class 2 book teaches students to count numbers in the format of tens. It explains the strategy of counting various objects in the bunch of ten, such as books, pens, pencils, etc.
From this segment, students can enhance their analytical skills and thinking abilities. It explains the counting strategy in the form of ten through interesting stories and activities.
Through this chapter, teachers can make students learn to divide objects in bundles of tens and count accordingly. Therefore, it makes the learning process more convenient for young minds.
You can also find this chapter in NCERT PDF format for your convenience. Just click on the relevant links and download the file for imparting knowledge to your child.
This chapter takes counting a step ahead for the students. From counting single digits to counting two-digit numbers. This prepares them for counting larger numbers ahead.
The fifth chapter of NCERT Maths book Class 2 explains the patterns or arrangement of various substances according to a specific order. It helps in enhancing the observation skills in students through the lesson. Moreover, it enables kids to study and identify different patterns that every object follows.
The lesson comprises exercises and numerous activities on studying the patterns of tiles, window grills, clothes, etc. Through filling in the boxes and solving the leafy patterns, kids can enhance their intuitive abilities and critical thinking abilities.
Further, this chapter also comprises number patterns allowing the kids to ace their learning abilities. Through this lesson, parents and teachers can encourage the students to solve problems quickly.
The next lesson of maths magic book Class 2 elaborates the process of interpreting the size and shapes of objects through their imprints. It comprises interesting facts and figures that enables young minds to comprehend and contrast between various substances.
Through this chapter, students can discover different thinking perspectives while drawing objects. Moreover, it enables them to draw various shapes of substances due to their ability to visualise things.
The lesson comprises multiple fun exercises, such as tracing things, evaluating the shapes of objects traced, jotting down the names of shapes, etc. Through this lesson, one can discuss various shapes and ascertain how these objects differ in shapes and sizes. It also focuses on improving the observation skills of your child.
The seventh chapter of NCERT Class 2 Maths textbook talks about the volume of drinking objects. Further, it shows how the size of these vessels can get replaced by the other for drinking purposes.
From this lesson, teachers can impart knowledge to the students about volumes of different vessels like jugs, glass, mugs and buckets, etc. Every vessel has a specific capacity of containing liquid, and this chapter covers this concept.
Through interactive questions, answers and fun activities of learning while filling vessels, kids can gather knowledge by comparing their volumes. It focuses on improving the observation power of the children and the ability to distinguish between various objects. You can also teach your kids this lesson from the NCERT PDF by saving the files from the downloadable links.
This chapter introduces a very concept of volumes to students in a very crude form which holds a lot of significance in higher standards.
The NCERT books for Class 2 Maths eighth lesson focuses on explaining the values of tens and ones. Moreover, it helps in enhancing a child’s ability to count the values mentally at a faster pace. It enables them to ascertain the value of money for specific objects.
The lesson contains interactive examples enabling students to comprehend the topic. Through multiple activities and exercises, they can learn counting strategies quickly. Besides, teachers can encourage them to make token cards for counting values and utilise them for other purposes as well.
The NCERT chapter allows students to understand the value of money and enhance their problem-solving abilities. You can also study this lesson from PDFs by clicking on the downloadable links.
The next lesson of NCERT books for Class 2 Maths explains the days of a month or week. In other words, it educates students about the calendar days of a year. The presence of a calendar while learning this chapter will help them ascertain the number of days a month holds.
Through this segment, your kids can also acquire knowledge about the short and long months in a calendar year. Moreover, it comprises fun question-answer sessions enabling pupils to learn engagingly.
This chapter develops a very basic understanding of calendars which form part of the advanced aptitude of various entrance exams in the future.
You can also teach your kids from the NCERT PDFs to enhance their knowledge.
The next chapter of NCERT Class 2 Maths textbooks describes the methods of counting values by adding the points. Through this lesson, students can add values quickly and perform addition. Addition forms part of basic numeracy skills and is further used to understand multiplication in the higher Classes.
It also improves a student’s problem-solving abilities and reasoning skills through this topic. Moreover, it explains these concepts through interesting stories and examples to keep the readers engaged.
From this lesson, teachers can make students comprehend the game of numbers by explaining a hundred charts. You can also read the NCERT Class 2 Mathematics books to ace your studies. Nowadays, PDFs formats are also readily available online as a ready reference.
