Question

Given that √5 is irrational, prove that 3- 2√7is an irrationalnumber
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Answer 1

Solution:-

i) prove that √5 is irrational

Proof

:- if possible, let √5 be rational and let its simplest form be a/b

Then , a and b are integers having no common factor other than 1 , and b is not equal to zero

Now , √5 = a/b => 5 = a² / b² [ squaring both sides ]

=> 5b² = a² ....(i)

=> 5 divides a²

=> 5 divides a

[ 5 is prime and 4 divides a² => 5 divides a ]

let a = 5c from some Integer c

putting a = 5c in (i) , we get

5b² = 25c² => b² = 5c²

=> 5 divides b² [ 5 divides 5c²

=>5 divides b

[ 5 is prime and 5 divides b² => 5 divides b ]

Thus , 5 is a common factor of a and b

But , this contradicts the fact that a and b have no common factor other than 1 .

The Contradiction arises by assuming that √5 is irrational

Hence, √5 is irrational

ii) Prove 3 - 2√7 is irrational

Proof

Let us assume, to the contrary that 3 - 2√7 is rational

then , there exist co - primes a and b ( b is not equal to zero ) such that

=> 3 - 2√7 = a/b

=> - 2√7 = a /b - 3 => (a -3b)/b

=> - √7= (a - 3b)/2b

=> Since a and b are integers, so (a - 3b)/2b is rational

thus √7 is also rational

But this contradicts the fact that √7 is irrational. So our assumption is incorrect

Hence (3 - 2√7 ) is irrational

Answer 2

Step-by-step explanation:

Given :

Swati walks in a circle of radius 20m and minal walks in a circle of radius 40m.

Both complete one circle in same time.

To Find :

Relation between their speeds.

Solution :

❒ Speed is defined as the ratio of distance travelled to the time taken.

It is a scalar quantity having only magnitude.

SI unit : m/s

Distance covered by body after completing one round of circle will be equal to the perimeter of circle.

Let time taken by them to complete one round of circle be t..

A. Speed of swati :

➛ v = 2πr / t

➛ v = 2π(20) / t

➛ v = 40π / t .......... (I)

B. Speed of minal :

➛ v' = 2πr' / t

➛ v' = 2π(40) / t

➛ v' = 80π / t .......... (II)

From equation (I) and (II),

➠ v / v' = (40π/t) / (80π/t)

➠ v / v' = 1 / 2

➠ v' = 2v

∴ (c) Speed of minal is twice the speed of swati.