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Date	May 2018	Marks available	6	Reference code	18M.1.SL.TZ1.S_6
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Find	Question number	S_6	Adapted from	N/A

Question

Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.
This is shown in the following diagram.

The vectors p , q and r are shown on the diagram.

Find p•(p + q + r).

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1 (using |p| |2q| cosθ)

finding p + q + r (A1)

eg 2q,

| p + q + r | = 2 × 3 (= 6) (seen anywhere) A1

correct angle between p and q (seen anywhere) (A1)

$\frac{π}{3}$ $\frac{\pi }{3}$ (accept 60°)

substitution of their values (M1)

eg 3 × 6 × cos $(\frac{π}{3})$ $\left( {\frac{\pi }{3}} \right)$

correct value for cos $(\frac{π}{3})$ $\left( {\frac{\pi }{3}} \right)$ (seen anywhere) (A1)

eg $\frac{1}{2}, 3 \times 6 \times \frac{1}{2}$ $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9 A1 N3

METHOD 2 (scalar product using distributive law)

correct expression for scalar distribution (A1)

eg p• p + p•q + p•r

three correct angles between the vector pairs (seen anywhere) (A2)

eg 0° between p and p, $\frac{π}{3}$ $\frac{\pi }{3}$ between p and q, $\frac{2 π}{3}$ $\frac{{2\pi }}{3}$ between p and r

Note: Award A1 for only two correct angles.

substitution of their values (M1)

eg 3.3.cos0 +3.3.cos $\frac{π}{3}$ $\frac{\pi }{3}$ + 3.3.cos120

one correct value for cos0, cos $(\frac{π}{3})$ $\left( {\frac{\pi }{3}} \right)$ or cos $(\frac{2 π}{3})$ $\left( {\frac{2\pi }{3}} \right)$ (seen anywhere) A1

eg $\frac{1}{2}, 3 \times 6 \times \frac{1}{2}$ $\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}$

p•(p + q + r) = 9 A1 N3

METHOD 3 (scalar product using relative position vectors)

valid attempt to find one component of p or r (M1)

eg sin 60 = $\frac{x}{3}$ $\frac{x}{3}$ , cos 60 = $\frac{x}{3}$ $\frac{x}{3}$ , one correct value $\frac{3}{2}, \frac{3 \sqrt{3}}{2}, \frac{- 3 \sqrt{3}}{2}$ $\frac{3}{2},\,\,\frac{{3\sqrt 3 }}{2},\,\,\frac{{ - 3\sqrt 3 }}{2}$

one correct vector (two or three dimensions) (seen anywhere) A1

eg $p = ⎛ ⎜ ⎜ ⎝ \begin{matrix} \frac{3}{2} \frac{3 \sqrt{3}}{2} \end{matrix} ⎞ ⎟ ⎟ ⎠, q = (\begin{matrix} 30 \end{matrix}), r = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} \frac{3}{2} - \frac{3 \sqrt{3}}{2} 0 \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$ $p = \left( \begin{gathered} \,\,\,\frac{3}{2} \hfill \\ \frac{{3\sqrt 3 }}{2} \hfill \\ \end{gathered} \right),\,\,q = \left( \begin{gathered} 3 \hfill \\ 0 \hfill \\ \end{gathered} \right),\,\,r = \left( \begin{gathered} \,\,\,\,\frac{3}{2} \hfill \\ - \frac{{3\sqrt 3 }}{2} \hfill \\ \,\,\,\,0 \hfill \\ \end{gathered} \right)$

three correct vectors p + q + r = 2q (A1)

p + q + r = $(\begin{matrix} 60 \end{matrix})$ $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ or $⎛ ⎝ \begin{matrix} 600 \end{matrix} ⎞ ⎠$ $\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ (seen anywhere, including scalar product) (A1)

correct working (A1)
eg $(\frac{3}{2} \times 6) + (\frac{3 \sqrt{3}}{2} \times 0), 9 + 0 + 0$ $\left( {\frac{3}{2} \times 6} \right) + \left( {\frac{{3\sqrt 3 }}{2} \times 0} \right),\,\,9 + 0 + 0$

p•(p + q + r) = 9 A1 N3

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.13—Scalar and vector products

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