Students can Download Class 10 Maths Chapter 8 Real Numbers Ex 8.3 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 10 Maths helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

## Karnataka State Syllabus Class 10 Maths Chapter 8 Real Numbers Ex 8.3

Question 1.

Prove that \(\sqrt{5}\) is irrational.

Answer:

Let us assume, to the contrary, that \(\sqrt{5}\) is rational.

∴ \(\sqrt{5}=\frac{a}{b}\)

∴ b × \(\sqrt{5}\) = a

By Squaring on both sides,

5b^{2} = a^{2} …………. (i)

∴ 5 divides a^{2}.

5 divides a.

∴ We can write a = 5c.

Substituting the value of ‘a’ in eqn. (i),

5b^{2} = (5c)^{2} = 25c^{2}

b^{2} = 5c^{2}

It means 5 divides b^{2}.

∴ 5 divides b.

∴ ‘a’ and ‘b’ have at least 5 as a common factor.

But this contradicts the fact that a’ and ‘b’ are prime numbers.

∴ \(\sqrt{5}\) is an irrational number.

Question 2.

Prove that \(3+2 \sqrt{5}\) is irrational.

Answer:

Let us assume that \(3+2 \sqrt{5}\) is an irrational number.

Here, p, q, ∈ z, q ≠ 0

\(\sqrt{5}\) is rational number.

∵ \(\frac{p-3 q}{2 q}\) is rational number.

But \(\sqrt{5}\) is not a rational number.

This contradicts the fact that,

∴ \(3+2 \sqrt{5}\) is an irrational number.

Question 3.

Prove that the following are irrationals:

i) \(\frac{1}{\sqrt{2}}\)

ii) \(7 \sqrt{5}\)

iii) \(6+\sqrt{2}\)

Answer:

i) Let \(\frac{1}{\sqrt{2}}\) is a rational number.

\(\frac{1}{\sqrt{2}}=\frac{\mathrm{p}}{\mathrm{q}}\)

\(\sqrt{2}=\frac{q}{p}\)

By Squaring on both sides,

2 × p^{2} = q^{2}

2, divides q^{2}.

∴ 2, divides q

∵ q is an even number.

Similarly ‘p’ is an even number.

∴ p and q are even numbers.

∴ Common factor of p and q is 2.

This contradicts the fact that p and q also irrational.

∴ \(\sqrt{2}\) is an irrational number.

∴ \(\frac{1}{\sqrt{2}}\) is an irrational number.

ii) Let \(7 \sqrt{5}\)is a rational number.

∴ \(7 \sqrt{5}=\frac{p}{q}\)

\(\sqrt{5}=\frac{p}{7 q}\)

Here,\(\frac{p}{7 q}\) is one rational number.

It means \(\sqrt{5}\) which is equal also a rational number.

This contradicts to the fact that \(\sqrt{5}\) is an irrational number.

This contradicts to the fact that \(7 \sqrt{5}\) is rational number.

∴ \(7 \sqrt{5}\) is a rational number.

iii) Let \(6+\sqrt{2}\) is a rational number.

\(\frac{a-6 b}{b}\) is a rational number, b

∴ \(\sqrt{2}\) is also rational number.

This contradicts to the fact that \(\sqrt{2}\) is an irrational number.

This contradicts to the fact that \(6+\sqrt{2}\) is a rational number.

∴ \(6+\sqrt{2}\) is an irrational number.