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CBSE Class 9 Mathematics Case Study Questions
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If you’re looking for a comprehensive and reliable study resource and case study questions for class 9 CBSE, myCBSEguide is the perfect door to enter. With over 10,000 study notes, solved sample papers and practice questions, it’s got everything you need to ace your exams. Plus, it’s updated regularly to keep you aligned with the latest CBSE syllabus . So why wait? Start your journey to success with myCBSEguide today!
Significance of Mathematics in Class 9
Mathematics is an important subject for students of all ages. It helps students to develop problem-solving and critical-thinking skills, and to think logically and creatively. In addition, mathematics is essential for understanding and using many other subjects, such as science, engineering, and finance.
CBSE Class 9 is an important year for students, as it is the foundation year for the Class 10 board exams. In Class 9, students learn many important concepts in mathematics that will help them to succeed in their board exams and in their future studies. Therefore, it is essential for students to understand and master the concepts taught in Class 9 Mathematics .
Case studies in Class 9 Mathematics
A case study in mathematics is a detailed analysis of a particular mathematical problem or situation. Case studies are often used to examine the relationship between theory and practice, and to explore the connections between different areas of mathematics. Often, a case study will focus on a single problem or situation and will use a variety of methods to examine it. These methods may include algebraic, geometric, and/or statistical analysis.
Example of Case study questions in Class 9 Mathematics
The Central Board of Secondary Education (CBSE) has included case study questions in the Class 9 Mathematics paper. This means that Class 9 Mathematics students will have to solve questions based on real-life scenarios. This is a departure from the usual theoretical questions that are asked in Class 9 Mathematics exams.
The following are some examples of case study questions from Class 9 Mathematics:
Class 9 Mathematics Case study question 1
There is a square park ABCD in the middle of Saket colony in Delhi. Four children Deepak, Ashok, Arjun and Deepa went to play with their balls. The colour of the ball of Ashok, Deepak, Arjun and Deepa are red, blue, yellow and green respectively. All four children roll their ball from centre point O in the direction of XOY, X’OY, X’OY’ and XOY’ . Their balls stopped as shown in the above image.
Answer the following questions:
Answer Key:
Class 9 Mathematics Case study question 2
- Now he told Raju to draw another line CD as in the figure
- The teacher told Ajay to mark ∠ AOD as 2z
- Suraj was told to mark ∠ AOC as 4y
- Clive Made and angle ∠ COE = 60°
- Peter marked ∠ BOE and ∠ BOD as y and x respectively
Now answer the following questions:
- 2y + z = 90°
- 2y + z = 180°
- 4y + 2z = 120°
- (a) 2y + z = 90°
Class 9 Mathematics Case study question 3
- (a) 31.6 m²
- (c) 513.3 m³
- (b) 422.4 m²
Class 9 Mathematics Case study question 4
How to Answer Class 9 Mathematics Case study questions
To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.
Students need to be careful while solving the Class 9 Mathematics case study questions. They should not make any assumptions and should always check their answers. If they are stuck on a question, they should take a break and come back to it later. With some practice, the Class 9 Mathematics students will be able to crack case study questions with ease.
Class 9 Mathematics Curriculum at Glance
At the secondary level, the curriculum focuses on improving students’ ability to use Mathematics to solve real-world problems and to study the subject as a separate discipline. Students are expected to learn how to solve issues using algebraic approaches and how to apply their understanding of simple trigonometry to height and distance problems. Experimenting with numbers and geometric forms, making hypotheses, and validating them with more observations are all part of Math learning at this level.
The suggested curriculum covers number systems, algebra, geometry, trigonometry, mensuration, statistics, graphing, and coordinate geometry, among other topics. Math should be taught through activities that include the use of concrete materials, models, patterns, charts, photographs, posters, and other visual aids.
CBSE Class 9 Mathematics (Code No. 041)
Class 9 Mathematics question paper design
The CBSE Class 9 mathematics question paper design is intended to measure students’ grasp of the subject’s fundamental ideas. The paper will put their problem-solving and analytical skills to the test. Class 9 mathematics students are advised to go through the question paper pattern thoroughly before they start preparing for their examinations. This will help them understand the paper better and enable them to score maximum marks. Refer to the given Class 9 Mathematics question paper design.
QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)
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Class 9 is an important milestone in a student’s life. It is the last year of high school and the last chance to score well in the CBSE board exams. myCBSEguide is the perfect platform for students to get started on their preparations for Class 9 Mathematics. myCBSEguide provides comprehensive study material for all subjects, including practice questions, sample papers, case study questions and mock tests. It also offers tips and tricks on how to score well in exams. myCBSEguide is the perfect door to enter for class 9 CBSE preparations.
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MATHS CLASS IX CASE STUDY BASED QUESTIONS
FOR ANNUAL EXAM 2020-21
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Case study questions for class 9 maths number system
Case study based questions class 9 | case study based on coordinate geometry | cbse class 9 Maths · Class 9 Maths | Chapter 2 | Introduction Part
Number System
To make your preparation ultra-pro-max easy, below given are the CBSE Case Study Questions for Class 9, of 3 of your very important subjects - English, Maths &
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Class 9 Maths: Case Study Questions of Chapter 1 Real Numbers PDF Download
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Case study Questions on Class 9 Mathematics Chapter 1 are very important to solve for your exam. Class 9 Maths Chapter 1 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving case study-based questions for Class 9 Maths Chapter 1 Real Numbers
In CBSE Class 9 Maths Paper, Students will have to answer some questions based on Assertion and Reason. There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.
Real Numbers Case Study Questions With answers
Here, we have provided case-based/passage-based questions for Class 9 Maths Chapter 1 Real Numbers
Case Study/Passage-Based Questions
Case Study 1:
A Mathematics Exhibition is being conducted in your school and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.
Observe the following factor tree and answer the following:
1. What will be the value of x?
Answer: b) 13915
2. What will be the value of y?
Answer: c) 11
3. What will be the value of z?
Answer: b) 23
4. According to the Fundamental Theorem of Arithmetic 13915 is a
a) Composite number
b) Prime number
c) Neither prime nor composite
d) Even number
Answer: a) Composite number
5. The prime factorization of 13915 is
a) 5 × 11 3 × 13 2
b) 5 × 11 3 × 23 2
c) 5 × 11 2 × 23
d) 5 × 11 2 × 13 2
Answer: c) 5 × 112 × 23
Case Study 2:
Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
(i) For what value of n, 4 n ends in 0?
(a) 10 (b) when n is even (c) when n is odd (d) no value of n
Answer: (d) no value of n3
(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, an is a rational number?
(a) when n is any even integer (b) when n is any odd integer (c) for all n > 1 (d) only when n=0
Answer: (c) for all n > 1
(iii) If x and y are two odd positive integers, then which of the following is true?
(a) x 2 +y 2 is even (b) x 2 +y 2 is not divisible by 4 (c) x 2 +y 2 is odd (d) both (a) and (b)
Answer: (d) both (a) and (b)
(iv) The statement ‘One of every three consecutive positive integers is divisible by 3’ is
(a) always true (b) always false (c) sometimes true (d) None of these
Answer:(a) always true
(v) If n is any odd integer, then n 2 – 1 is divisible by
(a) 22 (b) 55 (c) 88 (d) 8
Answer: (d) 8
Hope the information shed above regarding Case Study and Passage Based Questions for Class 9 Mathematics Chapter 1 Real Numbers with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 9 Maths Real Numbers Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate
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Worksheets Class 9 Mathematics Number System Pdf Download
Students can download here free printable Worksheets Class 9 Mathematics Number System Pdf Download . These Worksheets for Grade 9 Mathematics Number System are really important as they have been prepared based on the current year’s NCERT Books for Class 9 Mathematics Number System. Our faculty has ensured that the printable worksheets for Mathematics Number System CBSE Class 9 cover all important points which are explained in various chapters. The questions which can come in your school tests have also been considered while designing these CBSE NCERT printable worksheets for Class 9 Mathematics Number System with solutions and answers. You can click on the links given below and download Pdf worksheets for Mathematics Number System class 9 for free. All Kendriya Vidyalaya Class 9 Mathematics Number System worksheets and test papers with answers are given below for free download for students.
Class 9 Mathematics Number System Worksheets Pdf Download
We have provided below links to CBSE NCERT KVS Worksheets for Class 9 Mathematics Number System . All worksheets contain subject topic-wise Mathematics Number System Class 9 important questions and answers designed based on the latest syllabus for this academic session. All test sheets and solved question banks for Class 9 and NCERT Worksheets for Mathematics Number System Class 9 have been suggested by various schools and can be really helpful for Class 9 students. Solving these practice sheets daily will help you to revise and practice all important topics and prepare for various tests and examinations.
Subjectwise Worksheets for Class 9 Mathematics Number System
Benefits of Solving Class 9 Mathematics Number System Worksheets
- These worksheets for Grade 9 Mathematics Number System contain important questions which can be asked in various school level tests and examinations.
- All concepts given in your NCERT book for Class 9 Mathematics Number System have been covered in these Pdf worksheets
- All Mathematics Number System worksheets for Grade 9 have been solved so that you can compare your answers with the solutions provided by our teachers.
- CBSE, NCERT and KVS Mathematics Number System students should download these practice sheets and improve your knowledge
- All worksheets for Class 9 Mathematics Number System have been provided for free
- With the help of Class 9 Mathematics Number System question banks and workbooks, you will be able to improve your understanding of various topics and get better score in exams
FAQs by Students of Class 9 Mathematics Number System
At https://www.ncertbooksolutions.com, all worksheets for Class 9 Mathematics Number System have been provided covering all topics which you can easily download in Pdf format.
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Case Study Questions of Class 9(IX) Science
Download PDF Case Study Questions for Class 9 Science to prepare for the upcoming CBSE Class 9 Exams Exam 2022-23. With the help of our well-trained and experienced faculty, we provide solved examples and detailed explanations for the recently added Class 9 Science case study questions .
Class 9 Science: Case Study Questions
The inclusion of case study questions in Class 9 science CBSE is a great way to engage students in critical thinking and problem-solving. By working through real-world scenarios, Class 9 Science students will be better prepared to tackle challenges they may face in their future studies and careers. Class 9 Science Case study questions also promote higher-order thinking skills, such as analysis and synthesis. In addition, case study questions can help to foster creativity and innovation in students. As per the recent pattern of the Class 9 Science examination, a few questions based on case studies/passages will be included in the CBSE Class 9 Science Paper. There will be a paragraph presented, followed by questions based on it.
Chapterwise Case Study Questions for Class 9 Science
- Case Study Questions for Chapter 1 Matter in Our Surroundings
- Case Study Questions for Chapter 2 Is Matter Around Us Pure?
- Case Study Questions for Chapter 3 Atoms and Molecules
- Case Study Questions for Chapter 4 Structure of Atom
- Case Study Questions for Chapter 5 The Fundamental Unit of Life
- Case Study Questions for Chapter 6 Tissues
- Case Study Questions for Chapter 7 Diversity in Living Organisms
- Case Study Questions for Chapter 8 Motion
- Case Study Questions for Chapter 9 Force and Laws of Motion
- Case Study Questions for Chapter 10 Gravitation
- Case Study Questions for Chapter 11 Work and Energy
- Case Study Questions for Chapter 12 Sound
- Case Study Questions for Chapter 13 Why do we Fall ill
- Case Study Questions for Chapter 14 Natural Resources
- Case Study Questions for Chapter 15 Improvement in Food Resources
You can find a wide range of solved case studies on cbseexperts, covering various topics and concepts. Class 9 Science case studies are designed to help you understand the application of various concepts in real-life situations.
A look at the Class 9 Science Syllabus
Unit I: Matter-Nature and Behaviour
Definition of matter; solid, liquid, and gas; characteristics – shape, volume, density; change of statementing (absorption of heat), freezing, evaporation (cooling by evaporation), condensation, sublimation.
Nature of matter: Elements, compounds, and mixtures. Heterogeneous and homogenous mixtures, colloids, and suspensions. Physical and chemical changes (excluding separating the components of a mixture).
Particle nature and their basic units: Atoms and molecules, Law of Chemical Combination, Chemical formula of common compounds, Atomic and molecular masses.
Structure of atoms: Electrons, protons and neutrons, Valency, Atomic Number and Mass Number, Isotopes and Isobars.
Unit II: Organization in the Living World
Cell – Basic Unit of life: Cell as a basic unit of life; prokaryotic and eukaryotic cells, multicellular organisms; cell membrane and cell wall, cell organelles and cell inclusions; chloroplast, mitochondria, vacuoles, endoplasmic reticulum, Golgi apparatus; nucleus, chromosomes – basic structure, number.
Tissues, Organs, Organ System, Organism: Structure and functions of animal and plant tissues (only four types of tissues in animals; Meristematic and Permanent tissues in plants).
Unit III: Motio n, Force, and Work
Motion: Distance and displacement, velocity; uniform and non-uniform motion along a straight line; acceleration, distance-time and velocity-time graphs for uniform motion and uniformly accelerated motion, elementary idea of uniform circular motion.
Force and Newton’s laws: Force and Motion, Newton’s Laws of Motion, Action and Reaction forces, Inertia of a body, Inertia and mass, Momentum, Force and Acceleration.
Gravitation: Gravitation; Universal Law of Gravitation, Force of Gravitation of the earth (gravity), Acceleration due to Gravity; Mass and Weight; Free fall. Floatation: Thrust and Pressure. Archimedes’ Principle; Buoyancy.
Work, Energy and Power: Work done by a Force, Energy, power; Kinetic and Potential energy; Law of conservation of energy (excluding commercial unit of Energy).
Sound: Nature of sound and its propagation in various media, speed of sound, range of hearing in humans; ultrasound; reflection of sound; echo.
Unit IV: Food Production
Plant and animal breeding and selection for quality improvement and management; Use of fertilizers and manures; Protection from pests and diseases; Organic farming.
Books for Class 9 Science Exams
The rationale behind Science
Science is crucial for Class 9 students’ cognitive, emotional, and psychomotor development. It encourages curiosity, inventiveness, objectivity, and aesthetic sense.
In the upper primary stage, students should be given a variety of opportunities to engage with scientific processes such as observing, recording observations, drawing, tabulating, plotting graphs, and so on, whereas in the secondary stage, abstraction and quantitative reasoning should take a more prominent role in science teaching and learning. As a result, the concept of atoms and molecules as matter’s building units, as well as Newton’s law of gravitation, emerges.
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CBSE Class 9 Maths Important Questions for Chapter 1 - Number System
- CBSE Class 9 Maths Important Q...
CBSE Class 9 Maths Important Questions Chapter 1 - Number System Free PDF Download
Chapter 1 of Mathematics Class 9 deals with an introduction to various other topics. Those who are planning to pursue a career in mathematics should prepare well for this chapter. Mathematics is the subject to deal with practical life calculations and Class 9 Maths Chapter 1 Important Questions will help set a good base for the students.
Based on these crucial questions, students can prepare for mathematics finals without any hassle. Class 9 is the base to prepare well for 10th boards. Hence students need to master their concepts and utilise their time efficiently. According to CBSE’s basic guidelines, these Important Questions for Class 9 Maths Chapter 1 Number System are prepared. Therefore, students need not worry and search for such questions anywhere else. Number System is the first chapter in Mathematics 9th standard that deals with whole numbers, rational and irrational numbers, and integers. Learn in detail about the topics covered under the Important Questions for Class 9 Maths number system and complete syllabus prepared by experts. Vedantu provides students with a Free PDF download option for all the CBSE Solutions of updated CBSE textbooks . Subjects like Science, Maths, English will become easy to study if you have access to NCERT Class 9 Science , Maths solutions, and solutions of other subjects that are available on Vedantu only.
