IMO Practice Test — Irrational Numbers
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
Which of the following is an irrational number?
\(\sqrt{4} + \sqrt{9}\)
\(\sqrt{2} + \sqrt{8}\)
\(\sqrt{2} \times \sqrt{8}\)
\(\frac{\sqrt{12}}{\sqrt{3}}\)
Explanation: Option A: 2+3=5 rational; B: \(\sqrt{2}+2\sqrt{2}=3\sqrt{2}\) irrational; C: \(\sqrt{16}=4\) rational; D: \(\sqrt{4}=2\) rational
Question 2 of 6
If \(x = 0.123456789101112...\) (natural numbers written consecutively), then \(x\) is:
Rational terminating
Rational repeating
Irrational
Integer
Explanation: This Champernowne constant has no repeating pattern → irrational
Question 3 of 6
Find the value of \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\)
2
4
8
\(\sqrt{15} - 1\)
Explanation: \((a+b)(a-b)=a^2-b^2=5-3=2\), which is rational
Question 4 of 6
How many irrational numbers lie between 1 and 2?
0
1
10
Infinitely many
Explanation: There are infinitely many irrational numbers between any two distinct real numbers
Question 5 of 6
Which of the following is true about \(\sqrt{2} + \sqrt{3}\)?
It is rational
It is irrational
It is an integer
It equals \(\sqrt{5}\)
Explanation: Sum of two irrationals can be irrational; \((1.414+1.732=3.146...)\) not equal to \(\sqrt{5}\approx2.236\)
Question 6 of 6
The product of \(\sqrt{2}\) and \(\sqrt{8}\) is:
Irrational
Rational
Integer
Both B and C
Explanation: \(\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4\), which is rational and an integer