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## CBSE Class 9 Mathematics Case Study Questions

## Significance of Mathematics in Class 9

## Case studies in Class 9 Mathematics

## Example of Case study questions in Class 9 Mathematics

The following are some examples of case study questions from Class 9 Mathematics:

## Class 9 Mathematics Case study question 1

Answer the following questions:

## 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:

## Class 9 Mathematics Case study question 3

Class 9 Mathematics Case study question 4

How to Answer Class 9 Mathematics Case study questions

## Class 9 Mathematics Curriculum at Glance

CBSE Class 9 Mathematics (Code No. 041)

## Class 9 Mathematics question paper design

## QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)

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## Case study questions for class 9 maths number system

## Number System

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## Maths Class IX Case Study Questions

## Class 9 Maths: Case Study Questions of Chapter 1 Real Numbers PDF Download

## 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

Observe the following factor tree and answer the following:

1. What will be the value of x?

2. What will be the value of y?

3. What will be the value of z?

4. According to the Fundamental Theorem of Arithmetic 13915 is a

c) Neither prime nor composite

5. The prime factorization of 13915 is

(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

(a) when n is any even integer (b) when n is any odd integer (c) for all n > 1 (d) only when n=0

(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)

(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

(v) If n is any odd integer, then n 2 – 1 is divisible by

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NCERT Books and Solutions for all classes

## Worksheets Class 9 Mathematics Number System Pdf Download

## Class 9 Mathematics Number System Worksheets Pdf Download

## Subjectwise Worksheets for Class 9 Mathematics Number System

## Benefits of Solving Class 9 Mathematics Number System Worksheets

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- All concepts given in your NCERT book for Class 9 Mathematics Number System have been covered in these Pdf worksheets
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- 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

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## Case Study Questions of Class 9(IX) Science

## Class 9 Science: Case Study Questions

## 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

## A look at the Class 9 Science Syllabus

## Unit I: Matter-Nature and Behaviour

## Unit II: Organization in the Living World

## Unit III: Motio n, Force, and Work

## Unit IV: Food Production

## Books for Class 9 Science Exams

## The rationale behind Science

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## CBSE Class 9 Maths Important Questions for Chapter 1 - Number System

## CBSE Class 9 Maths Important Questions Chapter 1 - Number System Free PDF Download

## 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

Real Number and Their Decimal Expansion

Representation of Real Number on Number Line

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.

Ans: We know that the square root of every positive integer will not yield an integer.

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: 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:

4. Which of the following rational numbers have terminating decimal representation?

5. How many rational numbers can be found between two distinct rational numbers?

6. The value of $\left( \text{2+}\sqrt{\text{3}} \right)\left( \text{2-}\sqrt{\text{3}} \right)$ in

7. ${{\left( \text{27} \right)}^{\text{-2/3}}}$ is equal to

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) $\frac{\text{63}}{\text{9}}$

11. $\text{0}\text{.83458456}......$ is

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) $\frac{\text{1}}{\text{8}}$

14. The value of $\sqrt[\text{3}]{\text{1000}}$ is

15. The sum of rational and an irrational number

(i) $\frac{\text{49}}{\text{75}}$

(ii) $\frac{\text{50}}{\text{75}}$

(iii) \[\frac{\text{47}}{\text{75}}\]

(iv) $\frac{\text{46}}{\text{75}}$

17. $\text{0}\text{.12}\overline{\text{3}}$ is equal to

(i) $\frac{\text{122}}{\text{90}}$

(ii) $\frac{\text{122}}{\text{100}}$

(iii) $\frac{\text{122}}{\text{99}}$

18. The number ${{\left( \text{1+}\sqrt{\text{3}} \right)}^{\text{2}}}$ is

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) $\text{10}\sqrt{\text{6}}$

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) $0.\overline{23}=0.232323....$

$0.\overline{22}=0.222222....$

$0.\overline{23}+0.\overline{22}=0.454545....$

(ii) $\frac{\text{1}}{\text{2}}$

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: $16\sqrt{13}\div 9\sqrt{52}$

$\frac{16\sqrt{13}}{9\sqrt{52}}=\frac{16}{9}\sqrt{\frac{13}{52}}=\frac{8}{9}$

Ans: (D) $\sqrt{8}$ is an irrational number

$\therefore \sqrt{4\times 2}=2\sqrt{2}$

Zero can be written as $\frac{0}{1},\frac{0}{2},\frac{0}{3},\frac{0}{4},\frac{0}{5}......$

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.