Download CBSE Class 9 Maths Important Questions 2022-23 PDF
Also, check CBSE Class 9 Maths Important Questions for other chapters:
Important Topics Covered in Class 9 Maths Chapter 1
Introduction to number system
Irrational Number
Real Number and Their Decimal Expansion
Representation of Real Number on Number Line
Operations on Real Number
Laws of Exponents for Real Number
Download PDF of CBSE Class 9 Maths Important Questions for Chapter 1 - Number System
Study important questions for class 9 maths chapter 1 – number systems.
1 Marks Questions
1. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is irrational number.
Ans: We know that the square root of every positive integer will not yield an integer.
We know that \[\sqrt{4}\] is $2$, which is an integer. But, $\sqrt{7}$ or $\sqrt{10}$ will give an irrational number.
Therefore, we conclude that the square root of every positive integer is not an irrational number.
2. Write three numbers whose decimal expansions are non-terminating non-recurring.
Ans: The three numbers that have their expansions as non-terminating on recurring decimals are given below.
0.04004000400004....
0.07007000700007....
0.13001300013000013....
3. Find three different irrational numbers between the rational numbers $\frac{\text{5}}{\text{11}}$ and $\frac{\text{9}}{\text{11}}$ .
Ans: Let us convert $\frac{5}{11}$ and $\frac{9}{11}$ into decimal form, to get
$\frac{5}{7}=0.714285....and\frac{9}{11}=0.818181....$
Three irrational numbers that lie between $0.714285....$ and $0.818181....$ are:
0.73073007300073...
0.74074007400074....
0.76076007600076....
4. Which of the following rational numbers have terminating decimal representation?
$(i)\frac{3}{5}$
$(ii)\frac{2}{13} $
$(iii)\frac{40}{27} $
$(iv)\frac{23}{7}$
Ans: $(i)\frac{3}{5}$
5. How many rational numbers can be found between two distinct rational numbers?
(iv) Infinite
Ans: (iv) Infinite
6. The value of $\left( \text{2+}\sqrt{\text{3}} \right)\left( \text{2-}\sqrt{\text{3}} \right)$ in
(i) $\text{1}$
(ii) $\text{-1}$
(iii) $\text{2}$
(iv) none of these
Ans: (i) $1$
7. ${{\left( \text{27} \right)}^{\text{-2/3}}}$ is equal to
(i) $\text{9}$
(ii) $\text{1/9}$
(iii) $\text{3}$
Ans: (ii) $1/9$
8. Every natural number is
(i) not an integer
(ii) always a whole number
(iii) an irrational number
(iv) not a fraction
Ans: (ii) always a whole number
9. Select the correct statement from the following
(i) $\frac{\text{7}}{\text{9}}\text{}\frac{\text{4}}{\text{5}}$
(ii) $\frac{\text{2}}{\text{6}}\text{}\frac{\text{3}}{\text{9}}$
(iii) $\frac{\text{-2}}{\text{3}}\text{}\frac{\text{-4}}{\text{5}}$
(iv)$\frac{\text{-5}}{\text{7}}\text{}\frac{\text{-3}}{\text{4}}$
Ans: (iii) $\frac{-2}{3}>\frac{-4}{5}$
10. $\text{7}\text{.}\overline{\text{2}}$ is equal to
(i) $\frac{\text{68}}{\text{9}}$
(ii) $\frac{\text{64}}{\text{9}}$
(iii) $\frac{\text{65}}{\text{9}}$
(iv) $\frac{\text{63}}{\text{9}}$
Ans: (iii) $\frac{65}{9}$
11. $\text{0}\text{.83458456}......$ is
(i) an irrational number
(ii) rational number
(iii) a natural number
(iv) a whole number
Ans: (i) an irrational number
12. A terminating decimal is
(i) a natural number
(ii) a rational number
(iii) a whole number
(iv) an integer.
Ans: (ii) a rational number
13. The $\frac{\text{p}}{\text{q}}$ form of the number $\text{0}\text{.8}$ is
(i) $\frac{\text{8}}{\text{10}}$
(ii) $\frac{\text{8}}{\text{100}}$
(iii) $\frac{\text{1}}{\text{8}}$
(iv) $\text{1}$
Ans: (i) $\frac{8}{10}$
14. The value of $\sqrt[\text{3}]{\text{1000}}$ is
Ans: (ii) $10$
15. The sum of rational and an irrational number
(i) may be natural
(ii) may be irrational
(iii) is always irrational
(iv) is always rational
Ans: (iii) is always rational
16. The rational number not lying between $\frac{\text{3}}{\text{5}}$ and $\frac{\text{2}}{\text{3}}$ is
(i) $\frac{\text{49}}{\text{75}}$
(ii) $\frac{\text{50}}{\text{75}}$
(iii) \[\frac{\text{47}}{\text{75}}\]
(iv) $\frac{\text{46}}{\text{75}}$
Ans: (B) $\frac{50}{75}$
17. $\text{0}\text{.12}\overline{\text{3}}$ is equal to
(i) $\frac{\text{122}}{\text{90}}$
(ii) $\frac{\text{122}}{\text{100}}$
(iii) $\frac{\text{122}}{\text{99}}$
(iv) None of these
Ans: (a) $\frac{122}{990}$
18. The number ${{\left( \text{1+}\sqrt{\text{3}} \right)}^{\text{2}}}$ is
(a) natural number
(b) irrational number
(c) rational number
(d) integer
Ans: (b) irrational number
19. The simplest form of $\sqrt{\text{600}}$ is
(i) $\text{10}\sqrt{\text{60}}$
(ii) $\text{100}\sqrt{\text{6}}$
(iii) $\text{20}\sqrt{\text{3}}$
(iv) $\text{10}\sqrt{\text{6}}$
Ans: (D) $10\sqrt{6}$
20. The value of $\text{0}\text{.}\overline{\text{23}}\text{+0}\text{.}\overline{\text{22}}$ is
(i) $\text{0}\text{.4}\overline{\text{5}}$
(ii) $\text{0}\text{.4}\overline{\text{4}}$
(iii) $\text{0}\text{.}\overline{\text{45}}$
(iv) $\text{0}\text{.}\overline{\text{44}}$
Ans: (A) $0.\overline{23}=0.232323....$
$0.\overline{22}=0.222222....$
$0.\overline{23}+0.\overline{22}=0.454545....$
$=0.\overline{45}$
21. The value of ${{\text{2}}^{\frac{\text{1}}{\text{3}}}}\text{ }\!\!\times\!\!\text{ }{{\text{2}}^{\text{-}\frac{\text{4}}{\text{3}}}}$ is
(i) $\text{2}$
(ii) $\frac{\text{1}}{\text{2}}$
Ans: (B) ${{2}^{\frac{1}{3}}}\times {{2}^{-\frac{4}{3}}}={{2}^{\frac{1}{3}-\frac{4}{3}}}={{2}^{\frac{1-4}{3}}}={{2}^{-\frac{3}{3}}}$
22. \[\text{16}\sqrt{\text{13}}\text{ }\!\!\div\!\!\text{ 9}\sqrt{\text{52}}\] is equal to
(i) $\frac{\text{3}}{\text{9}}$
(ii) $\frac{\text{9}}{\text{8}}$
(iii) \[\frac{\text{8}}{\text{9}}\]
Ans: $16\sqrt{13}\div 9\sqrt{52}$
$\frac{16\sqrt{13}}{9\sqrt{52}}=\frac{16}{9}\sqrt{\frac{13}{52}}=\frac{8}{9}$
23. $\sqrt{\text{8}}$ is an
(i) natural number
(iii) integer
(iv) irrational number
Ans: (D) $\sqrt{8}$ is an irrational number
$\therefore \sqrt{4\times 2}=2\sqrt{2}$
2 Marks Questions
1. Is zero a rational number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$?
Ans: Consider the definition of a rational number. A rational number is the one that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.
Zero can be written as $\frac{0}{1},\frac{0}{2},\frac{0}{3},\frac{0}{4},\frac{0}{5}......$
So, we arrive at the conclusion that $0$ can be written in the form $\frac{p}{q}$, where $q$is any integer.
Therefore, zero is a rational number.
2. Find six rational numbers between $3$ and $4$.
Ans: We know that there are infinite rational numbers between any two numbers.
A rational number is the one that can be written in the form of $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.
We know that the numbers $3.1,3.2,3.3,3.4,3.5$ and $3.6$ all lie between $3$ and $4$.
We need to rewrite the numbers $3.1,3.2,3.3,3.4,3.5$ and $3.6$ in $\frac{p}{q}$ form to get the rational numbers between $3$ and $4$.
So, after converting we get $\frac{32}{10},\frac{32}{10},\frac{33}{10},\frac{34}{10},\frac{35}{10},$ and $\frac{36}{10},$ into lowest fractions.
On converting the fractions into lowest fractions, we get $\frac{16}{5},\frac{17}{5},\frac{7}{2}$ and $\frac{18}{5}$.
Therefore, six rational numbers between $3$ and $4$are \[\frac{31}{10},\frac{16}{5},\frac{33}{10},\frac{17}{5},\frac{7}{2}\] and $\frac{18}{5}$.
3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.
We know that the numbers $\frac{3}{5}$ and $\frac{4}{5}$ can also be written as $0.6\text{ and }0.8$.
We can conclude that the numbers$0.61,0.62,0.63,0.64$ and $0.65$ in \[\frac{p}{q}\] form to get the rational numbers between $3\text{ and }4$.
So, after converting, we get $\frac{61}{100},\frac{62}{100},\frac{63}{100},\frac{64}{100}\text{ and }\frac{65}{100}$.
We can further convert the rational numbers $\frac{62}{100},\frac{64}{100}\text{ and }\frac{65}{100}$ into lowest fractions.
On converting the fractions, we get $\frac{31}{50},\frac{16}{25}\text{ and }\frac{13}{20}$.
Therefore, six rational numbers between $3\text{ and }4$ are $\frac{61}{100},\frac{31}{50},\frac{63}{100},\frac{16}{50}\text{ and }\frac{13}{50}$.
4. Show how $\sqrt{5}$ can be represented on the number line.
Ans: According to Pythagoras theorem, we can conclude that
${{\left( \sqrt{5} \right)}^{2}}={{\left( 2 \right)}^{2}}+{{\left( 1 \right)}^{2}}$.
We need to draw a line segment $AB\text{ of }1$unit on the number line. Then draw a straight line segment $BC\text{ of }2$ units. Then join the points $C$ and $A$, to form a line segment $BC$.
Then draw the arc $ACD$, to get the number $\sqrt{5}$ on the number line.
5. You know that $\frac{1}{7}=0.142857....$. Can you predict what the decimal expansion of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ are, without actually doing the long division? If so, how?
(Hint: Study the remainder while finding the value of $\frac{1}{7}$ carefully.)
Ans: We are given that $\frac{1}{7}=0.\overline{142857}$ or $\frac{1}{7}=0.142857....$
We need to find the value of \[\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\text{ and }\frac{6}{7}\], without performing long division.
We know that \[\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\text{ and }\frac{6}{7}\] can be rewritten as
$2\times \frac{1}{7},3\times \frac{1}{7},4\times \frac{1}{7},5\times \frac{1}{7}\text{ and }6\times \frac{1}{7}$.
On substituting value of $\frac{1}{7}$ as $0.142857....$, we get
$2\times \frac{1}{7}=2\times 0.142857....=0.285714.... $
$3\times \frac{1}{7}=3\times 0.142857....=0.428571.... $
$4\times \frac{1}{7}=4\times 0.142857....=0.571428....$
$5\times \frac{1}{7}=5\times 0.142857....=0.714285.... $
$6\times \frac{1}{7}=6\times 0.142857....=0.857142.... $
Therefore, we conclude that, we can predict the values of \[\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7}\text{ and }\frac{6}{7}\], without performing long division, to get
\[\frac{2}{7}=0.\overline{285714},\frac{3}{7}=0.\overline{428571},\frac{4}{7}=0.\overline{571428},\frac{5}{7}=0.\overline{714285},\frac{6}{7}=0.\overline{857142}\]
6. Express $0.99999....$in the form $\frac{p}{q}$. Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmates.
Ans: Let $x=0.99999....\text{ }......(a)$
We need to multiply both sides by $10$ to get
$10x=9.9999....\text{ }......(b)$
We need to subtract $(a)\text{ from }(b)$, to get
10x=9.99999....
x=0.99999....
_____________
We can also write $9x=9\text{ as }x=\frac{9}{9}\text{ or }x=1$.
Therefore, on converting $0.99999....$ in the $\frac{p}{q}$ form, we get the answer as $1$.
Yes, at a glance we are surprised at our answer. But the answer makes sense when we observe that $0.99999....$ goes on forever. So there is no gap between $1$ and $0.9999....$ and hence they are equal.
7. Visualize $3.765$ on the number line using successive magnification.
Ans: We know that the number $3.765$ will lie between $3.764\text{ and }3.766$.
We know that the number $3.764$and $3.766$ will lie between $3.76\text{ and }3.77$.
We know that the number $3.76\text{ and }3.77$. will lie between $3.7\text{ and }3.8$.
We know that the number $3.7\text{ and }3.8$ will lie between $3\text{ and }~4$.
Therefore, we can conclude that we need to use the successive magnification, after locating numbers $3\text{ and }~4$ on the number line
(Image will be uploaded soon)
8. Visualize $4.\overline{26}$ on the number line, upto $4$decimal places.
Ans: We know that the number $4.\overline{26}$ can also be written as$4.262....$ .
We know that the number $4.262....$ will lie between $4.261\text{ and }4.263$.
We know that the number $4.261\text{ and }4.263$ will lie between $4.26\text{ and }4.27$.
We know that the number $4.26\text{ and }4.27$ will lie between $4.2\text{ and }4.3$.
We know that the number $4.2\text{ and }4.3$ will lie between $4\text{ and }5$.
Therefore, we can conclude that we need to use the successive magnification, after locating numbers $4\text{ and }5$ on the number line.
9. Recall, $\pi $is defined as the ratio of the circumference (say $c$) of a circle of its diameter (say $d$). That is, $\pi =\frac{c}{d}$. This seems to contradict the fact that $\pi $ is irrational. How you resolve this contradiction?
Ans: We know that when we measure the length of the line or a figure by using a scaleneory device, we do not get an exact measurement. In fact, we get an approximate rational value. So, we are not able to realize that either the circumference ($c$) or diameter ($d$) of a circle is irrational.
Therefore, we can conclude that as such there is not any contradiction regarding the value of $\pi $ and we realize that the value of $\pi $ is irrational.
10. Represent $9.3$ on the number line.
Ans: Mark the distance $9.3$ units from a fixed point $A$ on a given line to obtain a point $B$ such that $AB=9.3$ units. From $B$ mark a distance of $1$ unit and call the new point as $C$. Find themid-point of $AC$ and call that point as $O$. Draw a semi-circle with centre $O$ and radius $OC=5.15$units. Draw a line perpendicular to $AC$ passing through $B$ cutting the semi-circle at $D$.
Then $BD=\sqrt{9.3}$.
11. Find (i) ${{64}^{\frac{1}{5}}}$ (ii) ${{32}^{\frac{1}{5}}}$ (iii) ${{125}^{\frac{1}{3}}}$
(i) ${{64}^{\frac{1}{2~}}}$
We know that${{a}^{\frac{1}{n}}}=\sqrt[n]{a},\text{ where }a>0$
We conclude that ${{64}^{\frac{1}{2~}}}$can also be written as $\sqrt[2]{64}=\sqrt[2]{8\times 8}$
$\sqrt[2]{64}=\sqrt[2]{8\times 8}$$=8$
Therefore, the value of ${{64}^{\frac{1}{2~}}}$will be $8$.
(ii) ${{32}^{\frac{1}{5}}}$
We conclude that ${{32}^{\frac{1}{5}}}$can also be written as $\sqrt[5]{32}=\sqrt[5]{2\times 2\times 2\times 2\times 2}$
$\sqrt[5]{32}=\sqrt[5]{2\times 2\times 2\times 2\times 2}=2$
Therefore, the value of ${{32}^{\frac{1}{5}}}$will be $2$.
