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Date May 2014 Marks available 4 Reference code 14M.1.sl.TZ1.10
Level SL only Paper 1 Time zone TZ1
Command term Find Question number 10 Adapted from N/A

Question

The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice, as shown in diagrams 2 and 3.

 


 

 

Let \({x_n}\) denote the length of one of the equal sides of each new triangle.

 

Let \({A_n}\) denote the area of each new triangle.

 

The following table gives the values of \({x_n}\) and \({A_n}\), for \(1 \leqslant n \leqslant 3\). Copy and complete the table. (Do not write on this page.)

 

\(n\) 1 2 3
\({x_n}\) 8   4
\({A_n}\) 32 16  
[4]
a.

The process described above is repeated. Find \({A_6}\).

[4]
b.

Consider an initial square of side length \(k {\text{ cm}}\). The process described above is repeated indefinitely. The total area of the shaded regions is \(k {\text{ c}}{{\text{m}}^2}\). Find the value of \(k\).

 

[7]
c.

Markscheme

valid method for finding side length     (M1)

 

eg   \({8^2} + {8^2} = {c^2},{\text{ }}45 - 45 - 90{\text{ side ratios, }}8\sqrt 2 ,{\text{ }}\frac{1}{2}{s^2} = 16,{\text{ }}{x^2} + {x^2} = {8^2}\)

 

correct working for area     (A1)

 

eg   \(\frac{1}{2} \times 4 \times 4\)

 

\(n\) 1 2 3
\({x_n}\) 8 \(\sqrt {32}\) 4
\({A_n}\) 32 16 8

     A1A1     N2N2

[4 marks]

a.

METHOD 1

recognize geometric progression for \({A_n}\)     (R1)

eg   \({u_n} = {u_1}{r^{n - 1}}\)

\(r = \frac{1}{2}\)     (A1)

correct working     (A1)

eg   \(32{\left( {\frac{1}{2}} \right)^5};{\text{ 4, 2, 1, }}\frac{1}{2},{\text{ }}\frac{1}{4},{\text{ }} \ldots \)

\({A_6} = 1\)     A1     N3

 

METHOD 2

attempt to find \({x_6}\)     (M1)

eg   \(8{\left( {\frac{1}{{\sqrt 2 }}} \right)^5},{\text{ }}2\sqrt 2 ,{\text{ 2, }}\sqrt 2 ,{\text{ 1, }} \ldots \)

\({x_6} = \sqrt 2 \)     (A1)

correct working     (A1)

eg   \(\frac{1}{2}{\left( {\sqrt 2 } \right)^2}\)

\({A_6} = 1\)     A1     N3

[4 marks]

b.

METHOD 1

recognize infinite geometric series     (R1)

eg   \({S_n} = \frac{a}{{1 - r}},{\text{ }}\left| r \right| < 1\)

area of first triangle in terms of \(k\)     (A1)

eg   \(\frac{1}{2}{\left( {\frac{k}{2}} \right)^2}\)

attempt to substitute into sum of infinite geometric series (must have \(k\))     (M1)

eg   \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}},{\text{ }}\frac{k}{{1 - \frac{1}{2}}}\)

correct equation     A1

eg   \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}} = k,{\text{ }}k = \frac{{\frac{{{k^2}}}{8}}}{{\frac{1}{2}}}\)

correct working     (A1)

eg   \({k^2} = 4k\)

valid attempt to solve their quadratic     (M1)

eg   \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)

\(k = 4\)     A1     N2

METHOD 2

recognizing that there are four sets of infinitely shaded regions with equal area     R1

area of original square is \({k^2}\)     (A1)

so total shaded area is \(\frac{{{k^2}}}{4}\)     (A1)

correct equation \(\frac{{{k^2}}}{4} = k\)     A1

\({k^2} = 4k\)    (A1)

valid attempt to solve their quadratic     (M1)

eg   \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)

\(k = 4\)     A1     N2

[7 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 1 - Algebra » 1.1 » Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.
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