Wellcome!

 

 

Asked/Added Questions From NCERT:


Question 1. Asked on :29 June 2021:11:08:46 AM

Prove that √5 is an irrational number.

-Added by Master Purushottam Mathematics » Real Numbers

Answer:

Master Purushottam

Let us assume that √5 is a rational number. 

 we can write it as √5 = pq 

Now we can reduce p and q and we get reduced value of p and q as a and b which are co-prime have no common factor other than 1. 

So we have √5 = ab 

⇒ √5b = a  

⇒ 5b2 = a2   [squaring both sides]  ............ (i) 

⇒ b2 = a25 

5 divides a2 so a is also divisible by 5   (theorem 1.3) ..... (ii) 

so 5 is a factor of a.

Now taking a = 5c for some integer c 

⇒ 5b2 = (5c)2  

⇒ 5b2 = 25 c2 

⇒ b2 = 5 c2 

⇒ c2 = b25 

5 divides b2 so b is also divisible by 5 (theorem 1.3)  ...... (iii) 

from (ii) and (iii) we get that 

a and b are divisible by 5 this implies that 5 is common factor of a and b, while we assume that a and b are co-prime having no common factor other than 1. 

This contradiction has risen due to our wrong assumption so √5 cann't be 

expressed as pq, Hence √5 is an irrational number. 

-Answered by Master Purushottam On 29 June 2021:11:10:35 AM


Question 2. Asked on :24 June 2021:06:26:16 PM

Show that any odd positive integer is of the form 4q + 1 or 4q + 3, where q is some integer.

-Added by Master Mind Mathematics » Real Numbers

Answer:

Master Mind

Let the a is the any odd positive interger

To Show that: a = 4q + 1 or 4q + 3 

Using Euclid's division algorithem 

a = bq + r  

∴ b = 4 

so, possible remainder will be 0, 1, 2, 3 ( 0 ≤ r < b) 

Hence, a is an odd positive integer therefore remainder will be also odd.

∴ r = 1 or 3 

Now expressing a in the form of bq + r 

a = 4q + 1 or 4q + 3 

Hence proved 

-Answered by Master Mind On 24 June 2021:06:28:42 PM


Question 3. Asked on :24 June 2021:06:10:31 PM

If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y.

-Added by Master Mind Mathematics » Real Numbers

Answer:

Master Mind
HCF of 210 and 55 using Euclid's division algorithem

210 = 55 × 3 + 45

  55 = 45 × 1 + 10 

  45 = 10 × 4 + 5

  10 = 5 × 2 + 0 

∴ HCF = 5 

Now 210 × 5 + 55y = 5 

       1050 + 55y = 5 

                   55y = 5 - 1050 

                   55y = - 1045       

                      y = - 104555 = - 19 

y = - 19 


-Answered by Master Mind On 24 June 2021:06:15:15 PM


Question 4. Asked on :17 August 2019:09:22:47 AM

 Prove that √2 is irrational?

-Added by Akki chauhan Mathematics » Real Numbers

Answer:

Himanshi Verma
Let √2 be a rational number 

Therefore, √2= p/q  [ p and q are in their least terms i.e., HCF of (p,q)=1 and q ≠ 0

On squaring both sides, we get 
                   p²= 2q²           ...(1)
Clearly, 2 is a factor of 2q²
⇒ 2 is a factor of p²          [since, 2q²=p²]
⇒ 2 is a factor of p

 Let p =2 m for all m ( where  m is a positive integer)

Squaring both sides, we get 
            p²= 4 m²          ...(2)
From (1) and (2), we get 
           2q² = 4m²      ⇒      q²= 2m²
Clearly, 2 is a factor of 2m²
⇒       2 is a factor of q²          [since, q² = 2m²]
⇒       2 is a factor of q 

Thus, we see that both p and q have common factor 2 which is a contradiction that H.C.F. of (p,q)= 1

 Therefore, Our supposition is wrong

Hence √2 is not a rational number i.e., irrational number.

-Answered by Himanshi Verma On 18 August 2019:11:09:55 AM


Question 5. Asked on :17 June 2019:11:25:45 PM

Show that one and only one out of n, n+2 or n+4 is divisible by 3, where n is any positive integer. 

-Added by ATP Admin Mathematics » Real Numbers

Answer:

ATP Admin

Solution: Using Euclid's division lemma any positive integer can be written in the form of a = bq + r where r = 0, 1, 2 ...... and q is quotients. 

Let the number which is divisible by 3 be 3q + 0 or 3q + 1 or 3q + 2 where [0 <= r < b] 

Now n = 3q or n = 3q + 1 or n = 3q + 2 

Case I, 

When n = 3q  .......... (i) 

⇒ n = 3(q) where n is divisible by 3

Adding 2 both sides in equ. (i) 

We have,

    n + 2 = 3q + 2 Where n + 2 is not divisible by 3 

Now adding 4 both side in equ. (i) 

We have,

    n + 4 = 3q + 4 Where n + 4 is not divisible by 3 


Case II  

  Taking n = 3q + 1 ........ (ii) where n is not divisible by 3

Adding 2 both sides in equ. (ii) 

We have, 

n + 2 = 3q + 1 + 2 = 3q + 3 

n + 2 = 3(q + 1) where n + 2 is divisible by 3 

Now adding 4 both sides in equ. (ii) 

n + 4 = 3q + 1 + 4 = 3q + 5 where n + 4 is not divisible by 3 


Case III  

taking n = 3q + 2 .... (iii) where n is not divisible by 3 

Adding 2 both sides in equ. (iii) 

We have,

n + 2 = 3q + 2 + 2 = 3q + 4 where n + 2 is not divisible by 3

Now adding 4 both sides in equ. (iii) 

We have, 

n + 4 = 3q + 2 + 4 = 3q +6 

n + 4 = 3(q + 2) where n + 4 is divisible by 3 

Hence in all these three cases we have seen that either one and only one n or n + 2 or n + 4 is divisible by 3. 

 

 



-Answered by ATP Admin On 17 June 2019:11:28:43 PM


 

You can see here all the solutions of this question by various user for NCERT Solutions. We hope this try will help you in your study and performance.

This Solution may be usefull for your practice and CBSE Exams or All label exams of secondory examination. These solutions or answers are user based solution which may be or not may be by expert but you have to use this at your own understanding of your syllabus.

 

 

What do you have in your Mind....

Ask Your Question? From your text book.

Our Expert Team reply with answer soon.

 

Ask Your Question

 

* Now You can earn points on every asked question and Answer by you. This points make you a valuable user on this forum. This facility is only available for registered user and educators.

Next moment you answer is ready .... go ahead ...

 

 

 

Search your Question Or Keywords

 

 

Do you have a question to ask?

 

Ask Your Question

User Earned Point: Select

 

 

 

All Tags by Subjects:

 

Science (2235)
History (278)
Geography (310)
Economics (258)
Political Science (96)
Mathematics (201)
General Knowledge (5686)
Biology (94)
Physical Education (20)
Chemistry (118)
Civics (114)
Home Science (12)
Sociology (9)
Hindi (45)
English (259)
Physics (1435)
Other (97)
Accountancy (378)
Business Study (71)
Computer Science (128)
1 (2)

 

 

Sponsers link