Wellcome!

Question 1. Asked on :17 August 2019:09:22:47 AM

#### Prove that √2 is irrational?

-Added by Akki chauhan Mathematics » Real Numbers

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 2. 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.

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.

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....

* 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.

## Search your Question Or Keywords

#### Do you have a question to ask?

User Earned Point: Select

## All Tags by Subjects:

Science (1906)
History (200)
Geography (296)
Economics (157)
Political Science (95)
Mathematics (195)
General Knowledge (5431)
Biology (93)
Physical Education (20)
Chemistry (118)
Civics (114)
Home Science (12)
Sociology (8)
Hindi (43)
English (247)
Physics (1422)
Other (92)
Accountancy (176)