We know that the numbers $3.1,3.2,3.3,3.4,3.5$ and $3.6$ all lie between $3$ and $4$.

3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

On converting the fractions, we get $\frac{31}{50},\frac{16}{25}\text{ and }\frac{13}{20}$.

4. Show how $\sqrt{5}$ can be represented on the number line.

Ans: According to Pythagoras theorem, we can conclude that

${{\left( \sqrt{5} \right)}^{2}}={{\left( 2 \right)}^{2}}+{{\left( 1 \right)}^{2}}$.

Then draw the arc $ACD$, to get the number $\sqrt{5}$ on the number line.

(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 $\frac{1}{7}=0.142857....$

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.... $

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

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 $1$.

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 $3\text{ and }~4$.

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 $4\text{ and }5$.

10. Represent $9.3$ on the number line.

11. Find (i) ${{64}^{\frac{1}{5}}}$ (ii) ${{32}^{\frac{1}{5}}}$ (iii) ${{125}^{\frac{1}{3}}}$

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 $8$.

$\sqrt[5]{32}=\sqrt[5]{2\times 2\times 2\times 2\times 2}=2$

Therefore, the value of ${{32}^{\frac{1}{5}}}$will be $2$.

$\sqrt[3]{125}=\sqrt[3]{5\times 5\times 5}=5$

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 $3\text{ and }4\text{ is }12$

$={{\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},\frac{5}{12}\text{ and }\frac{1}{3} $

Second rational number 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 $\sqrt{6}$.

$\sqrt{6}\text{ and }\sqrt{24}$ 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) $\sqrt[4]{1250}$

(ii) $\sqrt[4]{1250}$$\begin{align}

$={{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 $\sqrt{3}$

Ans: $\frac{1}{\sqrt{5}+\sqrt{2}}\times \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

$=\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)

$x$ 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 ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$.

23. Evaluate $\frac{{{11}^{\frac{5}{2}}}}{{{11}^{\frac{3}{2}}}}$.

$ ={{11}^{\frac{5-3}{2}}}={{11}^{\frac{2}{2}}} $

24. Find four rational numbers between $\frac{3}{7}$ and $\frac{4}{7}$.

25. Write the following in decimal form (i) $\frac{36}{100}$ (ii) $\frac{2}{11}$

(ii) $\frac{2}{11}=0.\overline{18}$

26. Express $2.417\overline{8}$ in the form $\frac{a}{b}$

$10x=24.\overline{178}$$......(1)$$[\text{Multiplying both sides by }10]$

$1000\times 10x=1000\times 24.178178178....$Multiplying both sides by 1000

$ 10000x=24178.\overline{178}\text{ }......(2) $

Subtracting $(1)\text{ from }(2)$

$10,000x-x=24178.\overline{178}-24.\overline{178} $

$ 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 $

$ ={{675}^{\frac{1}{6}}}=\sqrt[6]{675} $

29. Convert $0.\overline{25}$ into rational number.

Ans: Let \[x=0.\overline{25}\] ......(i)

$100x=25.\overline{25}$ ......(ii)

$100x-x=25.\overline{25}-0.\overline{25} $

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) $

$ =30+\left( 9+4 \right)\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{1}{{{9}^{\frac{2}{2}}}}=\frac{1}{9} $

1. State whether the following statements are true or false. Give

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 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.

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.

We know that when we take the square root of any number, we cannot receive a negative value.

iii. Every real number is an irrational number.

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 $q\ne 0$.

$\Rightarrow x=0.6666$ ......(a)

Multiplying both sides by 10 we get

We need to subtract (a) from (b), to get

We can also write $9x=6$ as $x=\frac{6}{9}$ or $x=\frac{2}{3}$.

