10th Class math’s solution Chapter 1.3

 

10th Class math’s solution Chapter 1

 

 

 NCERT Solutions For Class 10 Maths Chapter 1 Real Numbers Ex1.3

NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 1.1 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 6 Triangles Exercise 1.3

           

Access Answers of Maths NCERT Class 10 Chapter 1 – Real Number Exercise 1.3

1. Prove that √5 is irrational.

Solutions: Let us, √5 is rational number.

i.e. √5 = x/y (where, x and y are co-primes)

y√5= x

Squaring both the sides, we get,

(y√5)² = x²

⇒5y² = x²……………………………….. (1)

Thus, x² is divisible by 5, so x is also divisible by 5.

Clearly, x and y are not co-primes. Thus, our assumption about √5 is rational is incorrect.

Hence, √5 is an irrational number.

2. Prove that 3 + 2√5 + is irrational.

Solutions: Let us, 3 + 2√5 is rational.

Then, 3 + 2√5 = x/y

Since, x and y are integers, thus,

is a rational number.

Therefore, √5 is also a rational number. But this contradicts the fact that √5 is irrational.

So, we conclude that 3 + 2√5 is irrational.

 

3. Prove that the following are irrationals:

(i) 1/√2

(ii) 7√5

(iii) 6 + √2

Solutions:

(i) 1/√2

Let us, 1/√2 is rational.

Then 1/√2 = x/y

So, √2 = y/x

Since, x and y are inteers, thus, √2 is a rational number, which contradicts the fact that √2 is irrational.

Hence, we can conclude that 1/√2 is irrational.

(ii) 7√5

Let us, 7√5 is a rational number.

Then 7√5 = x/y

So, √5 = x/7y

Since, x and y are integers, thus, √5 is a rational number, which contradicts the fact that √5 is irrational.

Hence, we can conclude that 7√5 is irrational.

(iii) 6 +√2

Let us, 6 +√2 is a rational number.

Then, 6 +√2 = x/y⋅

So, √2 = (x/y) – 6

Since, x and y are integers, thus (x/y) – 6 is a rational number and therefore, √2 is rational. This contradicts the fact that √2 is an irrational number.Hence, we can conclude that 6 +√2 is irrational.

 

 

 

Class 10 Math's Real Numbers

Rational numbers and irrational numbers are taken together form the set of real numbers. The set of real numbers is denoted by R. Thus every real number is either a rational number or an irrational number. In either case, it has a non–terminating decimal representation. In the case of rational numbers, the decimal representation is repeating (including repeating zeroes) and if the decimal representation is non–repeating, it is an irrational number. For every real number, there corresponds a unique point on the number line ‘l’ or we may say that every point on the line ‘l’ corresponds to a real number (rational or irrational).

From the above discussion we may conclude that:
To every real number there corresponds a unique point on the number line and conversely, to every point on the number line there corresponds a real number. Thus we see that there is one–to–one correspondence between the real numbers and points on the number line ‘l’, that is why the number line is called the ‘real number line’.