(iii) ${{125}^{\frac{1}{3}}}$
We conclude that ${{125}^{\frac{1}{3}}}$can also be written as $\sqrt[3]{125}=\sqrt[3]{5\times 5\times 5}$
$\sqrt[3]{125}=\sqrt[3]{5\times 5\times 5}=5$
Therefore, the value of ${{125}^{\frac{1}{3}}}$will be $5$.
12. Simplify $\sqrt[3]{2}\times \sqrt[4]{3}$
Ans: $\sqrt[3]{2}\times \sqrt[4]{3}$
${{2}^{\frac{1}{3}}}\times {{3}^{\frac{1}{4}}}$
The LCM of $3\text{ and }4\text{ is }12$
$\therefore {{2}^{\frac{1}{3}}}={{2}^{\frac{4}{12}}}={{\left( {{2}^{4}} \right)}^{\frac{1}{12}}}={{16}^{\frac{1}{12}}} $
${{3}^{\frac{1}{4}}}={{3}^{\frac{3}{12}}}={{\left( {{3}^{3}} \right)}^{\frac{1}{12}}}={{27}^{\frac{1}{12}}} $
${{2}^{\frac{1}{3}}}\times {{3}^{\frac{1}{4}}}={{16}^{\frac{1}{12}}}\times {{27}^{\frac{1}{12}}}={{\left( 16\times 27 \right)}^{\frac{1}{12}}} $
$={{\left( 432 \right)}^{\frac{1}{12}}} $
13. Find the two rational numbers between$\frac{1}{2}$ and $\frac{1}{3}$.
Ans: First rational number between $\frac{1}{2}$ and $\frac{1}{3}$
$=\frac{1}{2}\left[ \frac{1}{2}+\frac{1}{3} \right]\Rightarrow \frac{1}{2}\left[ \frac{3+2}{6} \right]\Rightarrow \frac{5}{12} $
$ =\frac{1}{2},\frac{5}{12}\text{ and }\frac{1}{3} $
Second rational number between $\frac{1}{2}$ and $\frac{1}{3}$
$=\frac{1}{2}\left[ \frac{1}{2}+\frac{5}{12} \right]\Rightarrow \frac{1}{2}\left[ \frac{6+5}{12} \right]\Rightarrow \frac{11}{24}$
$=\frac{5}{12}\text{ and }\frac{11}{24}$ are two rational numbers between $\frac{1}{2}$ and $\frac{1}{3}$.
14. Find two rational numbers between $2$ and $3$.
Ans: Irrational numbers between $2$ and $3$ is $\sqrt{2\times 3}=\sqrt{6}$
Irrational number between $2$ and $3$ is $\sqrt{6}$.
$\sqrt{2\times \sqrt{6}}={{2}^{\frac{1}{2}}}\times {{6}^{\frac{1}{4}}}={{2}^{2\times \frac{1}{4}}}\times {{6}^{\frac{1}{4}}} $
$ ={{\left( {{2}^{2}} \right)}^{\frac{1}{4}}}\times {{6}^{\frac{1}{4}}}={{4}^{\frac{1}{4}}}\times {{6}^{\frac{1}{4}}}={{\left( 24 \right)}^{\frac{1}{4}}}=\sqrt[4]{24} $
$\sqrt{6}\text{ and }\sqrt{24}$ are two rational numbers between $2$ and $3$.
15. Multiply $\left( 3-\sqrt{5} \right)$ by $\left( 6+\sqrt{2} \right)$.
Ans: $\left( 3-\sqrt{5} \right)$$\left( 6+\sqrt{2} \right)$
$ =3\left( 6-\sqrt{2} \right)-\sqrt{5}\left( 6+\sqrt{2} \right) $
$=18+3\sqrt{2}-6\sqrt{5}-\sqrt{5}\times \sqrt{2} $
$ =18+3\sqrt{2}-6\sqrt{5}-\sqrt{10} $
16. Evaluate (i) $\sqrt[3]{125}$ (ii) $\sqrt[4]{1250}$
(i) $\sqrt[3]{125}$$={{\left( 5\times 5\times 5 \right)}^{\frac{1}{3}}}={{\left( {{5}^{3}} \right)}^{\frac{1}{3}}}=5$
(ii) $\sqrt[4]{1250}$$\begin{align}
$={{\left( 2\times 5\times 5\times 5\times 5 \right)}^{\frac{1}{4}}}={{\left( 2\times {{5}^{4}} \right)}^{\frac{1}{4}}} $
$={{2}^{\frac{1}{4}}}\times {{\left( {{5}^{4}} \right)}^{\frac{1}{4}}}=5\times \sqrt[4]{2} $
17. Find rationalizing factor of $\sqrt{300}$.
Ans: $\sqrt{300}=\sqrt{2\times 2\times 3\times 5\times 5}$
$ =\sqrt{{{2}^{2}}\times 3\times {{5}^{2}}} $
$ =2\times 5\sqrt{3}=10\sqrt{3} $
Rationalizing factor is $\sqrt{3}$
18. Rationalizing the denominator $\frac{1}{\sqrt{5}+\sqrt{2}}$ and subtract it from $\sqrt{5}-\sqrt{2}$.
Ans: $\frac{1}{\sqrt{5}+\sqrt{2}}\times \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}$
$=\frac{\sqrt{5}-\sqrt{2}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}=\frac{\sqrt{5}-\sqrt{2}}{5-2}=\frac{\sqrt{5}-\sqrt{2}}{3}$
Difference between $\left( \sqrt{5}-\sqrt{2} \right)\text{ and }\left( \sqrt{5}-\frac{\sqrt{2}}{3} \right)$
$=\sqrt{5}-\sqrt{2}-\left( \frac{\sqrt{5}-\sqrt{2}}{3} \right) $
$ =\sqrt{5}-\sqrt{2}-\frac{\sqrt{5}}{3}+\frac{\sqrt{2}}{3} $
$=\left( \sqrt{5}-\frac{\sqrt{5}}{3} \right)-\left( \sqrt{2}-\frac{\sqrt{2}}{3} \right) $
$=\frac{2\sqrt{5}}{3}-\frac{2\sqrt{2}}{3}=\frac{2}{3}\left( \sqrt{5}-\sqrt{2} \right) $
19. Show that $\sqrt{7}-3$ is irrational.
Ans: Suppose $\sqrt{7}-3$ is rational
Let $\sqrt{7}-3=x$ ($x$ is a rational number)
$\sqrt{7}=x+3$
$x$ is a rational number $3$ is also a rational number
$\therefore x+3$ is a rational number
But is $\sqrt{7}$ irrational number which is contradiction
$\therefore \sqrt{7}-3$ is an irrational number.
20. Find two rational numbers between $7$ and $5$.
Ans: First rational number $=\frac{1}{2}\left[ 7+5 \right]=\frac{12}{2}=6$
Second rational number $=\frac{1}{2}\left[ 7+6 \right]=\frac{1}{2}\times 13=\frac{13}{2}$
Two rational numbers between $7\text{ and }5\text{ are }6\text{ and }\frac{13}{2}$.
21. Show that $5+\sqrt{2}$ is not a rational number.
Ans: Let $5+\sqrt{2}$ is a rational number.
Say $5+\sqrt{2=x}$ i.e., $\sqrt{2}=x-5$
$x$ is a rational number $5$ is also rational number
$\therefore x-5$ is also a rational number.
But $\sqrt{2}$ is irrational number which is a contradiction
$\therefore 5+\sqrt{2}$ is an irrational number.
22. Simplify ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$.
Ans: ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}={{\left( \sqrt{5} \right)}^{2}}+{{\left( \sqrt{2} \right)}^{2}}+2\sqrt{5}\times \sqrt{2}=5+2+2\sqrt{10}=7+2\sqrt{2}$
23. Evaluate $\frac{{{11}^{\frac{5}{2}}}}{{{11}^{\frac{3}{2}}}}$.
Ans: $\frac{{{11}^{\frac{5}{2}}}}{{{11}^{\frac{3}{2}}}}={{11}^{\frac{5}{2}-\frac{3}{2}}}\left[ \because \frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}} \right]$
$ ={{11}^{\frac{5-3}{2}}}={{11}^{\frac{2}{2}}} $
$=11 $
24. Find four rational numbers between $\frac{3}{7}$ and $\frac{4}{7}$.
$\frac{3}{7}\times \frac{10}{10}=\frac{30}{70}\text{ and }\frac{4}{7}\times \frac{10}{10}=\frac{40}{70}$
Take any four rational numbers between $\frac{30}{70}\text{ and }\frac{40}{70}$ i.e., rational numbers between $\frac{3}{7}$ and $\frac{4}{7}$ are $\frac{31}{70},\frac{32}{70},\frac{33}{70},\frac{34}{70},\frac{35}{70}$
25. Write the following in decimal form (i) $\frac{36}{100}$ (ii) $\frac{2}{11}$
(i) $\frac{36}{100}=0.36$
(ii) $\frac{2}{11}=0.\overline{18}$
26. Express $2.417\overline{8}$ in the form $\frac{a}{b}$
Ans: $x=2.4\overline{178}$
$10x=24.\overline{178}$$......(1)$$[\text{Multiplying both sides by }10]$
$10x=24.178178178.... $
$1000\times 10x=1000\times 24.178178178....$Multiplying both sides by 1000
$10,000x=24178.178178.... $
$ 10000x=24178.\overline{178}\text{ }......(2) $
Subtracting $(1)\text{ from }(2)$
$10,000x-x=24178.\overline{178}-24.\overline{178} $
$9990x=24154 $
$x=\frac{24154}{9990} $
$ 2.4\overline{178}=\frac{24154}{9990}+\frac{12077}{4995} $
27. Multiply $\sqrt{3}$ by $\sqrt[3]{5}$.
Ans: $\sqrt{3}\text{ and }\sqrt[3]{5}$
Or ${{3}^{\frac{1}{2}}}\text{ and }{{5}^{\frac{1}{3}}}$
$LCM\text{ of }2\text{ and }3\text{ is }6 $
${{3}^{\frac{1}{2}}}={{3}^{\frac{1}{2}\times \frac{3}{3}}}={{\left( {{3}^{3}} \right)}^{\frac{1}{6}}}={{\left( 27 \right)}^{\frac{1}{6}}} $
${{5}^{\frac{1}{3}}}={{5}^{\frac{1}{3}\times \frac{2}{2}}}={{\left( {{5}^{2}} \right)}^{\frac{1}{6}}}={{\left( 25 \right)}^{\frac{1}{6}}} $
$\sqrt{3}\times \sqrt[3]{5}={{\left( 27 \right)}^{\frac{1}{6}}}\times {{\left( 25 \right)}^{\frac{1}{6}}}={{\left( 27\times 25 \right)}^{\frac{1}{6}}} $
$ ={{675}^{\frac{1}{6}}}=\sqrt[6]{675} $
28. Find the value of $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{5}}$ if $\sqrt{5}=2.236$ and $\sqrt{10}=3.162$.
Ans: $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{10}+5}{5}=\frac{8.162}{5}=1.6324$
29. Convert $0.\overline{25}$ into rational number.
Ans: Let \[x=0.\overline{25}\] ......(i)
$x=0.252525....$
Multiply both sides by 100
$100x=25.252525....$
$100x=25.\overline{25}$ ......(ii)
Subtract (i) from (ii)
$100x-x=25.\overline{25}-0.\overline{25} $
$x=\frac{25}{99} $
30. Simplify $\left( 3\sqrt{3}+2\sqrt{2} \right)\left( 2\sqrt{3}+3\sqrt{2} \right)$.
Ans: By multiplying each terms in the given product we have,
$ \left( 3\sqrt{3}+2\sqrt{2} \right)\left( 2\sqrt{3}+3\sqrt{2} \right) $
$=3\sqrt{3}\left( 2\sqrt{3}+3\sqrt{2} \right)+2\sqrt{2}\left( 2\sqrt{3}+3\sqrt{2} \right) $
$=18+9\sqrt{6}+4\sqrt{6}+12 $
$ =30+\left( 9+4 \right)\sqrt{6} $
$=30+13\sqrt{6} $
31. Simplify $\frac{{{9}^{\frac{3}{2}}}\times {{9}^{-\frac{4}{2}}}}{{{9}^{\frac{1}{2}}}}$.
Ans: By using the formulas of exponents with same base we get,
$\frac{{{9}^{\frac{3}{2}}}\times {{9}^{-\frac{4}{2}}}}{{{9}^{\frac{1}{2}}}}=\frac{{{9}^{\frac{3}{2}-\frac{4}{2}}}}{{{9}^{\frac{1}{2}}}}\left[ {{a}^{m}}.{{a}^{n}}={{a}^{m-n}} \right]$
$\frac{{{9}^{-\frac{1}{2}}}}{{{9}^{\frac{1}{2}}}}=\frac{1}{{{9}^{\frac{1}{2}+\frac{1}{2}}}}\left[ {{a}^{-m}}=\frac{1}{{{a}^{m}}} \right] $
$ =\frac{1}{{{9}^{\frac{2}{2}}}}=\frac{1}{9} $
3 Marks Questions
1. State whether the following statements are true or false. Give
reasons for your answers.
i. Every natural number is a whole number.
Separately, consider whole numbers and natural numbers.
We know that the whole number series is 0,1,2,3,4,5....
We know that the natural number series is 0,1,2,3,4,5....
As a result, every number in the natural number series may be found in the whole number series.
Therefore, we can safely conclude that any natural number is a whole number.
ii. Every integer is a whole number.
Ans: Separately, consider whole numbers and integers.
We know that integers are those numbers that can be written in the form of $\frac{p}{q}$ where q=1.
In the case of an integer series, we now have.... 4,3,2,1,0,1,2,3,4....
We can conclude that all whole number series numbers belong to the integer series.
However, the whole number series does not contain every number of integer series.
As a result, we can conclude that no integer is a whole number.
iii. Every rational number is a whole number.
Ans: Separately, consider whole numbers and rational numbers.
We know that integers are those numbers that can be written in the form of $\frac{p}{q}$ where $q\ne 0$.
We know that every number of whole number series can be written in the form of $\frac{p}{q}$ as $\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1},\frac{5}{1}...$
We conclude that every number of the whole number series is a rational number.
But, every rational number does not appear in the whole number series.
2. State whether the following statements are true or false. Justify your answers.
i. Every irrational number is a real number.
Ans: Separately, consider irrational numbers and real numbers.
We know that irrational numbers are the numbers that cannot be converted in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.
A real number is made up of both rational and irrational numbers, as we all know.
As a result, we might conclude that any irrational number is, in fact, a real number.
ii. Every point on the number line is of the form $\sqrt{m}$, where m is a natural number.
Ans: Consider a number line. We know that we can express both negative and positive numbers on a number line.
We know that when we take the square root of any number, we cannot receive a negative value.
Therefore, we conclude that not every number point on the number line is of the form $\sqrt{m}$, where m is a natural number.
iii. Every real number is an irrational number.
As a result, we can deduce that any irrational number is actually a real number. However, not every real number is irrational.
Therefore, we conclude that every real number is not a rational number.
3. Express the following in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$.
i. $0.\overline{6}$
Let $x=0.\overline{6}$
$\Rightarrow x=0.6666$ ......(a)
Multiplying both sides by 10 we get
$10x=6.6666$ ......(b)
We need to subtract (a) from (b), to get
We can also write $9x=6$ as $x=\frac{6}{9}$ or $x=\frac{2}{3}$.
