This page provides comprehensive Real Numbers � Exercise 1.2. Free NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1-2. Step-by-step explained answers for CBSE Board exams. Download PDF and practice now.
NCERT Exercise Practice questions with detailed step-by-step solutions
Prove that v5 is an irrational number.
Step 1: Assume v5 is a rational number.
Then it can be written as:
v5 = a / b, where a and b are integers having no common factor and b ? 0.
Step 2: Squaring both sides:
5 = a� / b�
Step 3: Multiply both sides by b�:
5b� = a�
Step 4: This shows a� is divisible by 5, so a is divisible by 5.
Let a = 5k
Step 5: Substitute back:
5b� = 25k� ? b� = 5k�
This implies b is also divisible by 5.
Contradiction: a and b cannot both be divisible by 5.
Therefore, v5 is irrational.
Prove that (3 + 2v5) is irrational.
Step 1: Assume 3 + 2v5 is rational.
Then:
3 + 2v5 = a / b
Step 2: Subtract 3 from both sides:
2v5 = (a / b) - 3
The right-hand side is rational.
Step 3: Divide both sides by 2:
v5 is rational
This contradicts the fact that v5 is irrational.
Therefore, 3 + 2v5 is irrational.
Prove that the following numbers are irrational:
(i) 1/v2 (ii) 7v5 (iii) 6 + v2
(i) 1/v2
Assume 1/v2 is rational.
Then v2 becomes rational, which is false.
Hence, 1/v2 is irrational.
(ii) 7v5
Assume 7v5 is rational.
Dividing both sides by 7 gives v5 rational.
This contradicts the fact that v5 is irrational.
Hence, 7v5 is irrational.
(iii) 6 + v2
Assume 6 + v2 is rational.
Subtract 6 from both sides:
v2 is rational � which is false.
Hence, 6 + v2 is irrational.