The next chapter of the NCERT 2 Class Maths book imparts the primary knowledge of geometry. Further, it elaborates on the types of lines and the method of drawing them. The lesson also shows how students can make a pattern by drawing lines.
From this segment, teachers can encourage the children to draw lines by connecting the dots. In the same manner, they can form shapes of various objects by joining dots, like circles, squares, rectangles, etc.
The concept of lines comes in handy when students go to higher Classes where high-level geometry is introduced. Therefore understanding this chapter lays a foundation for future geometry.
Moreover, this lesson explains the methods of drawing slanting, straight and curved lines by interesting examples. Get the NCERT textbooks for Class 2 Mathematics to enhance the knowledge of your kids. You can also get the PDF as a ready reference for imparting knowledge to students anytime.
Apart from imparting knowledge to students on measuring strategy and geometric patterns, NCERT Maths books have a lot more in store. Their simple and engaging elaborations keep the readers hooked to the chapters.
Furthermore, the NCERT std 2 Maths book contains this lesson which explains the concepts of subtraction and addition of values. With the help of pictorial representation and examples, it describes these concepts more simply.
Teachers can also encourage students to engage in interactive learning activities by making bundles of 10 for every material like beads, sticks, pencils. It enables them to gain practical experiences increasing their abilities to comprehend things.
Through this lesson, they can also encourage students to solve arithmetic problems of addition and subtraction. Moreover, it comprises practical questions which help students to comprehend these concepts quickly.
In the previous lesson, concepts of addition and subtraction of arithmetic problems get elucidated. The next chapter of NCERT books for Class 2 Maths focuses on the methods of measuring the distance between two substances consisting of different units. It explains these methods through the use of fingers, sticks and handspans.
This chapter focuses on making students understand the concepts of distance and length.
The lesson starts with a pictorial example describing these concepts beautifully. Through this lesson, teachers can educate students about measuring the distance among objects correctly. Teachers can also encourage them to utilise reference objects to analyse the distance accurately.
There are an endless number of activities like guessing the correct height, making a mud house and telling a story from a picture, etc. You can also make a team of friends and by drawing lines, discover whose steps are the longest in the group.
The next chapter of NCERT Class 2 Maths textbooks enables teachers to get into a conversation with the students about the movement of birds. It talks about the migration pattern of the birds from one place to the other.
Through this lesson, students can get a count of the number of birds residing in a particular place. Moreover, teachers can encourage remembering and comprehending the patterns in which these different types of birds move.
Also, from this lesson, you can evaluate the number of birds migrating from one place to another by counting them through cards. For instance, one card can get used for a group of birds flocking together.
The lesson further unfolds many interesting exercises and activities using token cards that have been taught in chapter 8. The same method teachers can follow to make students count the number of cranes and swans.
Parents can also engage students in their learning activity by starting simple additions with them and gradually moving towards harder problems. They can follow the same process with their kids for subtraction purposes.
The last chapter of Class 2 Maths NCERT book PDF enables the students in learning the counting strategy for various objects. Through this lesson, the teachers can inspire kids to write down the number of objects available near them.
Through this lesson, one can improve their ability to observe and identify objects at a faster pace. There are also activities like finding letters in the names, counting numbers and writing, guessing the number of shoes, etc. for students to practice.
For instance, some activities are as simple as finding out the glasses of water your Classmates drink every day. These activities keep the readers interested in the topic and increase their knowledge on the subject.
You can also save the PDF format of these lessons to keep your child updated about these concepts.
NCERT publishes books and educational kits to help students excel in their careers. Most professors follow these textbooks and study papers in CBSE schools to explain various concepts to students. Moreover, these books are available in numerous subjects to enhance the learning process, such as Biology, Economics, Hindi and Sanskrit.
You also purchase these textbooks from any website or download their PDF versions to learn anytime. NCERT textbooks are indispensable for students for multiple reasons.
Follows the Latest CBSE Guidelines: NCERT is accountable for publishing educational papers, textbooks and other materials as per the CBSE regulations. Their textbooks get designed as per the syllabus of CBSE. Therefore, you get compact study materials according to the latest norms in your hands.
Use of Simple Language: NCERT textbooks follow a lucid language making the materials comprehensive for every student. They avoid jargon and complex examples in their lessons to make the learning process smooth and hassle-free.
