Class 9 Rational and Irrational Numbers Mock Test (ICSE)

Topics in Rational and Irrational Numbers

• Rational numbers between two numbers
• Irrational numbers and surds
• Proving irrationality
• Rationalising the denominator
• Irrational numbers between two numbers

Rational and Irrational Numbers practice questions with answers

1. Q1Multiple choice·1 mark
   Which of the following is the simplest form of the surd $\sqrt{108}$?

   1. (a)$2\sqrt{27}$
   2. (b)$6\sqrt{3}$
   3. (c)$9\sqrt{2}$
   4. (d)$3\sqrt{12}$
   Show answer and step marks

   1. Correct option: (b) $6\sqrt{3}$+1

   Answer: (b)
2. Q2Multiple choice·1 mark
   Which of the following statements is used as the initial assumption when proving that $\sqrt{5}$ is irrational by contradiction?

   1. (a)$\sqrt{5}$ can be expressed as a terminating decimal number.
   2. (b)$\sqrt{5}$ has a non-terminating, non-recurring decimal expansion.
   3. (c)$\sqrt{5} = \frac{p}{q}$, where $p, q$ are co-prime integers and $q \neq 0$.
   4. (d)$\sqrt{5} = \frac{p}{q}$, where $p$ and $q$ are real numbers and $q \neq 0$.
   Show answer and step marks

   1. Correct option: (c) $\sqrt{5} = \frac{p}{q}$, where $p, q$ are co-prime integers and $q \neq 0$. A proof by contradiction starts by assuming $\sqrt{5}$ is rational, written in lowest terms.+1

   Answer: (c)
3. Q3Multiple choice·1 mark
   When rationalising the denominator of $\frac{5}{\sqrt{6} - 1}$, the rationalising factor is:

   1. (a)$5(\sqrt{6} - 1)$
   2. (b)$\sqrt{6}$
   3. (c)$\sqrt{6} + 1$
   4. (d)$\sqrt{6} - 1$
   Show answer and step marks

   1. Correct option: (c) $\sqrt{6} + 1$. $(\sqrt{6} - 1)(\sqrt{6} + 1) = 6 - 1 = 5$, which is rational; the other factors leave a surd.+1

   Answer: (c)
4. Q4Multiple choice·1 mark
   A rational number lying between $\frac{1}{3}$ and $\frac{1}{2}$ is:

   1. (a)$\frac{7}{12}$
   2. (b)$\frac{3}{5}$
   3. (c)$\frac{5}{12}$
   4. (d)$\frac{1}{4}$
   Show answer and step marks

   1. Correct option: (c) $\frac{5}{12}$. Convert the given fractions to a common denominator: $\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{2} = \frac{6}{12}$. The fraction $\frac{5}{12}$ lies strictly between $\frac{4}{12}$ and $\frac{6}{12}$.+1

   Answer: (c)
5. Q5Multiple choice·1 mark
   Which of the following is an irrational number lying between $2$ and $3$?

   1. (a)$2.55$
   2. (b)$\sqrt{5}$
   3. (c)$\sqrt{9}$
   4. (d)$\frac{5}{2}$
   Show answer and step marks

   1. Correct option: (b) $\sqrt{5}$. Evaluate the options: $\frac{5}{2} = 2.5$ (rational), $\sqrt{9} = 3$ (rational), $2.55 = \frac{255}{100}$ (rational), and $\sqrt{5} \approx 2.236$ (irrational).+1

   Answer: (b)
6. Q6Multiple choice·1 mark
   An irrational number lying between $3$ and $4$ is:

   1. (a)$\sqrt{17}$
   2. (b)$\sqrt{10}$
   3. (c)$\sqrt{7}$
   4. (d)$3.5$
   Show answer and step marks

   1. Correct option: (b) $\sqrt{10}$+1

   Answer: (b)

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