Therefore, on converting $0.\overline{6}$ in the $\frac{p}{q}$ form, we get the answer as $\frac{2}{3}$.
ii. $0.4\overline{7}$
Ans: Let $x=0.4\overline{7}\Rightarrow x=0.47777$ ......(a)
Multiplying both sides by 10 we get
$10x=4.7777$ ......(b)
We can also write $9x=4.3$ as $x=\frac{4.3}{9}$ or $x=\frac{43}{90}$
Therefore, on converting $0.4\overline{7}$ in the $\frac{p}{q}$ form, we get the answer as $\frac{43}{90}$.
iii. $0.\overline{001}$
Ans: Let $x=0.\overline{001}\Rightarrow x=0.001001$ ......(a)
Multiplying both sides by 1000 we get
$1000x=1.001001$ ......(b)
We can also write $999x=1$ as $x=\frac{1}{999}$
Therefore, on converting $0.\overline{001}$ in the $\frac{p}{q}$ form, we get the answer as $\frac{1}{999}$.
4. What can the maximum number of digits be in the recurring block of digits in the decimal expansion of $\frac{1}{17}$ ? Perform the division to check your answer.
Ans: The number of digits in the recurring block of $\frac{1}{17}$ must be determined.
To acquire the repeating block of $\frac{1}{17}$ we'll use long division.
We need to divide 1 by 17, to get 0.0588235294117647.... and we got the remainder as 1, which will continue to be 1 after carrying out 16 continuous divisions.
Therefore, we conclude that
\[\frac{1}{17}=0.0588235294117647\] or \[\frac{1}{17}=0.\overline{0588235294117647}\] which is a non-terminating decimal and recurring decimal.
5. Look at several examples of rational numbers in the form $\frac{p}{q}\left( q\ne 0 \right)$ where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
Ans: Let us consider the examples of the form $\frac{p}{q}$ that are terminating decimals .
$ \frac{5}{2}=2.5 $
$ \frac{5}{4}=1.25 $
$ \frac{2}{5}=0.4 $
$ \frac{5}{16}=0.3125 $
It can be observed that the denominators of the above rational numbers have powers of 2,5 or both.
Therefore, we can conclude that property, which $q$ must satisfy in $\frac{p}{q}$ , so that the rational number $\frac{p}{q}$ is a terminating decimal is that q must have powers of 2,5 or both.
6. Classify the following numbers as rational or irrational:
i. $2-\sqrt{5}$
Ans: $2-\sqrt{5}$
We know that $\sqrt{5}=2.236....$, which is an irrational number.
$2-\sqrt{5}=2-2.236....$
$=-0.236...$, which is also an irrational number.
As a result, we can deduce that $2-\sqrt{5}$ is an irrational number.
ii. $\left( 3+\sqrt{23} \right)-\sqrt{23}$
Ans: $\left( 3+\sqrt{23} \right)-\sqrt{23}$
$\left( 3+\sqrt{23} \right)-\sqrt{23}=3+\sqrt{23}-\sqrt{23}=3$
As a result, we can deduce that $\left( 3+\sqrt{23} \right)-\sqrt{23}$ is a rational number.
iii. $\frac{2\sqrt{7}}{7\sqrt{7}}$
Ans: $\frac{2\sqrt{7}}{7\sqrt{7}}$
We can cancel $\sqrt{7}$ in the numerator and denominator to get $\frac{2\sqrt{7}}{7\sqrt{7}}=\frac{2}{7}$, because $\sqrt{7}$ is a common number in both the numerator and denominator.
iv. $\frac{1}{\sqrt{2}}$
Ans: $\frac{1}{\sqrt{2}}$
We know that $\sqrt{2}=1.4142...$, which is an irrational number.
$\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
$=\frac{1.4142...}{2}=0.707...$ which is also an irrational number.
As a result, we can deduce that $\frac{1}{\sqrt{2}}$ is an irrational number.
Ans: $2\pi $
We know that $\pi =3.1415....,$ which is an irrational number.
We can conclude that $2\pi $ will also be an irrational number.
As a result, we can deduce that $2\pi $ is an irrational number.
7. Simplify each of the following expression:
i. $\left( 3+3\sqrt{3} \right)\left( 2+\sqrt{2} \right)$
$(3+3\sqrt{3})(2+\sqrt{2})$
Applying distributive law,
\[(3+3\sqrt{3})(2+\sqrt{2})=3(2+\sqrt{2})\sqrt{3}(2+\sqrt{2})\]
\[=6+3\sqrt{2}+2\sqrt{3}+\sqrt{6}\]
ii. \[\left( 3+3\sqrt{3} \right)3-\sqrt{3}\]
Ans: $(3+3\sqrt{3})(3-\sqrt{3})$
$ (3+3\sqrt{3})(3-\sqrt{3})=(3-\sqrt{3})+\sqrt{3}(3-\sqrt{3}) $
$ =9-3\sqrt{3}+3\sqrt{3}-3 $
iii. ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$
Ans: ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$
Applying the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}={{\left( \sqrt{5} \right)}^{2}}+2\times \sqrt{5}\times \sqrt{2}+{{\left( \sqrt{2} \right)}^{2}} $
$ =5+2\sqrt{10}+2 $
$=7+2\sqrt{10}$
iv. $\left( 5+\sqrt{2} \right)\left( 5+\sqrt{2} \right)$
Ans: $\left( 5+\sqrt{2} \right)\left( 5+\sqrt{2} \right)$
Applying the formula $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$
$ \left( 5+\sqrt{2} \right)\left( 5+\sqrt{2} \right)={{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}} $
8. Find
i. ${{9}^{\frac{3}{2}}}$
Ans: We know that ${{a}^{\frac{1}{n}}}=\sqrt[n]{a},a>0$
As a result, we can deduce that ${{9}^{\frac{3}{2}}}$ can also be written as
$\sqrt[2]{{{\left( 9 \right)}^{3}}}=\sqrt[2]{9\times 9\times 9}=\sqrt[2]{3\times 3\times 3\times 3\times 3\times 3}$
$=3\times 3\times 3 $
Therefore, the value of ${{9}^{\frac{3}{2}}}$ will be $27$ .
ii. ${{32}^{\frac{2}{5}}}$
As a result, we can deduce that ${{32}^{\frac{2}{5}}}$ can also be written as
$ \sqrt[5]{{{\left( 32 \right)}^{2}}}=\sqrt[5]{\left( 2\times 2\times 2\times 2\times 2 \right)\left( 2\times 2\times 2\times 2\times 2 \right)} $
$=2\times 2 $
Therefore, the value of ${{32}^{\frac{2}{5}}}$ will be $4$.
iii. ${{16}^{\frac{3}{4}}}$
As a result, we can deduce that ${{16}^{\frac{3}{4}}}$ can also be written as
$\sqrt[4]{{{\left( 16 \right)}^{3}}}=\sqrt[4]{\left( 2\times 2\times 2\times 2 \right)\left( 2\times 2\times 2\times 2 \right)\left( 2\times 2\times 2\times 2 \right)} $
$ =2\times 2\times 2 $
& =8 $
Therefore, the value of ${{16}^{\frac{3}{4}}}$ will be $8$ .
iv. ${{125}^{-\frac{1}{3}}}$
Ans: We know that ${{a}^{-n}}=\frac{1}{{{a}^{n}}}$
As a result, we can deduce that ${{125}^{-\frac{1}{3}}}$ can also be written as $\frac{1}{{{125}^{\frac{1}{3}}}},or{{\left( \frac{1}{125} \right)}^{\frac{1}{3}}}$
We know that ${{a}^{\frac{1}{n}}}=\sqrt[n]{a},a>0$
$ \sqrt[3]{\frac{1}{125}}=\sqrt[3]{\left( \frac{1}{5}\times \frac{1}{5}\times \frac{1}{5} \right)} $
$=\frac{1}{5} $
Therefore, the value of ${{125}^{-\frac{1}{3}}}$ will be $\frac{1}{5}$.
9. Simplify
i. ${{2}^{\frac{2}{3}}}{{.2}^{\frac{1}{5}}}$
Ans: We know that ${{a}^{m}}.{{a}^{n}}={{a}^{\left( m+n \right)}}$
As a result, we can deduce that ${{2}^{\frac{2}{3}}}{{.2}^{\frac{1}{5}}}={{\left( 2 \right)}^{\frac{2}{3}+\frac{1}{5}}}$
${{2}^{\frac{2}{3}}}{{.2}^{\frac{1}{5}}}=\left( 2 \right)\frac{10+3}{15}={{\left( 2 \right)}^{\frac{13}{15}}}$
Therefore, the value of ${{2}^{\frac{2}{3}}}{{.2}^{\frac{1}{5}}}$ will be ${{\left( 2 \right)}^{\frac{13}{15}}}$.
ii. ${{\left( {{3}^{\frac{1}{3}}} \right)}^{7}}$
As a result, we can deduce that ${{\left( {{3}^{\frac{1}{3}}} \right)}^{7}}$ can also be written as ${{3}^{\frac{7}{3}}}$
iii. $\frac{{{11}^{\frac{1}{2}}}}{{{11}^{\frac{1}{4}}}}$
Ans: We know that $\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{\left( m-n \right)}}$
As a result, we can deduce that $\begin{align}
$ \frac{{{11}^{\frac{1}{2}}}}{{{11}^{\frac{1}{4}}}}={{11}^{\frac{1}{2}}}-{{11}^{\frac{1}{4}}} $
$={{11}^{\frac{2-1}{4}}}={{11}^{\frac{1}{4}}} $
Therefore, the value of $\frac{{{11}^{\frac{1}{2}}}}{{{11}^{\frac{1}{4}}}}$ will be ${{11}^{\frac{1}{4}}}$.
iv. ${{7}^{\frac{1}{2}}}{{.8}^{\frac{1}{2}}}$
Ans: We know that ${{a}^{m}}.{{b}^{m}}={{\left( a\times b \right)}^{m}}$
As a result, we can deduce that ${{7}^{\frac{1}{2}}}{{.8}^{\frac{1}{2}}}={{\left( 7\times 8 \right)}^{\frac{1}{2}}}.$
${{7}^{\frac{1}{2}}}{{.8}^{\frac{1}{2}}}={{\left( 7\times 8 \right)}^{\frac{1}{2}}}={{\left( 56 \right)}^{\frac{1}{2}}}.$
Therefore, the value of ${{7}^{\frac{1}{2}}}{{.8}^{\frac{1}{2}}}$ will be ${{\left( 56 \right)}^{\frac{1}{2}}}$.
10. Express $0.8888....$ in the form $\frac{p}{q}$.
Ans: Let us assume that the given decimal as,
$x=0.\overline{8}......\left( 1 \right)$
$10x=10\times 0.8888$ (Multiply both sides by 10)
$ 10x=8.8888 $
$10x=8.\overline{8}.....\left( 2 \right) $
$10x-x=8.\overline{8}-0.\overline{8}$ (Subtracting (1) from (2))
$x=\frac{8}{9} $
11. Simplify by rationalizing denominator $\frac{7+3\sqrt{5}}{7-3\sqrt{5}}$.
Ans: We are given the fraction to rationalize. By rationalizing the denominator we get,
$\frac{7+3\sqrt{5}}{7-3\sqrt{5}}=\frac{7+3\sqrt{5}}{7-3\sqrt{5}}\times \frac{7+3\sqrt{5}}{7+3\sqrt{5}}$
$=\frac{{{\left( 7+3\sqrt{5} \right)}^{2}}}{{{7}^{2}}-{{\left( 3\sqrt{5} \right)}^{2}}} $
$ =\frac{{{7}^{2}}+{{\left( 3\sqrt{5} \right)}^{2}}+2\times 7\times 3\sqrt{5}}{49-{{3}^{2}}\times 5} $
$=\frac{49+9\times 5+42\sqrt{5}}{49-45} $
$=\frac{49+45+42\sqrt{5}}{4} $
$ =\frac{94+42\sqrt{5}}{4} $
$ =\frac{94}{4}+\frac{42}{4}\sqrt{5} $
$ =\frac{47}{2}+\frac{21}{2}\sqrt{5} $
12 . Simplify ${{\left\{ {{\left[ {{625}^{-}}^{\frac{1}{2}} \right]}^{-\frac{1}{4}}} \right\}}^{2}}$.
Ans: Let us take the given expression to simplify and using the exponents formulas we get,
\[{{\left\{ {{\left[ {{625}^{-}}^{\frac{1}{2}} \right]}^{-\frac{1}{4}}} \right\}}^{2}}\]
$ ={{\left\{ {{\left( \frac{1}{{{625}^{\frac{1}{2}}}} \right)}^{-\frac{1}{4}}} \right\}}^{2}} $
$={{\left\{ {{\left( \frac{1}{{{\left( {{25}^{2}} \right)}^{\frac{1}{2}}}} \right)}^{-\frac{1}{4}}} \right\}}^{2}}\left( {{a}^{-m}}=\frac{1}{{{a}^{m}}} \right) $
$ =\left\{ {{\left( \frac{1}{25} \right)}^{-\frac{1}{4}\times 2}} \right\} $
$=\left( \frac{1}{{{25}^{-\frac{1}{2}}}} \right)=\frac{1}{{{\left( {{5}^{2}} \right)}^{-\frac{1}{2}}}}=\frac{1}{{{5}^{-1}}}=5 $
13. Visualize 3.76 on the number line using successive magnification.
14. Prove that $\frac{1}{1+{{x}^{b-a}}+{{x}^{c-a}}}+\frac{1}{1+{{x}^{a-b}}+{{x}^{c-b}}}+\frac{1}{1+{{x}^{a-c}}+{{x}^{b-c}}}=1$
Ans: We are asked to prove the expression,
$\frac{1}{1+{{x}^{b-a}}+{{x}^{c-a}}}+\frac{1}{1+{{x}^{a-b}}+{{x}^{c-b}}}+\frac{1}{1+{{x}^{a-c}}+{{x}^{b-c}}}=1$
Let us take the LHS of the given expression that is,
$LHS=\frac{1}{1+{{x}^{b}}.{{x}^{-a}}+{{x}^{c}}.{{x}^{-a}}}+\frac{1}{1+{{x}^{a}}.{{x}^{-b}}+{{x}^{c}}.{{x}^{-b}}}+\frac{1}{1+{{x}^{a}}.{{x}^{-c}}+{{x}^{b}}.{{x}^{-c}}} $
$=\frac{1}{{{x}^{-a}}.{{x}^{a}}+{{x}^{b}}.{{x}^{-a}}+{{x}^{c}}.{{x}^{-a}}}+\frac{1}{{{x}^{b}}.{{x}^{-b}}+{{x}^{a}}.{{x}^{-b}}+{{x}^{c}}.{{x}^{-b}}}+\frac{1}{{{x}^{c}}.{{x}^{-c}}+{{x}^{a}}.{{x}^{-c}}+{{x}^{b}}.{{x}^{-c}}} $
$ =\frac{1}{{{x}^{-a}}\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}+\frac{1}{{{x}^{-b}}\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}+\frac{1}{{{x}^{-c}}\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)} $
$=\frac{{{x}^{a}}}{\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}+\frac{{{x}^{b}}}{\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}+\frac{{{x}^{c}}}{\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)} $ $ =\frac{\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}{\left( {{x}^{a}}+{{x}^{b}}+{{x}^{c}} \right)}=1 $
15. Represent $\sqrt{3}$ on number line.
Ans: Consider a number line $\text{OD}$ such that the construction to form two triangles is done as shown below.
Take $OA=AB=1$ unit.
And $\angle A=90{}^\circ $
In $\Delta OAB$, by using the Pythagorean theorem we get,
$O{{B}^{2}}={{1}^{2}}+{{1}^{2}}$
$O{{B}^{2}}=2 $
$ OB=\sqrt{2}$
Now from triangle $\text{ }\!\!\Delta\!\!\text{ OBD}$, using the Pythagorean theorem we get,
$O{{D}^{2}}=O{{B}^{2}}+B{{D}^{2}} $
$ O{{D}^{2}}={{\left( \sqrt{2} \right)}^{2}}+{{\left( 1 \right)}^{1}} $
$O{{D}^{2}}=2+1=3 $
$OD=\sqrt{3} $
Now, if the point $\text{O}$ is $0$ units then the point $\text{D}$ represents $\sqrt{3}$units.