Due to this reason, understanding Mathematics becomes an easy thing even for 2 nd standard kids. With the availability of NCERT PDFs, students can quickly learn things on a digital platform.
Concise and Point-Wise Explanations: NCERT books offer clear elaborations to the readers, leaving no room for confusion. It is the reason why even CBSE recommends its use for school education.
Interactive Questions and the Presence of Multiple Fun Activities: The NCERT books for Class 2 Maths comprises interesting questions and activities that hold the attention of the readers.
Pictorial Representations: These textbooks published by NCERT explain concepts beautifully via pictorial examples. Kids find these representations engaging and find interest in the lesson immediately.
Facilitates Time Management: NCERT textbooks are a one-stop point for students willing to learn diverse concepts. You can find exercises, activities and summarised points in a single place which saves time for students and allows them to focus on other activities too.
Engaging Way of Writing: The reason why NCERT books are sought after is due to the interesting way of expression. They explain facts in a way that becomes easier to comprehend for 1 st or 2 nd standard kids.
Authentic Facts and Figures: The NCERT books for Class 2 Maths comprises authentic information and elaborations of complex notions. Therefore, reading these facts enhances your child’s skills and helps them score well.
Vedantu has been providing expert guidance to students over the years. With a team of highly-qualified professionals and subject matter experts, it is the best online learning platform for students.
Vedantu eases learning cumbersome topics and resolves doubts from the comfort of your homes. You can join their live Classroom to learn about vivid concepts in Mathematics, History and English. Apart from study materials, you can also find the NCERT Maths book Class 2 PDF to ace your studies.
Visit the Play Store and install Vedantu’s mobile application today to enhance your child’s knowledge!
Other Free CBSE Class 2 Maths Study Material
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1. Why is Studying Mathematics Necessary for Scholars?
Mathematics is a subject that improves the analytical and reasoning skills of a student. It also helps in enhancing their ability to observe things and manage their finances in the long term. There are various chapters in Maths that enable them to use counting strategies and improve a student’s problem-solving abilities. Reading NCERT books for Class 2 Maths can be your introduction to these important concepts. NCERT Class 2 Maths textbook forms a strong base for various topics that will be discussed in the higher standards. You can also refer to Vedantu’s NCERT Solutions for Class 2 on Vedantu’s website.
2. How many Chapters do Class 2 Maths Books Contain?
The NCERT books for Class 2 Maths complies with the CBSE guidelines. A Mathematics book of Class 2 standard comprises 15 chapters in total as per the curriculum. They range from counting in tens with other 13 chapters for study. All the chapters are explained in such a way that Class 2 students find them easy to understand. Each concept is explained with relatable examples and illustrations. To make sure books remain interesting and interactive for students many activities have been mentioned in the book. NCERT Class 3 textbooks ensure that learning and fun go hand in hand. NCERT publishes these lessons with the aim to help the students to improve their knowledge and critical thinking abilities.
3. Which Books Can You refer to Studying the Mathematics of Class 2?
There is an ample number of books available in the market for Mathematics of Class 2 standard. However, CBSE approves NCERT textbooks and recommends their use for educational purposes in schools. Besides, NCERT offers textbooks and educational kits to students for engaging in an improved learning process. Also, NCERT books are available in free PDFs on Vedantu, which enhances the e-learning process. Download your NCERT Class 2 Maths book PDF anytime and learn new concepts. This ensures even during the pandemic, the learning process is not hindered.
4. How Can Students Score Well in the Mathematics Class 2 Examination?
You can increase your academic marks in Mathematics through rigorous practice and hard work. A comprehensive explanation of the topics enables young minds to get a better hold of the subject. You can also solve the end number of questions and exercises to achieve a better score in the Class 2 Mathematics exams. Make sure apart from aiming for better marks your goal is to retain what you are learning for a long time because these concepts are to be remembered forever. To ensure this you can refer to Vedantu's NCERT Revision Notes . Click on the downloadable links of NCERT books for Class 2 Maths to save lessons in PDF formats.
You can download NCERT Class 2 Maths book PDF for free from the Vedantu website as well as the mobile application. You need not pay any charges. You can create an account on Vedantu for free. Following that, you simply need to navigate through the webpage, click on the download button, fill in your Login details and get these verified via an OTP and the pdf will be made available to you. Ensure that you download this pdf from the official handles only. Apart from textbooks, Vedantu also supplies NCERT Class 2 Maths Solutions, Maths Class 2 Worksheets and a lot more.
