Create New Account
Create New Account

Sign in to Innings2

or

New Account     Reset Password     

Real Numbers

    Exercise 1.2

Powered by Innings 2

Reset Progress

This will delete your progress and chat data for all chapters in this course, and cannot be undone!

Glossary

Select one of the keywords on the left…

10th class > Real Numbers > Exercise 1.2

Exercise 1.2

Prove that 5 is irrational.

Solution

Assume 5 is irrational: Suppose sqrt(5) = ab,where a and b are coprime integers(i.e., gcd (a,b) =1 and b ≠ 0 )

Square both sides:

5 =

5b2 =

This means a2 is divisible by , so a must be divisible by .

Let a = 5k for some integer k.

Substitute a = 5k into the equation: [Eq: 5b2 = a2]

⁡5b2 =

⁡5b2 = k2

b2 = k2

This means b2 is divisible by 5, so b must be divisible by 5.

Since both a and b are divisible by 5, they have a common factor of 5, contradicting the assumption that they are .

Therefore, the assumption that √5 is rational is false, so √5 is irrational.

Prove that 3 + 25 is irrational

Solution

Let's assume that 3 + 25 is rational.

If 3 + 25 is rational that means it can be written in the form of ab where a and b are integers that have no common factor other than 1 and b ≠ 0.

+ 25 =

b( + 25) =

+ 25b = a

25b = a -

5 = (a - 3b)2b

Since (a - 3b)2b is a number, then 5 is also a number.

But, we know that 5 is .

Therefore, our assumption was wrong that 3 + 25 is .

Hence, 3 + 25 is .

Prove that the following are irrationals :

(i)

(i)12

Solution

Let us assume that 12 is a number.

Then, 12 = ab, where a and b have no common factors other than .

√2 × a =

2 =

Since b and a are integers, ba is a number and so, √2 is .

But we know that √2 is .

So, our assumption was wrong.

Therefore, 12 is an irrational number.

(ii)

(ii)75

Solution

Let us assume that 75 is a rational number.

Then, 75= ab, where a and b have no common factors other than .

75 b =

5=

__{.m-orange}Since, a, 7, and b are integers, so, a7b is a number.

This means 5 is . But this contradicts the fact that 5 is .__

So, our assumption was wrong.

Therefore, 75 is an irrational number.

(iii)

(iii)6+2

Solution

Let us assume that 6+2 is rational.

Then, 6+2 = ab, where a and b have no common factors other than .

2 = ab -

__{.m-orange}Since, a, b, and 6 are integers, so, ab - 6 is a number.

This means 2 is also a number.

But this contradicts the fact that 2 is .

So, our assumption was wrong.

Therefore, 6+2 is an irrational number.