Mathematics · Real Numbers · NCERT Exercises

Exercise 1.2

Complete, independently verified solutions for NCERT Exercise 1.2.

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Mathematics · Chapter 1 · NCERT Exercises

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Question 1

irrational numbers · proof by contradiction

Prove that √5 is irrational.

Solution

Suppose √5 = a/b in lowest terms, where b ≠ 0.

a2=5b2a^2=5b^2

Hence 5 divides a, so let a = 5k. Then b² = 5k², and 5 divides b as well. This contradicts a and b being coprime.

Final answer

Therefore, √5 is irrational.

Question

Question 2

irrational numbers · proof by contradiction

Prove that 3 + 2√5 is irrational.

Solution

Suppose 3 + 2√5 = r for a rational number r.

5=r−32\sqrt5=\frac{r-3}{2}

The right side would be rational, contradicting the irrationality of √5.

Final answer

Therefore, 3 + 2√5 is irrational.

Question

Question 3

irrational numbers · proof by contradiction

Prove that the following are irrational.

  1. 12\frac1{\sqrt2}
  2. 757\sqrt5
  3. 6+26+\sqrt2

Solution

  1. If 1/√2 were rational, its non-zero reciprocal √2 would be rational, a contradiction.

  2. If 7√5 were rational, division by 7 would make √5 rational, a contradiction.

  3. If 6 + √2 were rational, subtracting 6 would make √2 rational, a contradiction.