exercise 1.2 Class 10 Maths | real numbers class 10

exercise 1.2 Class 10 Maths

1. Prove that √5 is irrational.

Sol. Let us assume to the contrary that √5 is an irrational number. So it can be written in the form of p/q where p and q are coprime integers and q is not equal to zero.

√5= p/q

squaring both sides

(√5)² = (p/q)²

5= p²/q²

p²= 5q² ,

so p² is divisible by 5 hence p is also divisible by 5. So put p = 5m in above equation where m is some positive integer.

(5m)² = 5q²

25m² = 5q²

q²= 5m² , so q² is divisible by 5 hence q is also divisible by 5. It means p and q both are divisible by 5. Hence 5 is common factor of p and q. But it contradicts that p and q are coprime. Therefore our assumption is wrong. Hence √5 is irrational.

2. Prove that 3 +2√5 is irrational.

let us assume to the contrary that 3+ 2√5 is a rational number. So

3 + 2√5 = p/q , where p and q are coprime integers and q is not equal to zero

2√5 = p/q -3

2√5 = (p-3q)/q

√5= (p-3q)/2q

since √5 is irrational but (p-3q)/2q is rational.

And rational number can not equal to irrational number. So there is a contradiction. Our assumption is wrong, hence 3+2√5 is irrational.

3. Prove that the following are irrational.

(i) 1/√2 (ii) 7√5 (iii) 6 + √2

Ncert Solutions Class 10 Maths Exercise 1.1 https://10thmathsguide.com/exercise-11-class-10-maths/

Ncert Solutions Class 10 Maths Exercise 1.2

exercise 1.2 Class 10 Maths | real numbers class 10

 

Ncert Solutions Class 10 Maths Exercise 2.1 https://10thmathsguide.com/exercise-21-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 2.2

exercise 2.2 class 10 maths | Polynomials class 10 Maths

 

Ncert Solutions Class 10 Maths Exercise 3.1 https://10thmathsguide.com/exercise-31-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 3.2 https://10thmathsguide.com/exercise-32-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 3.3 https://10thmathsguide.com/exercise-33-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 4.1 https://10thmathsguide.com/exercise-41-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 4.2 https://10thmathsguide.com/exercise-42-class-10-maths/

 

Ncert Solutions Class 10 Maths Exercise 4.3 https://10thmathsguide.com/exercise-43-class-10-maths/

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exercise 1.2 Class 10 Maths exercise 1.2 Class 10 Maths