Real Number System Class 9th Mathematics Term 1 Tamilnadu Board Solution

Class 9th Mathematics Term 1 Tamilnadu Board Solution
Exercise 2.1
  1. State whether the following statements are true or false. i. Every natural…
  2. Is zero a rational number? Give reasons for your answer.
  3. Find any two rational numbers between - 5/7 and - 2/7
Exercise 2.2
  1. Convert the following rational numbers into decimals and state the kind of…
  2. Without actual division, find which of the following rational numbers have…
  3. Express the following decimal expansions into rational numbers. i. 0. bar 18 ii.…
  4. Explain 1/13 in decimal form. Find the number of digits in the repeating block.…
  5. Find the decimal expansions of 1/7 and 2/7 by division method. Without using the…
Exercise 2.3
  1. Locate √5 on the number line.
  2. Find any three irrational numbers between √3 and √5.
  3. Find any two irrational numbers between 3 and 3.5.
  4. Find any two irrational numbers between 0.15 and 0.16.
  5. Insert any two irrational numbers between 4/7 and 5/7 .
  6. Find any two irrational numbers between √3 and 2.
  7. Find a rational number and also an irrational number between 1.1011001110001...…
  8. Find any two rational numbers between 0.12122122212222... and 0.2122122212222...…
Exercise 2.5
  1. A number having non-terminating and recurring decimal expansion is -A. an…
  2. If a number has a non-terminating non-recurring decimal expansion then the…
  3. Decimal form of - 3/4 isA. −0.75 B. −0.50 C. −0.25 D. −0.125
  4. The p/q form of 0. bar 3 isA. 1/7 B. 2/7 C. 1/3 D. 2/3
  5. Which one of the following is not true?A. Every natural number is a rational…
  6. Which one of the following has a terminating decimal expansion?A. 5/32 B. 7/9 C.…
  7. Which one of the following is an irrational number?A. π B. root 9 C. 1/4 D. 1/5…
  8. Which of the following are irrational numbers? i. root 2 + root 3 ii. root 4 +…

Exercise 2.1
Question 1.

State whether the following statements are true or false.

i. Every natural number is a whole number.

ii. Every whole number is a natural number.

iii. Every integer is a rational number.

iv. Every rational number is a whole number.

v. Every rational number is an integer.

vi. Every integer is a whole number.


Answer:

(i) Yes, because natural numbers are 1, 2, 3, …, and so on


And whole numbers are 0, 1, 2, 3, …,


Therefore, Every natural number is a whole number.


(ii) No,


0 is a whole number, but not a natural number.


(iii) Yes, Any integer 'p' can be represented in the form  and 1≠0,


Therefore, every integar is a rational number.


(iv) No,  is a rational number, but not a whole number.


(v) No,  is a rational number, but not an integer.


(vi) No, negative integers are not whole numbers,


For example, -1 is an integer but not a whole number.



Question 2.

Is zero a rational number? Give reasons for your answer.


Answer:

Yes, 0 can be re-written as , where q is any non-negative integer and therefore 0 is a rational number.



Question 3.

Find any two rational numbers between  and 


Answer:


As,


-5 < -4 < -3 < -2





Exercise 2.2
Question 1.

Convert the following rational numbers into decimals and state the kind of decimal expansion.

i.  ii. 

iii.  iv. 

v.  vi. 

vii.  viii. 


Answer:

i.  , the decimal expansion is terminating.


ii. , the decimal expansion is non-terminating and recurring.


iii. , the decimal expansion is non-terminating and recurring.


iv. , the decimal expansion is terminating.


v. , the decimal expansion is non-terminating and recurring.


the decimal expansion is non-terminating and recurring.


vi.  the decimal expansion is non-terminating and recurring.


vii. , the decimal expansion is non-terminating and recurring.


viii. , the decimal expansion is terminating.



Question 2.

Without actual division, find which of the following rational numbers have terminating decimal expansion.

i.  ii. 

iii.  iv. 


Answer:

i.  , As the rational number can be represented in form , therefore its terminating.


ii. , As the rational number can't be represented in form , therefore its non-terminating and recurring.


iii.  , As the rational number can be represented in form , therefore its terminating.


iv. , As the rational number can't be represented in form , therefore its non-terminating and recurring.



Question 3.

Express the following decimal expansions into rational numbers.

i.  ii. 

iii.  iv. 

v.  vi. 