16. Simplify ${{\left( 3\sqrt{2}+2\sqrt{3} \right)}^{2}}{{\left( 3\sqrt{2}-2\sqrt{3} \right)}^{2}}$.
Ans: We are given the expression as,
${{\left( 3\sqrt{2}+2\sqrt{3} \right)}^{2}}{{\left( 3\sqrt{2}-2\sqrt{3} \right)}^{2}}$
Now, by regrouping the terms in the above expression we have,
$ =\left( 3\sqrt{2}+2\sqrt{3} \right)\left( 3\sqrt{2}+2\sqrt{3} \right)\left( 3\sqrt{2}-2\sqrt{3} \right)\left( 3\sqrt{2}-2\sqrt{3} \right) $
$ =\left( 3\sqrt{2}+2\sqrt{3} \right)\left( 3\sqrt{2}-2\sqrt{3} \right)\left( 3\sqrt{2}+2\sqrt{3} \right)\left( 3\sqrt{2}-2\sqrt{3} \right) $
$=\left[ {{\left( 3\sqrt{2} \right)}^{2}}-{{\left( 2\sqrt{3} \right)}^{2}} \right]\left[ {{\left( 3\sqrt{2} \right)}^{2}}-{{\left( 2\sqrt{3} \right)}^{2}} \right] $
$ =\left[ 9\times 2-4\times 3 \right]\left[ 9\times 2-4\times 3 \right] $
$ =\left[ 18-12 \right]\left[ 18-12 \right] $
$=6\times 6=36 $
17. Express $2.\overline{4178}$ in the form $\frac{p}{q}$.
Ans: Let $\frac{p}{q}=2.\overline{4178}$
$\frac{p}{q}=2.4178178178$
Multiply by 10
$10\frac{p}{q}=24.178178$
Multiply by 1000
$10000\frac{p}{q}=1000\times 24.178178 $
$1000\frac{p}{q}-\frac{p}{q}=24178.178178-14.178178 $
$9999\frac{p}{q}=24154 $
$\frac{p}{q}=\frac{24154}{9999} $
18. Simplify ${{\left( 27 \right)}^{-\frac{2}{3}}}\div {{9}^{\frac{1}{2}}}{{.3}^{-\frac{3}{2}}}$.
Ans: ${{\left( 27 \right)}^{-\frac{2}{3}}}\div {{9}^{\frac{1}{2}}}{{.3}^{-\frac{3}{2}}}$
$ =\frac{{{\left( 3\times 3\times 3 \right)}^{-\frac{2}{3}}}\times {{3}^{\frac{3}{2}}}}{{{\left( 3\times 3 \right)}^{\frac{1}{2}}}}\left[ {{a}^{-m}}=\frac{1}{{{a}^{m}}} \right] $
$ =\frac{{{\left( {{3}^{3}} \right)}^{-\frac{2}{3}}}\times {{3}^{\frac{3}{2}}}}{{{\left( {{3}^{2}} \right)}^{\frac{1}{2}}}} $
$=\frac{{{3}^{\frac{3}{2}-2}}}{3}=\frac{{{3}^{-\frac{1}{3}}}}{3} $
$=\frac{1}{{{3}^{\frac{4}{3}}}}=\frac{1}{\sqrt[3]{81}} $
19. Find three rational numbers between $2.\overline{2}$ and $2.\overline{3}.$
Ans: The irrational numbers are the numbers that do not end after the decimal point nor repeat its numbers in a sequence.
Representing the given numbers in decimal form we have,
$ 2.\overline{2}=2.222222222...... $
$ 2.\overline{3}=2.333333333....... $
So any numbers between these two numbers that do not end nor repeat in any sequence gives the required irrational numbers.
Three rational numbers between $2.\overline{2}$ and $2.\overline{3}$ are $2.222341365....$, $2.28945187364....$ and $2.2321453269....$
20. Give an example of two irrational numbers whose
i. Sum is a rational number
Ans: The required two irrational numbers are $2+\sqrt{2}$ and $2-\sqrt{2}$
Sum $2+\sqrt{2}+2-\sqrt{2}=4$ which is a rational number.
ii. Product is a rational number
Ans: The required two irrational numbers are $3\sqrt{2}$ and $6\sqrt{2}$
Product $3\sqrt{2}\times 6\sqrt{2}=18\times 2=36$ which is rational.
iii. Quotient is a rational number
Ans: The required two irrational numbers are $2\sqrt{125}$ and $3\sqrt{5}$
Quotient $\frac{2\sqrt{125}}{3\sqrt{5}}=\frac{2}{3}\sqrt{\frac{125}{5}}=\frac{2}{3}\times 5=\frac{10}{3}$
21 . If $\sqrt{2}=1.414$ and $\sqrt{3}=1.732$, find the value of $\frac{5}{\sqrt{2}+\sqrt{3}}$.
Ans: First let us take the given expression and by rationalizing the denominator we get,
$\frac{5}{\sqrt{2}+\sqrt{3}}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}$
$\frac{5\left( \sqrt{2}-\sqrt{3} \right)}{{{\left( \sqrt{2} \right)}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} $
$=\frac{5\left( \sqrt{2}-\sqrt{3} \right)}{2-3} $
Now, substituting the required values of irrational numbers we get,
$=-5\left[ 1.414-1.732 \right] $
$ =-5\times -0.318 $
$ =1.59 $
22. Visualize 2.4646 on the number line using successive magnification.
23. Rationalizing the denominator of $\frac{1}{4+2\sqrt{3}}$.
Ans: First let us take the given expression and rationalizing the denominator by multiplying the numerator and denominator with its conjugate we get,
$\frac{1}{4+2\sqrt{3}}=\frac{1}{4+2\sqrt{3}}\times \frac{4-2\sqrt{3}}{4+2\sqrt{3}} $
$ =\frac{4-2\sqrt{3}}{{{\left( 4 \right)}^{2}}-{{\left( 2\sqrt{3} \right)}^{2}}} $
$=\frac{4-2\sqrt{3}}{16-{{\left( 2\sqrt{3} \right)}^{2}}} $
$=\frac{4-2\sqrt{3}}{16-12} $
$ =\frac{4-2\sqrt{3}}{4} $
$ =\frac{2\left( 2-\sqrt{3} \right)}{4} $
$ =\frac{2-\sqrt{3}}{2} $
24. Visualize the representation of $5.3\overline{7}$ on the number line up to 3 decimal places.
Ans: The representation of $5.3\overline{7}$ on the number line is given below:
25. Show that \[5\sqrt{2}\] is not a rational number.
Ans: Let us assume that \[5\sqrt{2}\] is a rational number.
Take \[x=5\sqrt{2}\] , with \[x\]being rational as well.
\[x=5\sqrt{2}\]
\[\Rightarrow \frac{x}{5}=\sqrt{2}\]
Let us compare the terms in LHS and RHS.
In LHS, we have\[\frac{x}{5}\] , with \[x\] and $5$ being rational numbers (Here \[x\] is rational, based on our assumption). So \[\frac{x}{5}\] is a rational number.
In RHS, we have$\sqrt{2}$, which is not a rational number, but an irrational number.
This is a contradiction, i.e. $LHS\ne RHS$.
So, we can conclude that \[5\sqrt{2}\] is not a rational number.
26. Simplify \[3\sqrt[3]{250}+7\sqrt[3]{16}-4\sqrt[3]{54}\].
Ans: Let us first find the cube roots of given numbers to their simplest forms by using the prime factorization then we get,
\[3\sqrt[3]{250}+7\sqrt[3]{16}-4\sqrt[3]{54}=3\sqrt[3]{5\times 5\times 5\times 2}+7\sqrt[3]{2\times 2\times 2\times 2}-4\sqrt[3]{3\times 3\times 3\times 2}\]
\[=\left( 3\times 5\sqrt[3]{2} \right)+\left( 7\times 2\sqrt[3]{2} \right)-\left( 4\times 3\sqrt[3]{2} \right)\]
\[=\left( 15\sqrt[3]{2} \right)+\left( 14\sqrt[3]{2} \right)-\left( 12\sqrt[3]{2} \right)\]
\[=\left( 15+14-12 \right)\sqrt[3]{2}\]
\[=17\sqrt[3]{2}\]
Thus, we get \[3\sqrt[3]{250}+7\sqrt[3]{16}-4\sqrt[3]{54}=17\sqrt[3]{2}\]
27. Simplify \[3\sqrt{48}-\frac{5}{2}\sqrt{\frac{1}{3}}+4\sqrt{3}\].
Ans: Let us first find the square roots of given numbers to their simplest forms by using the prime factorization then we get,
\[3\sqrt{48}-\frac{5}{2}\sqrt{\frac{1}{3}}+4\sqrt{3}=\left( 3\sqrt{2\times 2\times 2\times 2\times 3} \right)-\left[ \frac{5}{2}\left( \sqrt{\frac{1}{3}}\times \frac{\sqrt{3}}{\sqrt{3}} \right) \right]+\left( 4\sqrt{3} \right)\]
\[=\left( 3\times 2\times 2\sqrt{3} \right)-\left[ \frac{5}{2}\left( \frac{\sqrt{3}}{3} \right) \right]+\left( 4\sqrt{3} \right)\]
\[=\left( 12\sqrt{3} \right)-\left( \frac{5\sqrt{3}}{6} \right)+\left( 4\sqrt{3} \right)\]
\[=\left( 12-\frac{5}{6}+4 \right)\sqrt{3}\]
\[=\left( \frac{72-5+24}{6} \right)\sqrt{3}\]
\[=\frac{91}{6}\sqrt{3}\]
Thus, we get \[3\sqrt{48}-\frac{5}{2}\sqrt{\frac{1}{3}}+4\sqrt{3}=\frac{91}{6}\sqrt{3}\]
28. If $\frac{1}{7}=0.\overline{142857}$. Find the value of $\frac{2}{7},\frac{3}{7},\frac{4}{7}$
Ans: It is given that – $\frac{1}{7}=0.\overline{142857}$
(i) $\frac{2}{7}=2\times \frac{1}{7}$
$=2\times 0.\overline{142857}$
$=0.\overline{285714}$
$\Rightarrow \frac{2}{7}=0.\overline{285714}$
(ii) $\frac{3}{7}=3\times \frac{1}{7}$
$=3\times 0.\overline{142857}$
$=0.\overline{428571}$
$\Rightarrow \frac{3}{7}=0.\overline{428571}$
(iii) $\frac{4}{7}=4\times \frac{1}{7}$
$=4\times 0.\overline{142857}$
$=0.\overline{571428}$
$\Rightarrow \frac{4}{7}=0.\overline{571428}$
29. Find $6$ rational numbers between $\frac{6}{5}$ and $\frac{7}{5}$
Ans: It is possible to divide the interval between $\frac{6}{5}$ and $\frac{7}{5}$ into $10$ equal parts.
Then we will have – $\frac{6}{5},\frac{6.1}{5},\frac{6.2}{5},\frac{6.3}{5},\frac{6.4}{5},\frac{6.5}{5},\frac{6.6}{5},\frac{6.7}{5},\frac{6.8}{5},\frac{6.9}{5},\frac{7}{5}$
i.e. $\frac{60}{50},\frac{61}{50},\frac{62}{50},\frac{63}{50},\frac{64}{50},\frac{65}{50},\frac{66}{50},\frac{67}{50},\frac{68}{50},\frac{69}{50},\frac{70}{50}$
From these fractions, it is possible to choose $6$ rational numbers between $\frac{6}{5}$ and $\frac{7}{5}$
Thus , $6$ rational numbers between $\frac{6}{5}$ and $\frac{7}{5}$ are $\frac{61}{50},\frac{62}{50},\frac{63}{50},\frac{64}{50},\frac{65}{50},\frac{66}{50}$
30. Show how $\sqrt{4}$ can be represented on the number line.
Ans: Take $AB=OA=1\text{ }unit$ on a number line.
Also, $\angle A={{90}^{\circ }}$
In $\vartriangle OAB$, apply Pythagoras Theorem,
$\therefore O{{A}^{2}}+A{{B}^{2}}=O{{B}^{2}}$
$\Rightarrow O{{B}^{2}}={{1}^{2}}+{{1}^{2}}$
$\Rightarrow O{{B}^{2}}=1+1$
$\Rightarrow O{{B}^{2}}=2$
$\Rightarrow OB=\sqrt{2}$
Now, draw $OB=O{{A}_{1}}=\sqrt{2}$
And, ${{A}_{1}}{{B}_{1}}=1\text{ unit}$ with$\angle {{A}_{1}}={{90}^{\circ }}$
In \[\vartriangle O{{A}_{1}}{{B}_{1}}\], apply Pythagoras Theorem,
$\therefore O{{A}_{1}}^{2}+{{A}_{1}}{{B}_{1}}^{2}=O{{B}_{1}}^{2}$
$\Rightarrow O{{B}_{1}}^{2}={{\left( \sqrt{2} \right)}^{2}}+{{1}^{2}}$
$\Rightarrow O{{B}_{1}}^{2}=2+1$
$\Rightarrow O{{B}_{1}}^{2}=3$
$\Rightarrow O{{B}_{1}}=\sqrt{3}$
Now, draw $O{{B}_{1}}=O{{A}_{2}}=\sqrt{3}$
And, \[{{A}_{2}}{{B}_{2}}=1\text{ unit}\] with$\angle {{A}_{2}}={{90}^{\circ }}$
In \[\vartriangle O{{A}_{2}}{{B}_{2}}\], apply Pythagoras Theorem,
$\therefore O{{A}_{2}}^{2}+{{A}_{2}}{{B}_{2}}^{2}=O{{B}_{2}}^{2}$
$\Rightarrow O{{B}_{2}}^{2}={{\left( \sqrt{3} \right)}^{2}}+{{1}^{2}}$
$\Rightarrow O{{B}_{2}}^{2}=3+1$
$\Rightarrow O{{B}_{2}}^{2}=4$
$\Rightarrow O{{B}_{2}}=\sqrt{4}$
Now, draw $O{{B}_{2}}=O{{A}_{3}}=\sqrt{4}$
Thus line segment $O{{A}_{3}}=\sqrt{4}$
Short Answer Questions (4 Marks)
1. Write the following in decimal form and say what kind of decimal expansion each has:
i. $\frac{36}{100}$
Ans: Performing long division of $36$ by $100$
$\begin{matrix} &{0.36}\\ 100&{\overline{)\;36\quad}}\\ &\underline{-0\quad}\\ &360\\ &\underline{-300\quad}\\ &\;\;600\\ &\underline{-600}\\ &\underline{\quad 0 \;\;} \end{matrix}$
Thus, $\frac{36}{100}=0.36$ - this is a terminating decimal.
ii. $\frac{1}{11}$
Ans: Performing long division of $1$ by $11$
$\begin{matrix} {} & 0.0909.. \\ 11 & \overline{)\text{ }1\text{ }} \\ {} & \underline{-0} \\ {} & 10 \\ {} & \underline{-0} \\ {} & 100 \\ {} & \underline{-99} \\ {} & 10 \\ {} & \underline{-0} \\ {} & 100 \\ {} & \underline{-99} \\ {} & 1 \\ \end{matrix}$
It can be seen that performing further division will produce a reminder of $1$ continuously.
