Surds, roots and irrational numbersIB MYP Maths Standard: Subtopic test

1. 1
   A square tile has an area of $50$ cm$^2$.

   (a)

   Find the side length of the tile, giving your answer in its simplest exact form.

   [1 mark]

   • A$25$ cm
   • B$5\sqrt2$ cm
   • C$2\sqrt5$ cm
   • D$25\sqrt2$ cm

   (b)

   Find the perimeter of the tile, giving your answer in its simplest exact form.

   [1 mark]

   • A$9\sqrt2$ cm
   • B$20$ cm
   • C$10\sqrt2$ cm
   • D$20\sqrt2$ cm

   (c)

   Show that the diagonal of the tile is $10$ cm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the numbers $\sqrt{18}$ and $\sqrt{8}$.

   (a)

   Simplify $\sqrt{18}$.

   [1 mark]

   • A$9\sqrt2$
   • B$2\sqrt9$
   • C$3\sqrt2$
   • D$9$

   (b)

   Find $\sqrt{18}+\sqrt{8}$ in its simplest form.

   [1 mark]

   • A$5\sqrt2$
   • B$\sqrt{26}$
   • C$12$
   • D$5$

   (c)

   Show that $\sqrt{18}-\sqrt{8}=\sqrt{2}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangle has length $(3+\sqrt5)$ cm and width $(3-\sqrt5)$ cm.

   (a)

   Find the exact area of the rectangle.

   [3 marks]

   (b)

   Find the exact length of the diagonal of the rectangle in the form $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is as small as possible.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A landscape designer lays square lawns of area $8$ m$^2$, $18$ m$^2$ and $32$ m$^2$. Sam notices that their side lengths are $\sqrt{8}=2\sqrt{2}$ m, $\sqrt{18}=3\sqrt{2}$ m and $\sqrt{32}=4\sqrt{2}$ m.

   (a)

   (i) Describe the pattern in the numbers under the root signs and in the coefficients.
   (ii) Write down the next lawn in the pattern and show that its side length is $5\sqrt2$ m.
   (iii) Write a general rule for the side length of the $n$th lawn (starting at $n=2$ for the $8$ m$^2$ lawn), and verify it for $n=6$.

   [6 marks]

   (b)

   The designer puts fencing around the edge of the $8$ m$^2$ lawn and the $18$ m$^2$ lawn.
   (i) Find the exact total length of fencing needed.
   (ii) The designer uses $\sqrt2\approx1.4$ and orders $28$ m of fencing. Calculate the exact total to 3 significant figures and justify whether $28$ m is enough.
   (iii) Explain how the designer could avoid this problem.

   [6 marks]

   Total for question 4: 12 marks

End of questions