Answer:

i. let 


⇒ 100x = 18.181818 … = 18 + 0.181818…


⇒ 100x = 18 + x


⇒ 99x = 18



ii. let 


⇒ 1000x = 427.427427 … = 18 + 0.427427427…


⇒ 1000x = 427 + x


⇒ 999x = 427



iii. let 


⇒ 10000x = 1.00010001 … = 1 + 0.00010001…


⇒ 10000x = 1 + x


⇒ 9999x = 1



iv. As, 


Let x = 0.454545…


⇒ 1 + x = 1 + 0.454545..


⇒ 1+ x = 1.45454545


⇒ 100 + 100x = 145.454545… = 145 + 0.454545…


⇒ 100 + 100x = 145 + x


⇒ 99x = 45



And



v. As, 


Let x = 0.3333…


⇒ 7 + x = 7 + 0.3333…


⇒ 7 + x = 7.3333…


⇒ 70 + 10x = 73.3333… = 73 + 0.3333…


⇒ 70 + 10x = 73 + x


⇒ 9x = 3



And



vi. let 


⇒ 1000x = 416.416416 … = 416 + 0.416416…


⇒ 1000x = 416 + x


⇒ 999x = 416




Question 4.

Explain  in decimal form. Find the number of digits in the repeating block.


Answer:


No of divisions in repeating block = 6



Question 5.

Find the decimal expansions of  and  by division method. Without using the long division method, deduce the decimal expressions of  from the decimal expansion of .


Answer:



Also,








Exercise 2.3
Question 1.

Locate √5 on the number line.


Answer:

Step 1:


Draw a number line. Mark points O and A such that O represents the number zero and


A represents the number 2. i.e., OA = 2 unit



Step 2:


Draw AB ⊥ OA such that AB = 1unit.



Step 3:


Join OB


In right triangle OAB, by Pythagorean theorem,


OB2 = OA2 + AB2


OB2 = 22 + 12


OB2 = 4 + 1


OB = √5



Step 4:


With O as centre and radius OB, draw an arc to intersect the number line at C on the right side of O. Clearly OC = OB = √5 . Thus, C corresponds to √5 on the number line.




Question 2.

Find any three irrational numbers between √3 and √5.


Answer:

As, √3 = 1.73205…


And √5 = 2.23606…


We have to find three non-terminating and non-recurring numbers between 1.73205… and 2.23606…


Three possible answers are


1.740400400040000…


1.750500500050000…


1.760600600060000…



Question 3.

Find any two irrational numbers between 3 and 3.5.


Answer:

We have to find two non-terminating and non-recurring numbers between 3 and 3.5


Two possible answers are


3.10100100010000…


3.20200200020000…



Question 4.

Find any two irrational numbers between 0.15 and 0.16.


Answer:

We have to find two non-terminating and non-recurring numbers between 0.15 and 0.16


Two possible answers are


0.15100100010000…


0.15200200020000…



Question 5.

Insert any two irrational numbers between  and .


Answer:

Clearly, 


and 


We have to find two non-terminating and non-recurring numbers between 0.571428571428… and 0.714285714285…


Two possible answers are


0.580800800080000…


0.590900900090000…



Question 6.

Find any two irrational numbers between √3 and 2.


Answer:

Clearly,


As, √3 = 1.73205…


We have to find two non-terminating and non-recurring numbers between 1.73205… and 2


Two possible answers are


1.740400400040000…


1.750500500050000…



Question 7.

Find a rational number and also an irrational number between 1.1011001110001... and 2.1011001110001...


Answer:

Clearly,


1.1011001110001… < 2 < 2.1011001110001…


As, 2 is a rational number.


Therefore, 2 is a required rational number.


Now, We have to find a non-terminating and non-recurring number between 1.1011001110001… and 2.1011001110001…


And one of the possible answer is


1.20200200020000…



Question 8.

Find any two rational numbers between 0.12122122212222... and 0.2122122212222...


Answer:

Clearly,


0.12122122212222…<0.13<0.14<0.2122122122212222…


And 0.13, 0.14 are terminating, therefore rational numbers.




Exercise 2.5
Question 1.

A number having non-terminating and recurring decimal expansion is –
A. an integer

B. a rational number

C. an irrational number

D. a whole number


Answer:

By definition, a rational number is one which follows either of the following 2 conditions:

1. It can be expressed inform where p and q are integers (q0).


2. Its decimal expansion is either terminating or non-terminating recurring.


Ex. – 0.35 (terminating decimal expansion) → Rational number


0.7777…. =  (non-terminating but recurring i.e. the number does not terminate but its digits follow a pattern.) → Rational number


π = 3.14…… (Non-terminating non recurring) → Irrational number


Hence, the answer.