Thus, $\frac{1}{11}=0.09090...$ i.e. $\frac{1}{11}=0.\overline{09}$, this is a non-terminating, but recurring decimal.
iii. $4\frac{1}{8}$
Ans: First convert the mixed fraction into an improper fraction –
$4\frac{1}{8}=\frac{(4\times 8)+1}{8}=\frac{33}{8}$
Performing long division of $33$ by $8$
$\begin{matrix} {} & 4.125 \\ 8 & \overline{\left){\text{ }33\text{ }}\right.} \\ {} & \underline{-32} \\ {} & 10 \\ {} & \underline{-8} \\ {} & 20 \\ {} & \underline{-16} \\ {} & 40 \\ {} & \underline{-40} \\ {} & 0 \\ \end{matrix}$
Thus, $4\frac{1}{8}=4.125$ - this is a terminating decimal.
iv. $\frac{3}{13}$
Ans: Performing long division of $3$ by $13$
$\begin{matrix} {} & 0.230769.. \\ 13 & \overline{\left){\text{ }3\text{ }}\right.} \\ {} & \underline{-0} \\ {} & 30 \\ {} & \underline{-26} \\ {} & 40 \\ {} & \underline{-39} \\ {} & 10 \\ {} & \underline{-0} \\ {} & 100 \\ {} & \underline{-91} \\ {} & 90 \\ {} & \underline{-78} \\ {} & 120 \\ {} & \underline{-117} \\ {} & 3 \\ \end{matrix}$
It can be seen that performing further division will produce a reminder of $3$ periodically, after every six divisions.
Thus, $\frac{3}{13}=0.230769...$ i.e. \[\frac{3}{13}=0.\overline{230769}\], this is a non-terminating, but recurring decimal.
v. $\frac{2}{11}$
Ans: Performing long division of $2$ by $11$
$\begin{matrix} {} & 0.1818.. \\ 11 & \overline{)\text{ 2 }} \\ {} & \underline{-0} \\ {} & 20 \\ {} & \underline{-11} \\ {} & 90 \\ {} & \underline{-88} \\ {} & 20 \\ {} & \underline{-11} \\ {} & 90 \\ {} & \underline{-88} \\ {} & 2 \\ \end{matrix}$
It can be seen that performing further division will produce a reminder of $2$followed by $9$ alternatively.
Thus, $\frac{2}{11}=0.181818...$ i.e. $\frac{2}{11}=0.\overline{18}$this is a non-terminating, but recurring decimal.
vi. $\frac{329}{400}$
Ans: Performing long division of $33$ by $8$
$\begin{matrix} {} & 0.8225 \\ 400 & \overline{)\text{ 329 }} \\ {} & \underline{-0} \\ {} & 3290 \\ {} & \underline{-3200} \\ {} & 900 \\ {} & \underline{-800} \\ {} & 1000 \\ {} & \underline{-800} \\ {} & 2000 \\ {} & \underline{-2000} \\ {} & 0 \\ \end{matrix}$
Thus, $\frac{329}{400}=0.8225$ - this is a terminating decimal.
2. Classify the following as rational or irrational:
i. $\sqrt{23}$
Ans: It is known that the root of $23$ will produce a non-terminating and non-recurring decimal number (it is not a perfect square value), also it cannot be represented as a fraction. Thus we can say that $\sqrt{23}$ is an irrational number.
ii. $\sqrt{225}$
Ans: It is known that $\sqrt{225}=15$, which is an integer.
Thus $\sqrt{225}$ is a rational number.
iii. $0.3796$
Ans: Here, $0.3796$ is a terminating decimal number, and also it can be expressed as a fraction.
i.e. $0.3796=\frac{3796}{10000}=\frac{949}{2500}$
Thus $0.3796$ is a rational number.
iv. $7.478478...$
Ans: Here, $7.478478...$ is a non-terminating, but recurring decimal number, and also it can be expressed as a fraction.
i.e. $7.478478...=7.\overline{487}$
Converting it into fraction
If $x=7.478478...\text{ (1)}$
Then $1000x=7478.478478...\text{ (2)}$
Subtract equations $(2)-(1)$
$ 1000x=7478.478478... $
$ \underline{-\text{ }x=\text{ }7.478478...} $
$ \text{ }999x=7471 $
Now, $999x=7471$
$\Rightarrow x=\frac{7471}{999}$
i.e. $7.\overline{478}=\frac{7471}{999}$
Thus $7.478478...$ is a rational number.
v. $1.101001000100001...$
Ans: Here, $1.101001000100001...$ is a non-terminating and non-recurring decimal number and also it cannot be represented as a fraction. Thus we can say that $1.101001000100001...$ is an irrational number.
3. Rationalize the denominator of the following:
(i) $\frac{1}{\sqrt{7}}$
Ans: In order to rationalize the denominator, we multiply and divide $\frac{1}{\sqrt{7}}$ by $\sqrt{7}$
$\frac{1}{\sqrt{7}}\times \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}$
Rationalizing the denominator of $\frac{1}{\sqrt{7}}$ produces $\frac{\sqrt{7}}{7}$.
ii. $\frac{1}{\sqrt{7}-\sqrt{6}}$
Ans: In order to rationalize the denominator, we multiply and divide $\frac{1}{\sqrt{7}-\sqrt{6}}$ by $\sqrt{7}+\sqrt{6}$
$\frac{1}{\sqrt{7}-\sqrt{6}}\times \frac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}=\frac{\sqrt{7}+\sqrt{6}}{\left( \sqrt{7}-\sqrt{6} \right)\left( \sqrt{7}+\sqrt{6} \right)}$
Using the identity - \[(a+b)(a-b)={{a}^{2}}-{{b}^{2}}\]
$=\frac{\sqrt{7}+\sqrt{6}}{{{\left( \sqrt{7} \right)}^{2}}-{{\left( \sqrt{6} \right)}^{2}}}$
$=\frac{\sqrt{7}+\sqrt{6}}{7-6}$
$=\frac{\sqrt{7}+\sqrt{6}}{1}$
$\Rightarrow \frac{1}{\sqrt{7}-\sqrt{6}}=\sqrt{7}+\sqrt{6}$
Rationalizing the denominator of $\frac{1}{\sqrt{7}-\sqrt{6}}$ produces $\sqrt{7}+\sqrt{6}$.
iii. $\frac{1}{\sqrt{5}+\sqrt{2}}$
Ans: In order to rationalize the denominator, we multiply and divide $\frac{1}{\sqrt{5}+\sqrt{2}}$ by $\sqrt{5}-\sqrt{2}$
$\frac{1}{\sqrt{5}+\sqrt{2}}\times \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}=\frac{\sqrt{5}-\sqrt{2}}{\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)}$
$=\frac{\sqrt{5}-\sqrt{2}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}$
$=\frac{\sqrt{5}-\sqrt{2}}{5-2}$
$=\frac{\sqrt{5}-\sqrt{2}}{3}$
$\Rightarrow \frac{1}{\sqrt{5}+\sqrt{2}}=\frac{\sqrt{5}-\sqrt{2}}{3}$
Rationalizing the denominator of $\frac{1}{\sqrt{5}+\sqrt{2}}$ produces $\frac{\sqrt{5}-\sqrt{2}}{3}$.
iv. $\frac{1}{\sqrt{7}-2}$
Ans: In order to rationalize the denominator, we multiply and divide $\frac{1}{\sqrt{7}-2}$ by $\sqrt{7}+2$
$\frac{1}{\sqrt{7}-2}\times \frac{\sqrt{7}+2}{\sqrt{7}+2}=\frac{\sqrt{7}+2}{\left( \sqrt{7}-2 \right)\left( \sqrt{7}+2 \right)}$
$=\frac{\sqrt{7}+2}{{{\left( \sqrt{7} \right)}^{2}}-{{\left( 2 \right)}^{2}}}$
$=\frac{\sqrt{7}+2}{7-4}$
$=\frac{\sqrt{7}+2}{3}$
$\Rightarrow \frac{1}{\sqrt{7}+2}=\frac{\sqrt{7}+2}{3}$
Rationalizing the denominator of $\frac{1}{\sqrt{7}-2}$ produces $\frac{\sqrt{7}+2}{3}$.
Long Answer Questions (5 Marks)
It can be seen that performing further divisions will produce a reminder of $3$ periodically, after every six divisions.
Thus, $\frac{329}{400}=s0.8225$ - this is a terminating decimal.
4. If $\sqrt{5}=2.236$ and $\sqrt{3}=1.732$. Find the value of $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}$.
Ans: It is given that –
$\sqrt{5}=2.236$
$\sqrt{3}=1.732$
Now, $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}$
$\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}=\left[ \frac{2}{\left( \sqrt{5}+\sqrt{3} \right)}\times \frac{\left( \sqrt{5}-\sqrt{3} \right)}{\left( \sqrt{5}-\sqrt{3} \right)} \right]+\left[ \frac{7}{\left( \sqrt{5}-\sqrt{3} \right)}\times \frac{\left( \sqrt{5}+\sqrt{3} \right)}{\left( \sqrt{5}+\sqrt{3} \right)} \right]$
$=\left[ \frac{2\left( \sqrt{5}-\sqrt{3} \right)}{\left( \sqrt{5}+\sqrt{3} \right)\left( \sqrt{5}-\sqrt{3} \right)} \right]+\left[ \frac{7\left( \sqrt{5}+\sqrt{3} \right)}{\left( \sqrt{5}-\sqrt{3} \right)\left( \sqrt{5}+\sqrt{3} \right)} \right]$
$=\left[ \frac{\left( 2\sqrt{5}-2\sqrt{3} \right)+\left( 7\sqrt{5}+7\sqrt{3} \right)}{\left( \sqrt{5}+\sqrt{3} \right)\left( \sqrt{5}-\sqrt{3} \right)} \right]$
$=\left[ \frac{2\sqrt{5}-2\sqrt{3}+7\sqrt{5}+7\sqrt{3}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} \right]$
$=\left[ \frac{(2+7)\sqrt{5}+(7-2)\sqrt{3}}{5-3} \right]$
$=\left[ \frac{9\sqrt{5}+5\sqrt{3}}{2} \right]$
Since, $\sqrt{5}=2.236$ and $\sqrt{3}=1.732$
$=\left[ \frac{(9\times 2.236)+(5\times 1.732)}{2} \right]$
$=\left[ \frac{20.124+8.66}{2} \right]$
$=\left[ \frac{28.784}{2} \right]$
Thus, $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}=14.392$
5. Find the value of $\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{7}{\sqrt{5}-\sqrt{2}}$, if $\sqrt{5}=2.236$ and $\sqrt{2}=1.414$.
$\sqrt{2}=1.414$
Now, $\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{7}{\sqrt{5}-\sqrt{2}}$
$\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{7}{\sqrt{5}-\sqrt{2}}=\left[ \frac{3}{\left( \sqrt{5}+\sqrt{2} \right)}\times \frac{\left( \sqrt{5}-\sqrt{2} \right)}{\left( \sqrt{5}-\sqrt{2} \right)} \right]+\left[ \frac{7}{\left( \sqrt{5}-\sqrt{2} \right)}\times \frac{\left( \sqrt{5}+\sqrt{2} \right)}{\left( \sqrt{5}+\sqrt{2} \right)} \right]$
$=\left[ \frac{3\left( \sqrt{5}-\sqrt{2} \right)}{\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)} \right]+\left[ \frac{7\left( \sqrt{5}+\sqrt{2} \right)}{\left( \sqrt{5}-\sqrt{2} \right)\left( \sqrt{5}+\sqrt{2} \right)} \right]$
$=\left[ \frac{\left( 3\sqrt{5}-3\sqrt{2} \right)+\left( 7\sqrt{5}+7\sqrt{2} \right)}{\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)} \right]$
$=\left[ \frac{3\sqrt{5}-3\sqrt{2}+7\sqrt{5}+7\sqrt{2}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}} \right]$
$=\left[ \frac{(3+7)\sqrt{5}+(7-3)\sqrt{2}}{5-2} \right]$
$=\left[ \frac{10\sqrt{5}+4\sqrt{2}}{3} \right]$
Since, $\sqrt{5}=2.236$ and $\sqrt{2}=1.414$
$=\left[ \frac{(10\times 2.236)+(4\times 1.414)}{3} \right]$
$=\left[ \frac{22.36+5.656}{3} \right]$
$=\left[ \frac{28.016}{3} \right]$
Thus, $\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{7}{\sqrt{5}-\sqrt{2}}=\frac{28.016}{3}$
6. Simplify $\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}$
Ans: $\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}$
\[\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}=\left[ \frac{2+\sqrt{5}}{2-\sqrt{5}}\times \frac{\left( 2+\sqrt{5} \right)}{\left( 2+\sqrt{5} \right)} \right]+\left[ \frac{2-\sqrt{5}}{2+\sqrt{5}}\times \frac{\left( 2-\sqrt{5} \right)}{\left( 2-\sqrt{5} \right)} \right]\]
\[=\left[ \frac{\left( 2+\sqrt{5} \right)\left( 2+\sqrt{5} \right)}{\left( 2-\sqrt{5} \right)\left( 2+\sqrt{5} \right)} \right]+\left[ \frac{\left( 2-\sqrt{5} \right)\left( 2-\sqrt{5} \right)}{\left( 2+\sqrt{5} \right)\left( 2-\sqrt{5} \right)} \right]\]
\[=\left[ \frac{{{\left( 2+\sqrt{5} \right)}^{2}}+{{\left( 2-\sqrt{5} \right)}^{2}}}{\left( 2-\sqrt{5} \right)\left( 2+\sqrt{5} \right)} \right]\]
Using the identities –
\[(a+b)(a-b)={{a}^{2}}-{{b}^{2}}\]
\[{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]
\[{{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]
\[=\left[ \frac{\left( {{\left( 2 \right)}^{2}}+{{\left( \sqrt{5} \right)}^{2}}+\left( 2\times 2\times \sqrt{5} \right) \right)+\left( {{\left( 2 \right)}^{2}}+{{\left( \sqrt{5} \right)}^{2}}-\left( 2\times 2\times \sqrt{5} \right) \right)}{{{\left( 2 \right)}^{2}}-{{\left( \sqrt{5} \right)}^{2}}} \right]\]
\[=\left[ \frac{\left( 4+5+\left( 4\sqrt{5} \right) \right)+\left( 4+5-\left( 4\sqrt{5} \right) \right)}{4-5} \right]\]
$=\left[ \frac{9+9}{-1} \right]$
$=\left[ \frac{18}{-1} \right]$
Thus, $\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}=\left( -18 \right)$
7. Find a and b, if $\frac{3-\sqrt{6}}{3+2\sqrt{6}}=a\sqrt{6}-b$
Ans: $\frac{3-\sqrt{6}}{3+2\sqrt{6}}=a\sqrt{6}-b$
Here,
$LHS=\frac{3-\sqrt{6}}{3+2\sqrt{6}}$
$RHS=a\sqrt{6}-b$
Start by rationalizing the denominator in LHS
In order to rationalize the denominator, we multiply and divide $\frac{3-\sqrt{6}}{3+2\sqrt{6}}$ by $3+2\sqrt{6}$
$\frac{3-\sqrt{6}}{3+2\sqrt{6}}\times \frac{3-2\sqrt{6}}{3-2\sqrt{6}}=\frac{\left( 3-\sqrt{6} \right)\left( 3-2\sqrt{6} \right)}{\left( 3+2\sqrt{6} \right)\left( 3-2\sqrt{6} \right)}$
$=\frac{\left( 3\times 3 \right)-\left( 3\times 2\sqrt{6} \right)-\left( \sqrt{6}\times 3 \right)+\left( \sqrt{6}\times 2\sqrt{6} \right)}{{{\left( 3 \right)}^{2}}-{{\left( 2\sqrt{6} \right)}^{2}}}$
$=\frac{\left( 9 \right)-\left( 6\sqrt{6} \right)-\left( 3\sqrt{6} \right)+\left( 12 \right)}{9-24}$
$=\frac{\left( 21 \right)-\left( 9\sqrt{6} \right)}{-15}$
$=\frac{\left( 21 \right)}{-15}-\frac{\left( 9\sqrt{6} \right)}{-15}$
They are all divisible by $3$
$=-\frac{7}{5}+\frac{\left( 3\sqrt{6} \right)}{5}$
Thus, $LHS=\frac{3}{5}\sqrt{6}-\frac{7}{5}$
Comparing with RHS, we get –
Thus,
$a=\frac{3}{5}$
$b=\frac{7}{5}$
Important Questions for Class 9 Maths Chapter 1 - Free PDF Download
Class 9 is like beginning to your academics career, which is right before board 10th. Thus students need to be very serious regarding their studies during preparation. No matter what we learn in class 9, it is important to clarify your concepts better. Hence, clearing Number System concepts will help students further apply in electronics physics and higher maths. Thus it is better to build a good base in mathematics with these Important Questions Maths Class 9 Chapter 1.