Ans: Let $x=0.4\overline{7}\Rightarrow x=0.47777$ ......(a)

Multiplying both sides by 10 we get

We can also write $9x=4.3$ as $x=\frac{4.3}{9}$ or $x=\frac{43}{90}$

Ans: Let $x=0.\overline{001}\Rightarrow x=0.001001$ ......(a)

Multiplying both sides by 1000 we get

We can also write $999x=1$ as $x=\frac{1}{999}$

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.

Ans: Let us consider the examples of the form $\frac{p}{q}$ that are terminating decimals .

It can be observed that the denominators of the above rational numbers have powers of 2,5 or both.

6. Classify the following numbers as rational or irrational:

We know that $\sqrt{5}=2.236....$, which is an irrational number.

$=-0.236...$, 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 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.

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. $\left( 3+3\sqrt{3} \right)\left( 2+\sqrt{2} \right)$

\[(3+3\sqrt{3})(2+\sqrt{2})=3(2+\sqrt{2})\sqrt{3}(2+\sqrt{2})\]

\[=6+3\sqrt{2}+2\sqrt{3}+\sqrt{6}\]

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}) $

iii. ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$

Ans: ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$

Applying the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$

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 $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$

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

Therefore, the value of ${{9}^{\frac{3}{2}}}$ will be $27$ .

As a result, we can deduce that ${{32}^{\frac{2}{5}}}$ can also be written as

Therefore, the value of ${{32}^{\frac{2}{5}}}$ will be $4$.

As a result, we can deduce that ${{16}^{\frac{3}{4}}}$ can also be written as

Therefore, the value of ${{16}^{\frac{3}{4}}}$ will be $8$ .

Ans: We know that ${{a}^{-n}}=\frac{1}{{{a}^{n}}}$

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)} $

Therefore, the value of ${{125}^{-\frac{1}{3}}}$ will be $\frac{1}{5}$.

i. ${{2}^{\frac{2}{3}}}{{.2}^{\frac{1}{5}}}$

Ans: We know that ${{a}^{m}}.{{a}^{n}}={{a}^{\left( m+n \right)}}$

ii. ${{\left( {{3}^{\frac{1}{3}}} \right)}^{7}}$

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}}} $

iv. ${{7}^{\frac{1}{2}}}{{.8}^{\frac{1}{2}}}$

Ans: We know that ${{a}^{m}}.{{b}^{m}}={{\left( a\times b \right)}^{m}}$

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.\overline{8}.....\left( 2 \right) $

$10x-x=8.\overline{8}-0.\overline{8}$ (Subtracting (1) from (2))

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{{{\left( 7+3\sqrt{5} \right)}^{2}}}{{{7}^{2}}-{{\left( 3\sqrt{5} \right)}^{2}}} $

$=\frac{49+9\times 5+42\sqrt{5}}{49-45} $

$=\frac{49+45+42\sqrt{5}}{4} $

$ =\frac{94}{4}+\frac{42}{4}\sqrt{5} $

$ =\frac{47}{2}+\frac{21}{2}\sqrt{5} $

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}{25} \right)}^{-\frac{1}{4}\times 2}} \right\} $

13. Visualize 3.76 on the number line using successive magnification.

Ans: We are asked to prove the expression,

Let us take the LHS of the given expression that is,

15. Represent $\sqrt{3}$ on number line.

In $\Delta OAB$, by using the Pythagorean theorem we get,

$O{{B}^{2}}={{1}^{2}}+{{1}^{2}}$

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}} $

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[ 9\times 2-4\times 3 \right]\left[ 9\times 2-4\times 3 \right] $

$ =\left[ 18-12 \right]\left[ 18-12 \right] $

17. Express $2.\overline{4178}$ in the form $\frac{p}{q}$.

Ans: Let $\frac{p}{q}=2.\overline{4178}$

$10000\frac{p}{q}=1000\times 24.178178 $

$1000\frac{p}{q}-\frac{p}{q}=24178.178178-14.178178 $

$\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{{{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 $2.\overline{3}.$

