Number systems and sets of numbersIB MYP Maths Extended: Subtopic test

1. 1
   Consider the numbers in the set $S=\{-3,\ 0,\ \frac{2}{5},\ \sqrt{16},\ \sqrt{7},\ 0.\dot{3},\ \pi\}$.

   (a)

   How many elements of $S$ are integers?

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$5$

   (b)

   Which elements of $S$ are irrational?

   [1 mark]

   • A$\sqrt{7}$ and $\pi$
   • B$\sqrt{16}$ and $\pi$
   • C$\sqrt{7}$, $\pi$ and $0.\dot{3}$
   • D$\pi$ only

   (c)

   Write down the elements of $S$ that are rational but not integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $A=\{x\in\mathbb{Z}:-2\le x<3\}$ and $B=\{0,\ 2,\ 4,\ 6\}$.

   (a)

   How many elements are in set $A$?

   [1 mark]

   • A$3$
   • B$4$
   • C$5$
   • D$6$

   (b)

   Find $A\cap B$.

   [1 mark]

   • A$\{2\}$
   • B$\{0,\ 1,\ 2\}$
   • C$\{-2,-1,0,1,2,4,6\}$
   • D$\{0,\ 2\}$

   (c)

   Write down $A\cup B$ and state the value of $n(A\cup B)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the numbers $\sqrt{49}$, $\sqrt{48}$, $-\frac{9}{4}$, $0.45$, $\frac{\pi}{2}$ and $-\sqrt{25}$.

   (a)

   For each of $\sqrt{49}$, $-\sqrt{25}$ and $\frac{\pi}{2}$, state the smallest of the sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ that contains it, or state that it is irrational.

   [3 marks]

   (b)

   (i) Write $0.45$ as a fraction in its simplest form. (ii) Explain why $\sqrt{48}$ is irrational.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A tiler lays square tiles of side $1$ m. She must cut a strip equal in length to the diagonal of one tile, whose exact length is $\sqrt{2}$ m. Her tape measure shows $1.41$ m and her calculator shows $1.414213562$.

   (a)

   (i) State whether $\sqrt{2}$ is rational or irrational and give a reason. (ii) Write $1.41$ as a fraction and explain why any length read from a tape measure is rational. (iii) Show by calculation whether $1.41$ is exactly equal to $\sqrt{2}$.

   [6 marks]

   (b)

   The strip must be at least $\sqrt{2}$ m long and shorter than $1.5$ m. (i) Show that a strip of length $1.42$ m is long enough. (ii) Write, in set notation, the set of all real lengths $x$ m that are acceptable. (iii) Describe how to show this set on a number line, giving a reason for each end-point.

   [6 marks]

   Total for question 4: 12 marks

End of questions