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Mathematics Grade 12 Latest Assignment and Memos for CAPS Curriculum (Syllabus) South Africa: A National Curriculum and Assessment Policy Statement (CAPS) is a single, comprehensive, and concise policy document introduced by the Department of Basic Education for all the subjects listed in the National Curriculum Statement for Grades R – 12. CAPS gives detailed guidance for teachers on what they should teach and how to assess.
On this page you will find Mathematics Grade 12 Assignment and Memos for Term 1, Term 2, Term 3, and Term 4.
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Extra questions of chapter 2 class 11 with answer and hints to the difficult questions. Important and useful math. assignment for the students of class 11
i) Find x and y if (x + 3, 5) = (6, 2x + y) Ans (x = 3, y = - 1)
[Ans a = 2, b = 1]
i) If A = {1, 3, 5, 6} and B = {2, 4}, find A × B and B × A
ii) If A = {1, 2, 3}, B = {3, 4} and C ={1, 3, 5}, find
(a) A × (B ∩ C) (b) (A × B) ∩ (A × C)
iii) If A = {1, 3, 5} , B = {x, y} then represent A × B and B × A in arrow diagrams
Find the domain and range of the relation R defined by R = {(x, x 3 ) : x is a prime number less than 10}
(i) n(A) = 3, n(B) = 4, then find n(A × A × B) Ans: 36
Ans : {(1, 4), (4, 4)}
[Ans: 2 2 ✕ 2 = 2 4 =16 ]
R = {(x, y) : |x - y| is odd}. Write R in roster form.
Ans: R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)
Express the following functions as set of ordered pair and determine their ranges
(i) f : A ➝ R, f(x) = x 2 + 1, where A = {-1, 0, 2, 4}
Ans(i): Range = {1, 2, 5, 17}
Ans(ii) : Range = {2, 4, 6, 8, 10, 12, 14}
Let f : R - {2} ➝ R be defined by
and g : R ➝ R be defined by g(x) = x + 2. Find whether f = g or not.
Also g(x) = x + 2 ⇒ f(x) = g(x)
But Domain of f(x) = R - {2} and the domain of g(x) = R and two functions
are equal only if their domains are equal.
Hence f(x) ≠ g(x)
Find the domain for which the functions f(x) = 2x 2 - 1 and g(x) = 1 - 3x are equal
Here f(x) = g(x)
2x 2 - 1 = 1 - 3x ⇒ 2x 2 + 3x – 2 = 0
⇒ x = -2, 1/2
⇒ f(x) and g(x) are equal on the set {-2, 1 / 2 }
If f, g, h are three functions defined from R to R as follows
(i) f(x) = x 2 (ii) f(x) = sinx (iii) f(x) = x 2 + 1
Find the range of each function
(i) For all values of x, f(x) takes only +ve value. Also if x = 0, then f(x) = 0 . So Range of f(x) = [0, ∞)
(ii) Since -1 ≤ sinx ≤ 1 for all values of x. So Range of f(x) = [-1,1]
(iii) Since x 2 is ≥ 0 ⇒ x 2 + 1 ≥ 1 ⇒ Range of f(x) = [1, ∞)
Find the domain of the function f(x) defined by
f(x) is defined for all x satisfying
4 - x ≥ 0 and x 2 - 1 > 0
x ≤ 4 and (x - 1)(x + 1) > 0
For (x - 1)(x + 1) = 0, the critical points are x = 1 and x = -1
x ∈ (- ∞, -1) U (1, ∞), But x ≤ 4
Therefore : D f = (- ∞, -1) U (1, 4]