Representing the given numbers in decimal form we have,

$ 2.\overline{2}=2.222222222...... $

$ 2.\overline{3}=2.333333333....... $

20. Give an example of two irrational numbers whose

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 $3\sqrt{5}$

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)}{2-3} $

Now, substituting the required values of irrational numbers we get,

$=-5\left[ 1.414-1.732 \right] $

22. Visualize 2.4646 on the number line using successive magnification.

23. Rationalizing the denominator of $\frac{1}{4+2\sqrt{3}}$.

$\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{2\left( 2-\sqrt{3} \right)}{4} $

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.

\[\Rightarrow \frac{x}{5}=\sqrt{2}\]

Let us compare the terms in LHS and RHS.

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 \[3\sqrt[3]{250}+7\sqrt[3]{16}-4\sqrt[3]{54}\].

\[=\left( 15\sqrt[3]{2} \right)+\left( 14\sqrt[3]{2} \right)-\left( 12\sqrt[3]{2} \right)\]

\[=\left( 15+14-12 \right)\sqrt[3]{2}\]

Thus, we get \[3\sqrt[3]{250}+7\sqrt[3]{16}-4\sqrt[3]{54}=17\sqrt[3]{2}\]

27. Simplify \[3\sqrt{48}-\frac{5}{2}\sqrt{\frac{1}{3}}+4\sqrt{3}\].

\[=\left( 12\sqrt{3} \right)-\left( \frac{5\sqrt{3}}{6} \right)+\left( 4\sqrt{3} \right)\]

\[=\left( 12-\frac{5}{6}+4 \right)\sqrt{3}\]

\[=\left( \frac{72-5+24}{6} \right)\sqrt{3}\]

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) $\frac{2}{7}=2\times \frac{1}{7}$

$=2\times 0.\overline{142857}$

$\Rightarrow \frac{2}{7}=0.\overline{285714}$

(ii) $\frac{3}{7}=3\times \frac{1}{7}$

$=3\times 0.\overline{142857}$

$\Rightarrow \frac{3}{7}=0.\overline{428571}$

(iii) $\frac{4}{7}=4\times \frac{1}{7}$

$=4\times 0.\overline{142857}$

$\Rightarrow \frac{4}{7}=0.\overline{571428}$

29. Find $6$ rational numbers between $\frac{6}{5}$ and $\frac{7}{5}$

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,

$\therefore O{{A}^{2}}+A{{B}^{2}}=O{{B}^{2}}$

$\Rightarrow O{{B}^{2}}={{1}^{2}}+{{1}^{2}}$

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:

Ans: Performing long division of $36$ by $100$

Thus, $\frac{36}{100}=0.36$ - this is a terminating decimal.

Ans: Performing long division of $1$ by $11$

It can be seen that performing further division will produce a reminder of $1$ continuously.

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 $8$

Thus, $4\frac{1}{8}=4.125$ - this is a terminating decimal.

Ans: Performing long division of $3$ by $13$

Ans: Performing long division of $2$ by $11$

Ans: Performing long division of $33$ by $8$

Thus, $\frac{329}{400}=0.8225$ - this is a terminating decimal.

2. Classify the following as rational or irrational:

Ans: It is known that $\sqrt{225}=15$, which is an integer.

Thus $\sqrt{225}$ is a rational number.

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.

i.e. $7.478478...=7.\overline{487}$

Then $1000x=7478.478478...\text{ (2)}$

$ \underline{-\text{ }x=\text{ }7.478478...} $

$\Rightarrow x=\frac{7471}{999}$

i.e. $7.\overline{478}=\frac{7471}{999}$

Thus $7.478478...$ is a rational number.