Question 2.

If a number has a non-terminating non-recurring decimal expansion then the number is –
A. a rational number

B. a natural number

C. an irrational number

D. an integer.


Answer:

As explained in the above question, for a number to be rational its decimal expansion has to be either terminating or non-terminating and recurring.


For example –


π = 3.14…… (Non-terminating non recurring) → Irrational number


Hence, the answer.


Question 3.

Decimal form of  is
A. −0.75

B. −0.50

C. −0.25

D. −0.125


Answer:

To convert into its equivalent decimal expansion, we can either proceed by long division method or multiply the denominator and (hence) numerator by a number such that the denominator turns out to be 100.


Now, let 4×a = 100


Solving, we get a = 25.


Hence multiplying numerator and denominator by 25.


∴ (-3/4) × (25/25) = (-75/100) = -0.75 (Placing decimal point before 2 digits).


Hence, the answer.


Question 4.

The  form of  is
A. 

B. 

C. 

D. 


Answer:

Let x = 0. = 0.3333... → (eq)1


Since there is only one repeating digit after decimal point, multiply equation 1 by 10.


10x = 3.3333… = 3 + 0.3333… = 3 + x → (eq)2


(eq)2 –(eq)1


10x-x = 3.3333… - 0.3333… = 3


∴ 9x = 3


∴ x = 1/3


Hence, the answer.


Question 5.

Which one of the following is not true?
A. Every natural number is a rational number

B. Every real number is a rational number

C. Every whole number is a rational number

D. Every integer is a rational number.


Answer:

Real numbers is the set which includes both rational and irrational numbers.


∴ Rational numbers and Irrational numbers are 2 independent subsets of Real numbers. Hence every real number is not a rational number (since real numbers also consists of irrational numbers.)


Option (A) is correct because every natural number can be expressed in  form (q≠0 i.e a rational number).


Option (C) is correct because every whole number can be expressed in  form (q≠0 i.e a rational number).


Option (D) is correct because every integer can be expressed in  form (q≠0 i.e a rational number).


Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real numbers


Real numbers ⊃ (Rational numbers + Irrational numbers).


Hence, only option B is incorrect.


Question 6.

Which one of the following has a terminating decimal expansion?
A. 

B. 

C.

D. 


Answer:

If a rational number  (q≠0) can be expressed in the form  , where p ∈ Z, and m,n ∈ W, then the rational number will have a terminating decimal expansion.


Otherwise, the rational number will have a non-terminating and recurring decimal


expansion.


Now, 32 = 25 × 50 Hence, it has terminating decimal expansion.Hence, option A is correct.


Option B is incorrect ∵ 9 = 32 and it cannot be expressed in the form 2m × 5n.


Option C is incorrect ∵ 15 = 3 × 5 and it cannot be expressed in the form 2m × 5n.


Option D is incorrect ∵ 12 = 3 × 22 and it cannot be expressed in the form 2m × 5n.


Hence, only option A is correct.


Question 7.

Which one of the following is an irrational number?
A. π

B. 

C.

D. 


Answer:

Option A is correct since π = 3.141592… i.e. the decimal expansion is non-terminating and non-recurring.


Option B is incorrect because  = ±3 and it is an integer (and hence a rational number, Integers ⊂ Rational Numbers).


Option C is incorrect as it is expressed in the form  (p = 1 and q = 4) and is a rational number (by definition, q≠0).


Option D is incorrect as it is expressed in the form  (p = 1 and q = 5) and is a rational number (by definition, q≠0).


Hence, option A is correct.


Question 8.

Which of the following are irrational numbers?

i.  ii. 

iii.  iv. 
A. (ii), (iii) and (iv)

B. (i), (ii) and (iv)

C. (i), (ii) and (iii)

D. (i), (iii) and (iv)


Answer:

i.√(2 + √3 ) is irrational ∵ √3 is irrational and 2 is rational and sum of a rational and irrational number is always irrational. Hence 2 + √3 is irrational. Hence, its square root is also irrational.


ii.√(4 + √25) is rational ∵ √25( = 5) is rational and 4 is also rational.


Hence, √(4 + 5) = √9 = 3 is rational.


iii. ∛(5 + √7) is irrational ∵ √7 is irrational and 5 is rational and sum of a rational and irrational number is always irrational. Hence 5 + √7 is irrational. Hence, its cube root is also irrational.


iv.√(8-∛8 ) is irrational. Here both  ( = 2) and 8 are rational. Hence


8 – 81/3 = 8 – 2 = 6 is rational. But √6 is irrational.