f(x) is defined ∀ x ∊ R ⇒ D f = R
By using quadratic formula we get
Now x is defined (real) if 1 - 4y 2 ≥ 0 and y ≠ 0
⇒ 4y 2 - 1 ≤ 0 ⇒ (2y + 1)(2y - 1) ≤ 0
Critical points are y = -1/2 and 1/2
(2y + 1)(2y - 1) ≤ 0 and y ≠ 0 ⇒ y ∈ [ -1/2, 1/2] - {0}
⇒ R f = [ -1/2, 1/2] - {0}
For range find x in terms of y we get
f (x) is defined for all real numbers except at x = 3
Therefore D f = R - 3
y = f(x) = x - 3
But x = 3 ∉ D f ⇒ y = 3 + 3 = 6 ∉ R f ⇒ R f = R - {6}
Find the domain and range of the following functions
i) f(x) = |x - 1| Ans : [D = R] R = [0, ∞)
Ans: Domain = R - {2}, Range = R - {-1}
Ans: Domain = [-4, 4], Range = [0, 4]
Ans: Domain = (-3, 3) R f = (-∞, -1] ⋃ [1, ∞)
f(x) is defined if 9 - x 2 > 0 ⇒ x < ± 3
- 3 < x < 3 ⇒ x ∊ (- 3, 3) ⇒ D f = (- 3, 3)
⇒ 9y 2 - x 2 y 2 = 9
⇒ x 2 y 2 = 9y 2 - 9
Critical points are : -1, 0, 1
x is defined if 9y 2 - 9 ≥ 0 and y ≠ 0
x is defined if y 2 - 1 ≥ 0 and y ≠ 0
x is defined if y 2 ≥ 1 ⇒ y ≥ ± 1 and y ≠ 0
x is defined if y ≤ -1, y ≥ 1 and y ≠ 0
⇒ y ∈ (-∞, -1] ⋃ [1, ∞) ⇒ R f = (-∞, -1] ⋃ [1, ∞)
Ans: D f = (-∞, - 3] ⋃ (-1, 3]
Draw the graph of the following write its range
Find the domain of the following function
Critical points are y = 5, y = 3
y = |x| = 5 and y = |x| = 3 ⇒ x = 土 5, x = 土 3
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Question 20 : (DAV SP 2023)
Find the inverse relation R -1 in each of the following
(i) R = {(1,2), (1,3), (2, 3), (3, 2), (5, 6)
Ans {(2,1), (3,1), (3,2), (2,3), (6,5)}
(ii) R = {(x, y) : x, y ∈ N, x + 2y = 8
Ans {(3, 2), (2, 4), (1,6)
Solution If x = 2, y = 3 ⇒ 2 + 2 × 3 = 8 ⇒ (2, 3) ∈ R
If x = 4, y = 2 ⇒ 4 + 2 × 2 = 8 ⇒ (4, 2) ∈ R
If x = 6, y = 1 ⇒ 6 + 2 × 1 = 8 ⇒ (6, 1) ∈ R
R = {(2, 3), (4, 2), (6, 1)} ⇒ {(3, 2), (2, 4), (1, 6)}
Let f : R ➝ R be a function given by f(x) = x 2 + 1. Find f -1 {10, 37}
If f(x) = y then x = f -1 (y)
f -1 (10) = x ⇒ f(x) = 10 ⇒ x 2 + 1 = 10
⇒ x 2 = 9 ⇒ x = 土 3
f -1 (37) = x ⇒ f(x) = 37 ⇒ x 2 + 1 = 37
⇒ x 2 = 36 ⇒ x = 土 6
f -1 {10, 37} = {-3, 3, - 6, 6}
Let A = {-2, -1, 0, 1, 2} and f : A ➝ Z be a function defined by f(x) = x 2 - 2x - 3. Find
(i) Range of f i.e. f(A) (ii) Pre - image of 6, - 3 and 5
(i) f(A) = {f(-2), f(-1), f(0), f(1), f(2) } = {5, 0, - 3, - 4, - 3} = {- 4,- 3, 0, 5}
(ii) Let Pre - image of 6 = x ⇒ f -1 (6) = x ⇒ f(x) = 6
⇒ x 2 - 2x - 3 = 6 ⇒ x 2 - 2x - 9= 0
There is no real value of x which satisfies this equation.
So Pre - image of 6 = Φ
Let Pre - image of - 3 = x ⇒ f -1 (-3) = x ⇒ f(x) = - 3
⇒ x 2 - 2x - 3 = - 3 ⇒ x 2 - 2x = 0 ⇒ x = {0, 2}
Let Pre - image of 5 = x ⇒ f -1 (5) = x ⇒ f(x) = 5
⇒ x 2 -2x - 3 = 5 ⇒ x 2 - 2x - 8 = 0 ⇒ x = {-2, 4}
(i) f(x) = 3x 4 - 5x 2 + 9, find f(x - 1)
Ans [3x 4 - 12x 3 + 13x 2 - 2x + 7]
[Ans x = -1, 2/3]
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