3. Rationalize the denominator of the following:

$\frac{1}{\sqrt{7}}\times \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}$

Rationalizing the denominator of $\frac{1}{\sqrt{7}}$ produces $\frac{\sqrt{7}}{7}$.

ii. $\frac{1}{\sqrt{7}-\sqrt{6}}$

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. $\frac{1}{\sqrt{5}+\sqrt{2}}$

$=\frac{\sqrt{5}-\sqrt{2}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}}$

$=\frac{\sqrt{5}-\sqrt{2}}{5-2}$

$=\frac{\sqrt{5}-\sqrt{2}}{3}$

$\Rightarrow \frac{1}{\sqrt{5}+\sqrt{2}}=\frac{\sqrt{5}-\sqrt{2}}{3}$

$=\frac{\sqrt{7}+2}{{{\left( \sqrt{7} \right)}^{2}}-{{\left( 2 \right)}^{2}}}$

$\Rightarrow \frac{1}{\sqrt{7}+2}=\frac{\sqrt{7}+2}{3}$

Rationalizing the denominator of $\frac{1}{\sqrt{7}-2}$ produces $\frac{\sqrt{7}+2}{3}$.

Long Answer Questions (5 Marks)

Thus, $\frac{329}{400}=s0.8225$ - this is a terminating decimal.

Now, $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}$

$=\left[ \frac{(2+7)\sqrt{5}+(7-2)\sqrt{3}}{5-3} \right]$

$=\left[ \frac{9\sqrt{5}+5\sqrt{3}}{2} \right]$

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, $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{7}{\sqrt{5}-\sqrt{3}}=14.392$

Now, $\frac{3}{\sqrt{5}+\sqrt{2}}+\frac{7}{\sqrt{5}-\sqrt{2}}$

$=\left[ \frac{(3+7)\sqrt{5}+(7-3)\sqrt{2}}{5-2} \right]$

$=\left[ \frac{10\sqrt{5}+4\sqrt{2}}{3} \right]$

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: $\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}$

\[(a+b)(a-b)={{a}^{2}}-{{b}^{2}}\]

\[{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]

\[{{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]

$=\left[ \frac{9+9}{-1} \right]$

$=\left[ \frac{18}{-1} \right]$

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: $\frac{3-\sqrt{6}}{3+2\sqrt{6}}=a\sqrt{6}-b$

$LHS=\frac{3-\sqrt{6}}{3+2\sqrt{6}}$

Start by rationalizing the denominator in LHS

$=\frac{\left( 9 \right)-\left( 6\sqrt{6} \right)-\left( 3\sqrt{6} \right)+\left( 12 \right)}{9-24}$

$=\frac{\left( 21 \right)-\left( 9\sqrt{6} \right)}{-15}$

$=\frac{\left( 21 \right)}{-15}-\frac{\left( 9\sqrt{6} \right)}{-15}$

$=-\frac{7}{5}+\frac{\left( 3\sqrt{6} \right)}{5}$

Thus, $LHS=\frac{3}{5}\sqrt{6}-\frac{7}{5}$

## Important Questions for Class 9 Maths Chapter 1 - Free PDF Download

## Number System Class 9 Important Questions

## Exercise 1.1

## Exercise 1.2

## Exercise 1.3

## Exercise 1.4

## Exercise 1.5

## Exercise 1.6

## Chapter 1 Maths Class 9 Important Questions

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.

A number line having representation in the form of √m has m as a natural number.

A real number is always an irrational number.

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

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.

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) \[(3 + \sqrt{3}) ( 2 + \sqrt{2})\] (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) \[64^{1}{2}\] (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

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## Chapter 1 Class 9 Number Systems

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)

Click on an NCERT Exercise below to get started.

Or you can also check the concepts from the Concept wise. Check it out now

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## Class 9 Math Number System Notes, Important Questions & Practice Paper

## Class 9 Math Number System Notes, Important Question & Practice Paper

## All Topics Maths Notes for Class 9

- Coordinate Geometry
- Surface Area and Volumes
- Herons Formula
- Probability
- Parallelograms And Triangles
- Construction
- Quadrilaterals
- Linear Equations in Two Variables
- Lines and Angles
- Introduction to Euclids Geometry
- Number System
- Polynomials

## Class 9 Math Polynomials Notes, Important Questions & Practice Paper

## ICSE Class 9 Sample Paper 2023 (PDF) – CISCE Sample Paper for Class 9

- CBSE Date Sheet
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## Class Wise Study Material

## Study Material

## FREE CBSE Online Coaching for Class 9 Maths. No Conditions

NCERT Class 9 Math / Number System Extra Questions

## What are rational numbers?

## What are irrational numbers?

## How to Rationalize Irrational Numbers?

## Important Laws of Exponents (Rules of Indices)

- 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

Rational numbers - Fractions: Find 5 rational numbers between \\frac{3}{4}) and \\frac{4}{5}).