Hence, the answer.


PDF FILE TO YOUR EMAIL IMMEDIATELY PURCHASE NOTES & PAPER SOLUTION. @ Rs. 50/- each (GST extra)

HINDI ENTIRE PAPER SOLUTION

MARATHI PAPER SOLUTION

SSC MATHS I PAPER SOLUTION

SSC MATHS II PAPER SOLUTION

SSC SCIENCE I PAPER SOLUTION

SSC SCIENCE II PAPER SOLUTION

SSC ENGLISH PAPER SOLUTION

SSC & HSC ENGLISH WRITING SKILL

HSC ACCOUNTS NOTES

HSC OCM NOTES

HSC ECONOMICS NOTES

HSC SECRETARIAL PRACTICE NOTES

2019 Board Paper Solution

HSC ENGLISH SET A 2019 21st February, 2019

HSC ENGLISH SET B 2019 21st February, 2019

HSC ENGLISH SET C 2019 21st February, 2019

HSC ENGLISH SET D 2019 21st February, 2019

SECRETARIAL PRACTICE (S.P) 2019 25th February, 2019

HSC XII PHYSICS 2019 25th February, 2019

CHEMISTRY XII HSC SOLUTION 27th, February, 2019

OCM PAPER SOLUTION 2019 27th, February, 2019

HSC MATHS PAPER SOLUTION COMMERCE, 2nd March, 2019

HSC MATHS PAPER SOLUTION SCIENCE 2nd, March, 2019

SSC ENGLISH STD 10 5TH MARCH, 2019.

HSC XII ACCOUNTS 2019 6th March, 2019

HSC XII BIOLOGY 2019 6TH March, 2019

HSC XII ECONOMICS 9Th March 2019

SSC Maths I March 2019 Solution 10th Standard11th, March, 2019

SSC MATHS II MARCH 2019 SOLUTION 10TH STD.13th March, 2019

SSC SCIENCE I MARCH 2019 SOLUTION 10TH STD. 15th March, 2019.

SSC SCIENCE II MARCH 2019 SOLUTION 10TH STD. 18th March, 2019.

SSC SOCIAL SCIENCE I MARCH 2019 SOLUTION20th March, 2019

SSC SOCIAL SCIENCE II MARCH 2019 SOLUTION, 22nd March, 2019

XII CBSE - BOARD - MARCH - 2019 ENGLISH - QP + SOLUTIONS, 2nd March, 2019

HSC Maharashtra Board Papers 2020

(Std 12th English Medium)

HSC ECONOMICS MARCH 2020

HSC OCM MARCH 2020

HSC ACCOUNTS MARCH 2020

HSC S.P. MARCH 2020

HSC ENGLISH MARCH 2020

HSC HINDI MARCH 2020

HSC MARATHI MARCH 2020

HSC MATHS MARCH 2020

SSC Maharashtra Board Papers 2020

(Std 10th English Medium)

English MARCH 2020

HindI MARCH 2020

Hindi (Composite) MARCH 2020

Marathi MARCH 2020

Mathematics (Paper 1) MARCH 2020

Mathematics (Paper 2) MARCH 2020

Sanskrit MARCH 2020

Sanskrit (Composite) MARCH 2020

Science (Paper 1) MARCH 2020

Science (Paper 2)

MUST REMEMBER THINGS on the day of Exam

Are you prepared? for English Grammar in Board Exam.

Paper Presentation In Board Exam

How to Score Good Marks in SSC Board Exams

Tips To Score More Than 90% Marks In 12th Board Exam

How to write English exams?

How to prepare for board exam when less time is left

How to memorise what you learn for board exam

No. 1 Simple Hack, you can try out, in preparing for Board Exam

How to Study for CBSE Class 10 Board Exams Subject Wise Tips?

JEE Main 2020 Registration Process – Exam Pattern & Important Dates

NEET UG 2020 Registration Process Exam Pattern & Important Dates

How can One Prepare for two Competitive Exams at the same time?

8 Proven Tips to Handle Anxiety before Exams!

BUY FROM PLAY STORE

DOWNLOAD OUR APP

HOW TO PURCHASE OUR NOTES?

S.P. Important Questions For Board Exam 2021

O.C.M. Important Questions for Board Exam. 2021

Economics Important Questions for Board Exam 2021

Chemistry Important Question Bank for board exam 2021

Physics – Section I- Important Question Bank for Maharashtra Board HSC Examination

Physics – Section II – Science- Important Question Bank for Maharashtra Board HSC 2021 Examination