Vedantu provides a free PDF to download for Class 9 Chapter 1 Important Questions such that students can prepare well according to the CBSE syllabus. CBSE is strict to its pattern and follows the same throughout the question paper set. Students need to understand these guidelines and find solutions with a proper explanation. This free PDF online will surely help students master their concepts and build a Number System base. This PDF covers all the important concepts in the form of question example to learn how to implement them during exams. Thus PDF proves to be magical for those who are weak in mathematics as they also get solutions to the concepts covered in the back exercise.
Number System Class 9 Important Questions
Before you begin practising Class 9 Chapter 1 Maths Important Questions, you need to know the different topics and subtopics to cover in the chapter. Chapter 1 of Mathematics Class 9 covers a total of 6 exercises with a small introduction of the number system, number lines, defining real numbers, natural numbers, whole numbers, rational, and irrational numbers. Also, students become familiar with the concepts of addition, subtraction, division, and multiplication of the real numbers. The last topic in the chapter will be covering the law of exponents in the real numbers.
Below are the mentioned section-wise topics and concepts that a student to prepare through important questions.
Exercise 1.1
Under exercise 1.1, students will become familiar with the basic understanding of rational and irrational numbers. There is also a revision on whole numbers, real number, integers, and natural numbers and definition. They will even know how to represent a number in the form of p/q, where q is not equal to 0.
Exercise 1.2
Further moving to in-depth study about rational and irrational numbers, there are questions on justification with true and false. Also, students become familiar with the concept of representing rational and irrational numbers on the number line. A new concept of constructing a square root spiral comes in exercise 1.2.
Exercise 1.3
Here the student will know how to represent fractions into decimal form and find if it is terminating or non-terminating. Thus the concepts of terminating and non-terminating fractions will further help to identify it is a rational or irrational number.
Exercise 1.4
A new concept of representing decimal expansion on the number line is introduced through exercise 1.4 mathematics class 9. Here students will learn about magnifying the number to the maximum requirement and representing it on the number line. These decimal places can be either terminating or non-terminating. Hence there are two different concepts in number line representation of decimal expansions.
Exercise 1.5
Now comes the basic calculations of different rational and irrational numbers. It includes addition, multiplication, subtraction, and division of the rational and irrational numbers. Questions will be based on such concepts, and thus you have to simplify the statement accordingly. Also, students become familiar with the concept of rationalising.
Exercise 1.6
Here students will learn to solve questions based with a number having power in fractional form. Also, it might cover the basic addition and subtraction of powers for in-depth conceptual and extra knowledge.
These are the six exercises which will be covered under Chapter 1 Maths Class 9 Important Questions. Hence students can prepare questions according to the concepts discussed above.
Chapter 1 Maths Class 9 Important Questions
According to the syllabus mentioned above exercise-wise, below are some important questions covered to let students prepare well for important questions for class 9 maths number system. These exercise-wise solutions will let students master each concept in detail. Below are some of the questions that are usually picked to set question paper as prepared by CBSE.
State if zero is a rational number. Justify your statement by representing it in p/q where q is not equal to 0 and p and q are both integers.
Find 5 different rational numbers between 5 and 6. Mention each step in detail.
Find out 5 different rational numbers between 12/11 and 10/11.
Justify your statement for the following terms stating true or false.
Rational numbers are real.
A number line having representation in the form of √m has m as a natural number.
A real number is always an irrational number.
Check if true or false. Square roots of all positive numbers will be irrational. Explain your answer statement with the help of an example.
Try to represent √5 on the number line.
Represent the following in the form of decimal expansion:
(i) 36/100 (ii) 1/11 (iii) 4⅛ (iv) 3/13 (v) 2/11 (vi) 329/400
Turn 2/7, 13/7, 4/7, 5/7, 6/7
into decimal expansions without actually doing any long division calculations. Here 1/7= 0.142857.
Express the following as fractional form p/q where q is not 0.
(i) \[0.\overline{6}\]
(ii) \[0.4 \overline{7}\]
(iii) \[0. \overline{001}\]
Represent 2.675 on the number line with number line magnification.
How will you visualise 6.2626…... on the number line up to 4 decimal places?
State if the numbers are rational or irrational.
(i) \[2 - \sqrt{5}\] (ii) \[(3 + \sqrt{23}) - \sqrt{23}\] (iii) \[\frac{2\sqrt{7}}{7\sqrt{7}}\] (iv) \[\frac{1}{\sqrt{2}}\] (v) \[2 \pi\]
Simplify the following terms:
(i) \[(3 + \sqrt{3}) ( 2 + \sqrt{2})\] (ii) \[(3 + \sqrt{3}) (3 + \sqrt{3})\]
(iii) \[(\sqrt{5} + \sqrt{2})^{2}\] (iv)\[(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})\]
Rationalise the following terms with their denominators:
(i) \[\frac{1}{7}\] (ii) \[\frac{1}{\sqrt{7} - \sqrt{6}}\] (iii) \[\frac{1}{\sqrt{5} + \sqrt{2}}\] (iv) \[\frac{1}{\sqrt{7} - 2}\]
(i) \[64^{1}{2}\] (ii) \[32^{1}{5}\] (iii) \[125^{1}{3}\]
Class 9 Maths Chapter 1 Extra Questions
Find three rational numbers between $\frac{1}{3}$ and $\frac{1}{2}$.
Express 0.4323232 in the form of $\frac{a}{b}$ where a and b are integers and b 0.
Simplify and find the value of $(729)^{1/6}$ .
Rationalise the denominator 1 9 + 5 + 6 .
Find 6 rational numbers between 4 and 6.
Simplify $\sqrt[3]{2}$+$\sqrt[4]{3}$ and $\sqrt{5}+\sqrt{2}$ .
Locate $\sqrt{5}$ on the number line.
Visualise the representation of 4.26 on the number line upto 3 decimal places.
Is 2 - 5 a rational number or irrational number?
Convert 0.45 into rational numbers.
Benefits of Important Questions for Class 9 Maths Number System
Number System Class 9 Important Questions PDF is prepared by Vedantu experts and is free of cost. Students can also schedule additional problems to prepare in-depth to clarify their concepts. These questions turn to be magical for those weak in mathematics or do not show interest in it. The questions are prepared precisely accordingly to CBSE guidelines for their question paper pattern such that students need not search them anywhere else.
Below are the mentioned reasons why students should refer to the Important Question of Maths Class 9 Chapter 1:
Students are free to access these important questions PDF. They need not pay for any study material on Vedantu website.
These Number System Class 9 Important Questions are helpful during exams and help students clarify their concepts of homework questions.
These crucial questions are available in PDF format, which can be easily downloaded through the website. Thus students need not unnecessarily waste their precious time in finding solutions to problems.
Students can print these PDF questions and solutions, which reduces the stress of preparing through soft copies.
All the questions and solutions to the questions are prepared according to CBSE guidelines. Thus it will help students to know question paper pattern.
There is a total of 100 marks for the Class 9 Mathematics paper where 20 marks are for internal assessment, and rest 80 is for the written exam. The marks weightage for Chapter 1 Maths Class 9 is 8 out of 80. Rest 72 marks are for rest of the syllabus of Class 9 Mathematics.
There are a total of 26 sums to be solved according to NCERT book for Chapter 1 Of Class 9 Maths in its six exercises. However, there are other Class 9 Maths Chapter 1 Important Questions prepared by the experts and solutions to each of them provided. Thus, students can build a good base in mathematics through this PDF of important questions available with Vedantu for free. According to exercise-wise, some questions describe different questions and situations that students generally encounter while sitting in the exam. Thus students need to struggle to find essential concepts in the chapter and solutions to each of them.
Important Related Links for CBSE Class 9
Faqs on cbse class 9 maths important questions for chapter 1 - number system.
1. What number of questions are there in each exercise of Chapter 1 of Class 9 Maths?
Class 9 Maths Chapter 1 is based on real numbers. There are six exercises. In Chapter 1, students will learn about the different concepts of real numbers. The exercises include Exercise 1.1, which contains four questions, in Exercise 1.2, there are four questions, in Exercise 1.3, there are nine questions, in Exercise 1.4, there are two questions, in Exercise 1.5 there are five questions and in Exercise 1.6 there are three questions. Students should practice all NCERT questions by Vedantu given for Chapter 1 to understand the entire Chapter 1 of Class 9 maths.
2. Is Chapter 1 of Class 9 Maths difficult to solve?
Students of Class 9 should understand the concepts of real numbers given in Chapter 1 of Class 9 Maths. It is not difficult to solve if students understand the basic concepts of real numbers. Students can take help from the NCERT solutions Class 9 Maths Chapter 1 given at Vedantu app and website. All concepts related to the real numbers are explained in a simple way for quick understanding of the students.
3. Why are Class 9 Maths NCERT Solutions of Chapter 1 important?
Students should practise Class 9 Maths NCERT Solutions for Chapter 1 to understand and practise questions based on real numbers. All NCERT Solutions are important as students can get similar questions in their exams. Students can easily solve the questions on real numbers in the exam well with the help of these questions. They can find all NCERT Solutions for Class 9 Maths Chapter 1 on Vedantu platform free of cost.
4. What are five rational numbers between ⅖ and ⅗?
Students can find five or more rational numbers between the two given rational numbers easily. They can follow the given steps to find five rational numbers between ⅖ and ⅗. We have to multiply the numerator and denominator of the given rational numbers with the same number.
⅖ x 6/6 = 12/30
⅗ x 6/6= 18/30
Now, we can write five rational numbers between 12/30 and 18/30. Thus, the five rational numbers are 13/30, 14/30, 15/30, 16/30, 17/30.
5. What are the main topics covered in Chapter 1 of Class 9 Maths?
Students will learn topics related to rational and irrational numbers in Chapter 1 of Class 9 Maths. They will study irrational numbers, representation of real numbers on a number line, expression of real numbers in the decimal form, different operations on real numbers, laws of exponents related to real numbers. Students should practice all NCERT questions given in the textbook to understand the different concepts of rational numbers in Class 9 Chapter 1.
CBSE Class 9 Maths Important Questions
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Chapter 1 Class 9 Number Systems
Get solutions of all NCERT Questions of Chapter 1 Class 9 Number System free at teachoo. Answers to all NCERT Exercises and Examples are solved for your reference. Theory of concepts is also made for your easy understanding
In this chapter, we will learn
- Different Types of numbers like Natural Numbers, Whole numbers, Integers, Rational numbers
- How to find rational numbers between two rational numbers
- What is an irrational number
- Checking if number is irrational or not
- And how to draw an irrational number on the number line
- Then, we will study What a real number is
- And find Decimal expansions - Terminating, Non terminating - repeating, Non terminating Non repeating
- Converting non-terminating repeating numbers into p/q form
- Finding irrational numbers between two numbers
- Representing real numbers on the number line (we use magnification)
- We will learn how to add , subtract and multiply numbers with square root (like 5√2 + 3√3 - 8√2)
- We will learn some identities of numbers with square root (like (√a + √b) 2 )
- How to rationalize numbers
- We will also do questions on Law of Exponents (here, the exponents can also be in fractions)
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Class 9 Math Number System Notes, Important Questions & Practice Paper
Class 9 Math Number System – Get here the Notes, Question & Practice Paper for Class 9 Number System. Candidates who are ambitious to qualify the Class 9 with good score can check this article for Notes, Question & Practice Paper. Below we provided the link to access the Notes, Important Question & Practice Paper of Class 9 Math for topic Number System. You can practice the questions and check your answers from the solutions given after question. By practicing this resources candidates definitely get the idea of which his/her weak areas and how to prepare well for the examination.
- Subject: Math
- Topic: Number System
- Resource: Notes, Important Question & Practice Paper
Class 9 Math Number System Notes, Important Question & Practice Paper
Candidates who are pursuing in the Class 9 are advised to solve the Question Paper and revised the notes from this post. With the help of Notes, candidates can plan their Strategy for particular weaker section of subject and study hard. So, go ahead and check the Important Question & Practice Paper for Class 9 Math Number System from the link given below in this article.
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Extra questions for class 9 maths chapter 1, number systems | real numbers | rational & irrational numbers.
NCERT Class 9 Math / Number System Extra Questions
C hapter 1 of CBSE NCERT Class 9 Math covers number systems. Concepts covered in chapter 1 include rational numbers, irrational numbers, rationalizing irrational numbers by multiplying with their conjugates, decimal expansion of real numbers, operations on real numbers and laws of exponents or rules of indices. The extra questions given below include questions akin to HOTS (Higher Order Thinking Skills) questions and exemplar questions of NCERT.
Here is a quick recap of the key concepts that are covered in this chapter in the CBSE NCERT Class 9 Math text book.
What are rational numbers?
A number that can be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is a rational number.
Possibility 1 : If the decimal expansion of the number is terminating it is a rational number. Note: Integers are terminating decimals and are therefore, rational numbers.
Possibility 2 : If the decimal expansion of the number is non-terminating but is recurring , it is rational. Example \\frac{1}{3}\\) = 0.333.. is a non-terminating recurring decimal and is a rational number.
What are irrational numbers?
A number that CANNOT be written in the form \\frac{p}{q}\\) where p and q are integers and p ≠ 0 is an irrational number.
If the decimal expansion of the number is non-terminating AND non-recurring it is an irrational number. Example: \\sqrt{2}\\), π
How to Rationalize Irrational Numbers?
For an irrational number of the form a + √b, a - √b is its conjugate. And for an irrational number of the from a - √b, a + √b is its conjugate.
Important Laws of Exponents (Rules of Indices)
If a > 0 is a real number and m and n are rational numbers, the following laws of exponents hold good.
- a m × a n = a m + n Example .: 10 3 × 10 2 = 10 3 + 2 = 10 5
- (a m ) n = a mn Example : (10 3 ) 2 = 10 (3 \\times\\) 2) = 10 6
- \\frac{a^m}{a^n}\\) = a (m - n) Example : \\frac{10^3}{10^2}\\) = 10 (3 - 2) = 10
- a m b m = (ab) m Example : 2 2 × 5 2 = (2 × 5) 2 = 10 2
Extra Questions for Class 9 Maths - Number Systems
Prime Factorise & Rationalise Denominator: \\frac{14}{{\sqrt {108}} - {\sqrt {96}} + {\sqrt {192}} - {\sqrt {54}}}\\)
Rational numbers - Fractions: Find 5 rational numbers between \\frac{3}{4}) and \\frac{4}{5}).
Express as Fractions Express 1.363636... in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.
Express in the form \\frac{p}{q}) Express 0.4323232… in the form \\frac{p}{q}), where p and q are integers and q ≠ 0.