Rationalise the denominator \\frac{1}{9 + {\sqrt{5} + \sqrt{6}}}\\)

## Free CBSE Online Coaching Class 9 Maths

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## NCERT Solutions For Class 9 Math

- Number Systems
- Polynomials
- Coordinate Geometry
- Linear Equations
- Euclid's Geometry
- Lines and Angles
- Quadrilaterals
- Areas: Parallelograms & Triangles
- Construction
- Heron's Formula
- Surface Areas & Volumes

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## Class 9 Maths Chapter 1 Number System MCQs

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

1.) Can we write 0 in the form of p/q?

Explanation: 0 is a rational number and hence it can be written in the form of p/q.

2.) The three rational numbers between 3 and 4 are:

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:

Explanation: Take the reference from question number 2 explained above.

c. Neither rational nor irrational

Hence, √9 is a rational number.

6.) Which of the following is an irrational number?

Hence, √12 cannot be simplified to a rational number.

Explanation: 3√6 + 4√6 = (3 + 4)√6 = 7√6

9.) Which of the following is equal to x 3 ?

Explanation: x 6 /x 3 = x 6 – 3 = x 3

10.) Which of the following is an irrational number?

Explanation: √23 = 4.79583152331…

But, √225 = 15, 0.3796 and 7.478478 are terminating.

11.) Which of the following is an irrational number?

Explanation: 0.4014001400014…is an irrational number as it is non-terminating and non-repeating.

Explanation: 2√3+√3 = (2+1)√3= 3√3.

14.) The number obtained on rationalising the denominator of 1/ (√7 – 2) is

15.) Which of the following is rational?

16.) The irrational number between 2 and 2.5 is

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

b. Either terminating or repeating

c. Either terminating or non-repeating

d. Neither terminating nor repeating

19.) Which of the following is a rational number?

20.) Which of the following is an irrational 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 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

## ☛ 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

- 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

## List of Formulas in NCERT Solutions Class 9 Maths Chapter 1

## Important Questions for Class 9 Maths NCERT Solutions Chapter 1

## Why are Class 9 Maths NCERT Solutions Chapter 1 Important?

## What are the Important Formulas in NCERT Solutions Class 9 Maths Chapter 1?

## How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 1 Real Numbers?

## What are the Important Topics Covered in NCERT Solutions Class 9 Maths Chapter 1?

## How CBSE Students can utilize NCERT Solutions Class 9 Maths Chapter 1 effectively?

## Case study questions for class 9 maths number system

## CBSE Case Study Questions for Class 9 (2021

## Number System

Case study questions chapter 1: real numbers.

In just 5 seconds, you can get the answer to your question.

## Student testimonials

## case study based on Number System class 9

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Class 9 Science Case Study Questions Class 9 Maths Syllabus 2022-23 UNIT I: NUMBER SYSTEMS 1. REAL NUMBERS (18 Periods) 1. Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers. 2.

Class 9 Mathematics Case study question 1 Read the Source/Text given below and answer any four questions: 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.

Case Study Based Questions | NUMBER SYSTEM | CLASS 9 MATHS CHAPTER 1 | NCERT Solutions | Math Infinity. This is a Super Amazing Session with Our Master Teacher Jyotsna mam, in this Session, mam...

MATHS CLASS IX CASE STUDY BASED QUESTIONS FOR ANNUAL EXAM 2020-21 S. No.QuestionChapterYouTube Link1Maths Case Study Question 01Linear Equations in two variables Case Study Question 02Linear Equati…

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.