TN 11th, Maths Chapter 2, Exercise 2.1

Welcome To

Class : XI

Subject : Maths

Chapter : 2

Exercise : 2.1

Que 1 :

Classify each element of {√7,-¼,0,3.14,4,(22/7)} as a member of ℕ, Q, ℝ, -Q or ℤ.

Solution :

√7 irrational number

Q = R-Q

√7 ∈ R-Q

-¼ negative rational number

-¼ ∈ Q

0 Integer & Rational Number

0 ∈ Z,Q

3.14 Rational Number

3.14 ∈ Q

4 Natural Number, Rational Number

4 ∈ N, Z, Q

22/7 Rational Number

22/7 ∈ Q            

Que 2 :

Prove That √3 is an irrational number.

Solution :

Let Assume That 

√3 is Rational Number 

i.e √3 = (m/n); m,n ∈ z ; n ∈ 0

n√3 = m

Squaring on Both Side;

(n√3)² = m²

3n² = m² ------------------------------ (1)

Let m² is even

m must be even

m = 2k

Substitute

m = 2k in (1)

3n² = m²

3n² = 4k²

n² = (4k²/3)

n² & n also an even number

m and n have common factor 2

This Contradiction that m and n have no factor greater than one 

∴ √3 is an irrational number.

Que 3 :

Are There Two Distinct irrational number, such that their difference is Rational Number. Justify.

Solution :

= (2+√2) - (1+√2)

= 2+√2-1-√2

= 2-1

= 1

1 Is an Rational Number

1 ∈ Q 

Que 4 :

Find 2 irrational Number Such That Their sum is a rational number, Can you find two irrational number, Whose product is a rational number.

Solution :

2+√2 ; 2-√2

Sum Of The Root

= (2+√2) + (2-√2)

= 2+2

= 4 ∈ Q

Product Of The Root

= (2+√2)(2-√2)

= 2²-(√2)²

= 4 - 2

= 2 ∈ Q 

Que 5 :

Find the positive number Smaller than 1/2¹⁰⁰⁰. Justify.

Solution :

Given number is 1/2¹⁰⁰⁰

Let 1000<1001

∴ 2¹⁰⁰⁰ < 2¹⁰⁰¹

1/2¹⁰⁰⁰ < 1/2¹⁰⁰¹

∴ Positive Number is smaller than 1/2¹⁰⁰⁰ is 1/2¹⁰⁰¹

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