Solved: Consider $w = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$ These four... [IB Mathematics Applications & Interpretation (Math AI)]

Question

HLPaper 1

Consider $w = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$

These four points form the vertices of a quadrilateral, Q.

Express $w^2$ and $w^3$ in modulus-argument form.

[3]

Verified

Solution

$w^2 = 4\text{cis}\left( {\frac{{2\pi }}{3}} \right);\,\,w^3 = 8{\text{cis}}\left( \pi \right)$ M1 can be awarded for either both correct moduli or both correct arguments. A1A1

Accept Euler form. Allow multiplication of correct Cartesian form for M1, final answers must be in modulus-argument form.

Sketch on an Argand diagram the points represented by $w^0$ , $w^1$ , $w^2$ and $w^3$ .

[2]

Verified

Solution

Markscheme

A1 for correctly sketching $w^0 = 1$ on the positive real axis

A1 for correctly sketching the following points:

$w^1 = 2\cos\left(\frac{\pi}{3}\right) + 2i\sin\left(\frac{\pi}{3}\right)$
$w^2 = 4\cos\left(\frac{2\pi}{3}\right) + 4i\sin\left(\frac{2\pi}{3}\right)$
$w^3 = -8$

\begin{array}{ccc} \text{Re} & \bullet & \text{Re}\\ & \curvearrowleft & \\ & \bullet & \bullet\\ & \curvearrowright & \\ \text{Im} & \bullet & \text{Im} \end{array}

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$ .

[3]

Verified

Solution

Method #1

\text{Area} = \frac{1}{2}ab\sin C

M1 for using the area formula

\frac{1}{2} \times 1 \times 2 \times \sin\frac{\pi}{3} + \frac{1}{2} \times 2 \times 4 \times \sin\frac{\pi}{3} + \frac{1}{2} \times 4 \times 8 \times \sin\frac{\pi}{3}

A1 for $C = \frac{\pi}{3}$ A1 for correct moduli

= \frac{21\sqrt{3}}{2}

Other methods of splitting the area may receive full marks.

Let $z = 2(\cos\frac{\pi}{n} + i\sin\frac{\pi}{n})$ , $n \in \mathbb{Z}^+$ . The points represented on an Argand diagram by $z^0$ , $z^1$ , $z^2$ , $\ldots$ , $z^n$ form the vertices of a polygon $P_n$ .

Show that the area of the polygon $P_n$ can be expressed in the form $a(b^n - 1)\sin\frac{\pi}{n}$ , where $a, b \in \mathbb{R}$ .

[6]

Verified

Solution

Method #1

\frac{1}{2} \times {2^0} \times {2^1} \times {\text{sin}}\frac{\pi}{n} + \frac{1}{2} \times {2^1} \times {2^2} \times {\text{sin}}\frac{\pi}{n} + \frac{1}{2} \times {2^2} \times {2^3} \times {\text{sin}}\frac{\pi}{n} + \, \ldots \, + \frac{1}{2} \times {2^{n - 1}} \times {2^n} \times {\text{sin}}\frac{\pi}{n}

M1A1

Award M1 for powers of 2, A1 for any correct expression including both the first and last term.

= {\text{sin}}\frac{\pi}{n} \times \left( {{2^0} + {2^2} + {2^4} + \, \ldots \, + {2^{n - 2}}} \right)

Identifying a geometric series with common ratio 2² (= 4) M1A1

= \frac{{1 - {2^{2n}}}}{{1 - 4}} \times {\text{sin}}\frac{\pi}{n}

Award M1 for use of formula for sum of geometric series.

= \frac{1}{3}\left( {{4^n} - 1} \right){\text{sin}}\frac{\pi}{n}

[6 marks]

Consider $$w = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)$$

Markscheme

Method #1

Method #1

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