Simplify the following (a) \({8 + \sqrt{5})}) \({8 - \sqrt{5})}) (b) \({10 + \sqrt{3})}) \({6 + \sqrt{2})}) (c) \{(\sqrt {3} + \sqrt {11})}^2) + \{(\sqrt {3} - \sqrt {11})}^2)
Rationalize the denominator: (a) \\frac{2}{\sqrt{3} - 1}) (b) \\frac{7}{\sqrt{12} - \sqrt{5}}) (c) \\frac{1}{8 + 3\sqrt{5}}) (d) \\frac{1}{4 + \sqrt{2} + \sqrt{5}})
Simplify and find the value of (a) \{(729)}^{\frac{1}{6}}) (b) \{(64)}^{\frac{2}{3}}) (c) \{(243)}^{\frac{6}{5}}) (d) \{(21)}^{\frac{3}{2}} \times {(21)}^{\frac{5}{2}}) (e) \\frac{{(81)}^{\frac{1}{3}}}{{(81)}^{\frac{1}{12}}})
Operation on real numbers & Algebraic identities If x = \\frac{3 - {\sqrt{13}}}{2}\\), what is the value of \x^2 + \frac{1}{x^2}\\)?
Rationalise & find value of cubic expression If x = \\frac{1}{8-\sqrt{60}}\\), what is the value of (x 3 - 5x 2 + 8x - 4) ?
Question 10
Rationalise the denominator \\frac{1}{9 + {\sqrt{5} + \sqrt{6}}}\\)
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- Class 9 Maths MCQs
- Chapter 1 Number System Mcq
Class 9 Maths Chapter 1 Number System MCQs
Class 9 Maths Chapter 1 Number System MCQs are available here with solutions. The MCQs are prepared as per the CBSE syllabus (2022-2023) and NCERT curriculum. These objective questions are given online, as per the latest exam pattern. Also, we have provided detailed explanations for some questions to help students understand the concept very well. Solving the chapter-wise MCQs for 9th Standard Maths subject will help students to boost their problem-solving skills and confidence. Also, check Important Questions for Class 9 Maths here.
Download the below PDF to get more MCQs on Class 9 Maths Chapter 1 Number System.
Class 9 Maths Chapter 1 Number System MCQs – Download PDF
MCQs on Class 9 Maths Chapter 1 Number System
Check the below multiple choice questions for 9th Class Maths chapter 1-Number system. All MCQs have four options, out of which only one is correct. Students have to choose the correct option and check the answer with the provided one.
1.) Can we write 0 in the form of p/q?
c. Cannot be explained
d. None of the above
Explanation: 0 is a rational number and hence it can be written in the form of p/q.
Example: 0/4 = 0
2.) The three rational numbers between 3 and 4 are:
a. 5/2, 6/2, 7/2
b. 13/4, 14/4, 15/4
c. 12/7, 13/7, 14/7
d.11/4, 12/4, 13/4
Explanation: There are many rational numbers between 3 and 4
To find 3 rational numbers, we need to multiply and divide both the numbers by 3+1 = 4
Hence, 3 x (4/4) = 12/4 and 4 x (4/4) = 16/4
Thus, three rational numbers between 12/4 and 16/4 are 13/4, 14/4 and 15/4.
3.) In between any two numbers, there are:
a. Only one rational number
b. Two rational numbers
c. Infinite rational numbers
d. No rational number
Explanation: Take the reference from question number 2 explained above.
4.) Every rational number is:
a. Whole number
b. Natural number
d. Real number
Explanation: Real number consist of all the rational and irrational numbers. A rational number is a number that is represented in the form of P/Q, where Q is not equal to zero and both P and Q are integers. For example, 1/2 is a rational number, but not a whole number, a natural number or an integer.
5.) √9 is __________ number.
a. A rational
b. An irrational
c. Neither rational nor irrational
Explanation: √9 = 3
Hence, √9 is a rational number.
6.) Which of the following is an irrational number?
Hence, √12 cannot be simplified to a rational number.
7.) 3√6 + 4√6 is equal to:
Explanation: 3√6 + 4√6 = (3 + 4)√6 = 7√6
8.) √6 x √27 is equal to:
Explanation:
= (3 × 3)√2
9.) Which of the following is equal to x 3 ?
a. x 6 – x 3
b. x 6 .x 3
c. x 6 /x 3
d. (x 6 ) 3
Explanation: x 6 /x 3 = x 6 – 3 = x 3
10.) Which of the following is an irrational number?
d. 7.478478
Explanation: √23 = 4.79583152331…
Since the decimal expansion of the number is non-terminating non-recurring. Hence, it is an irrational number.
But, √225 = 15, 0.3796 and 7.478478 are terminating.
11.) Which of the following is an irrational number?
d. 0.4014001400014…
Explanation: 0.4014001400014…is an irrational number as it is non-terminating and non-repeating.
12.) 2√3+√3 =
Explanation: 2√3+√3 = (2+1)√3= 3√3.
14.) The number obtained on rationalising the denominator of 1/ (√7 – 2) is
a. (√7+2)/3
b. (√7-2)/3
c. (√7+2)/5
d. (√7+2)/45
\(\begin{array}{l}\frac{1}{\sqrt{7}-2} = \frac{1}{\sqrt{7}-2}\times \frac{\sqrt{7}+2}{\sqrt{7}+2} = \frac{\sqrt{7}+2}{(\sqrt{7})^{2}-(2)^{2}} = \frac{\sqrt{7}+2}{3}\end{array} \)
15.) Which of the following is rational?
Explanation: 0/4 is a rational number that is equal to 0. Whereas π and √3 are irrational numbers and 4/0 is undefined.
16.) The irrational number between 2 and 2.5 is
Explanation: The irrational number between 2 and 2.5 is √5 because the approximate value of √5 is 2. 23606…
17.) The value of √10 times √15 is equal to
Explanation: √10 × √15 =(√2.√5) ×( √3. √5) = (√5 × √5) (√2 × √3) = 5√6.
18.) The decimal representation of the rational number is
a. Always terminating
b. Either terminating or repeating
c. Either terminating or non-repeating
d. Neither terminating nor repeating
Explanation: As per the definition of rational number, its decimal representation is either terminating or repeating.
19.) Which of the following is a rational number?
Explanation: 0 is a rational number, and it can be written as 0/1 or 0/2 or 0/3 etc. Whereas 2√3, 2+√3, and π are irrational numbers.
20.) Which of the following is an irrational number?
Explanation: √7 is an irrational number. Because other options given are simplified into a rational number.
Related Articles for Class 9
- Number System for Class 9
- Number System Questions
- Number System PDF
- Important Questions Class 9 Maths Chapter 1 Number System
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NCERT Solutions Class 9 Maths Chapter 1 Number Systems
NCERT solutions for class 9 maths chapter 1 number systems consists of an introduction about the number system and the different kinds of numbers in it. The number system has been classified into different types of numbers like natural numbers, whole numbers , integers, rational numbers, irrational numbers , etc. The NCERT solutions class 9 maths chapter 1 covers all the basics of the number system which will be helpful in forming the basic foundation of mathematics.
Class 9 maths chapter 1 number systems will help the students in differentiating between rational and irrational numbers, wherein irrational numbers cannot be expressed in the form of a ratio, and also about real numbers. Class 9 maths NCERT solutions chapter 1 number systems sample exercises can be downloaded from the links below and also you can find some of these in the exercises given below.
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.1
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.2
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.3
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.4
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.5
- NCERT Solutions Class 9 Maths Chapter 1 Ex 1.6
NCERT Solutions for Class 9 Maths Chapter 1 PDF
These NCERT solutions for class 9 maths involving the important concepts of real numbers , rational and irrational numbers, are available for free pdf download. The questions involving real numbers and their decimal form, the law of exponents are given below:
☛ Download Class 9 Maths NCERT Solutions Chapter 1 Number Systems
NCERT Class 9 Maths Chapter 1 Download PDF
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
It is advisable for the students to practice the questions in the above links as this will give them better clarity on the kind of numbers and their properties. An exercise-wise detailed analysis of NCERT Solutions Class 9 Maths Chapter 1 number systems is given below for reference.
- Class 9 Maths Chapter 1 Ex 1.1 - 4 Questions
- Class 9 Maths Chapter 1 Ex 1.2 - 4 Questions
- Class 9 Maths Chapter 1 Ex 1.3 - 9 Questions
- Class 9 Maths Chapter 1 Ex 1.4 - 2 Questions
- Class 9 Maths Chapter 1 Ex 1.5 - 5 Questions
- Class 9 Maths Chapter 1 Ex 1.6 - 11 Questions
☛ Download Class 9 Maths Chapter 1 NCERT Book
Topics Covered: The important topics focussed upon are irrational numbers, real numbers, and real numbers when expanded in the decimal form. The class 9 maths NCERT solutions chapter 1 covers the representation of real numbers on a number line, methods to perform operations on real numbers, and laws of exponents when dealing with real numbers.
Total Questions: Class 9 maths chapter 1 Number Systems consists of total 35 questions of which 30 are easy, 2 are moderate and 3 are long answer-type questions.
List of Formulas in NCERT Solutions Class 9 Maths Chapter 1
NCERT solutions class 9 maths chapter 1 covers important facts about the number systems which will help strengthen the math foundation. Like if a number ‘a’ is rational, and ‘b’ represents an irrational number, then ‘a+b’, and ‘a-b’ are irrational numbers, and ‘ab’ and ‘a/b’ are supposed to be irrational numbers, and ‘b’ is not equal to zero. For ‘a’ and ‘b’ positive real numbers the following formula or entities will be true:
- √ab = √a √b
- √(a/b) = √a / √b
Important Questions for Class 9 Maths NCERT Solutions Chapter 1
Video solutions for class 9 maths ncert chapter 1, faqs on ncert solutions class 9 maths chapter 1, do i need to practice all questions provided in ncert solutions class 9 maths number systems.
Practicing the NCERT solutions class 9 maths number systems and exercises on real numbers, rational numbers will help in exploring the number systems in a better way. The NCERT Solutions Class 9 Maths Number Systems will also provide a good insight into the solving of problems.
Why are Class 9 Maths NCERT Solutions Chapter 1 Important?
Since the number systems chapter deals with rational and irrational numbers, real numbers, and their expansion, their decimal form, also covering the law of exponents. Hence, this makes the NCERT solutions class 9 maths important for examinations.
What are the Important Formulas in NCERT Solutions Class 9 Maths Chapter 1?
There are several formulas or entities for positive real numbers which will be helpful in learning mathematics even for higher grades. Like if one wants to rationalize the denominator of 1/ ( √a + b ), then we can multiply and divide by its algebraic conjugate which is √a - b
How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 1 Real Numbers?
The questions in the NCERT Solutions Class 9 Maths Chapter 1 are a great way for learning real numbers. There are around 35 questions dealing with number systems with 25 of them being simple and have straightforward logic, 6 of them are with medium complexity and 4 are elaborative questions.
What are the Important Topics Covered in NCERT Solutions Class 9 Maths Chapter 1?
The NCERT Solutions Class 9 Maths Chapter 1 deal with integers, real numbers, rational and irrational numbers. Apart from these the important topics covered are the real numbers, and what happens when they are expanded in decimal form, the law of exponents in the case of real numbers, how to differentiate between rational and irrational numbers etc.
How CBSE Students can utilize NCERT Solutions Class 9 Maths Chapter 1 effectively?
The students should first practice all the examples to understand the logic and problem solving technique and should try to solve all the exercise questions. The CBSE itself recommends the NCERT Solutions Class 9 Maths for the board exam studies.
Case study questions for class 9 maths number system
Case Study Questions: Question 1: Himanshu has made a project on real numbers, where he finely explained the applicability of exponential laws
CBSE Case Study Questions for Class 9 (2021
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Case study questions chapter 1: real numbers.
Full syllabus notes, lecture & questions for Number System- Case Based Type Questions - Notes | Study Mathematics (Maths) Class 9 - Class 9 - Class 9 | Plus
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case study based on Number System class 9
Class 9 Mathematics Case study question 2 What is the value of x? 48 96 100 120 What is the value of y? 48 96 100 24 What is
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Chapter wise extra questions for class 9 maths pdf are give below. Extra Questions for Class 9 Maths Chapter 1 Number Systems. Extra Questions for Class 9 Maths Chapter 2 Polynomials. Extra Questions for Class 9 Maths Chapter 3 Coordinate Geometry. Extra Questions for Class 9 Maths Chapter 4 Linerar Equation in Two Variables.
Class 9 Case Study Questions Maths Polynomials CBSE(NCERT): CLASS IX MATHS. CASE STUDY. QUESTION 24. By. M. S. Kumar Swamy taking the number of children as 'x' and the number of adults as 'y'?
Here, we have provided case-based/passage-based questions for Class 9 Maths Chapter 1 Real Numbers Case Study/Passage-Based Questions Case Study 1: A Mathematics Exhibition is being conducted in your school and one of your friends is making a model of a factor tree.
These worksheets for Grade 9 Mathematics Number System contain important questions which can be asked in various school level tests and examinations. All concepts given in your NCERT book for Class 9 Mathematics Number System have been covered in these Pdf worksheets
maths-ix-case-study-question-22.pdf Like Share Views Add to classroom R Renaissance Das I am experienced Maths teacher. Class Details IXC Maths More from Renaissance Das (20) Test Heron's Formula, Linear Equations In Two Variables class-9th Maths 0 Likes 163 Views R Renaissance Das Jul 28, 2022 Test Integers class-6th Maths 0 Likes 15 Views R
Download PDF Case Study Questions for Class 9 Science to prepare for the upcoming CBSE Class 9 Exams Exam 2022-23. With the help of our well-trained and experienced faculty, we provide solved examples and detailed explanations for the recently added Class 9 Science case study questions. Class 9 Science: Case Study Questions
CBSE | Central Board of Secondary Education : Academics
Chapter 1 of Mathematics Class 9 covers a total of 6 exercises with a small introduction of the number system, number lines, defining real numbers, natural numbers, whole numbers, rational, and irrational numbers. Also, students become familiar with the concepts of addition, subtraction, division, and multiplication of the real numbers.
Below given important Number system questions for 9th class students will help them to get acquainted with a wide variation of questions and thus, develop problem-solving skills. Q.1: Find five rational numbers between 1 and 2. Solution: We have to find five rational numbers between 1 and 2. So, let us write the numbers with denominator 5 + 1 = 6
Case study based questions for class 9.Case study based on number systemrational numbers and irrational numbersnatural numbers. and whole numbers, integers,c...
Get solutions of all NCERT Questions of Chapter 1 Class 9 Number System free at teachoo. Answers to all NCERT Exercises and Examples are solved for your reference. Theory of concepts is also made for your easy understanding. In this chapter, we will learn. Different Types of numbers like Natural Numbers, Whole numbers, Integers, Rational numbers.
Class 9 Number System study Material New: File Size: 606 kb: File Type: pdf: Download File. 9th Numbers System Solved Problems: File Size: 615 kb: File Type: pdf: Download File. IX Maths-Real Numbers guess paper - 01[SA-1] File Size: 462 kb: File Type: pdf: Download File.
Below we provided the link to access the Notes, Important Question & Practice Paper of Class 9 Math for topic Number System. You can practice the questions and check your answers from the solutions given after question. By practicing this resources candidates definitely get the idea of which his/her weak areas and how to prepare well for the ...
Extra Questions for Class 9 Maths - Number Systems Question 1 Prime Factorise & Rationalise Denominator: 14 108 − 96 + 192 − 54 Explanation Video Solution Question 2 Rational numbers - Fractions: Find 5 rational numbers between 3 4 and 4 5. Explanation Video Solution Question 3 Express as Fractions
MCQs on Class 9 Maths Chapter 1 Number System Check the below multiple choice questions for 9th Class Maths chapter 1-Number system. All MCQs have four options, out of which only one is correct. Students have to choose the correct option and check the answer with the provided one. 1.) Can we write 0 in the form of p/q? a. Yes b. No
The class 9 maths NCERT solutions chapter 1 covers the representation of real numbers on a number line, methods to perform operations on real numbers, and laws of exponents when dealing with real numbers. Total Questions: Class 9 maths chapter 1 Number Systems consists of total 35 questions of which 30 are easy, 2 are moderate and 3 are long ...
Case study questions for class 9 maths number system. This Case study questions for class 9 maths number system helps to fast and easily solve any math problems.