EduNinja

[Maximum number: 6]

Nanthana is completing an exploration on multiplication using complex numbers. She considers a geometric sequence where $u_{1}=9$ and $r=\frac{2}{3}+\frac{2}{3} \mathrm{i}$ .

(a)

Show that Nanthana's claim is correct, stating the exact value of the common ratio for this sequence.

[ 4 ]

attempt to find |r|

\begin{aligned} & |r|=\sqrt{\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{2}} \\ & |r|=\frac{2 \sqrt{2}}{3} \end{aligned}

statement that multiplying a number by a complex number changes the modulus of the number by a factor of |r| (may be seen at any stage)

Note: Award at most (M1)A1A0R1 if common ratio is given as $0.942809 \ldots \ldots$ .
therefore it is a geometric sequence

(b)

Hence, find the sum of the infinite sequence $\left|u_{1}\right|,\left|u_{2}\right|,\left|u_{3}\right|, \ldots$ .

[ 2 ]

substitution of their r and 9 into the sum formula for infinite sequence
$S_{\infty}=\frac{9}{1-\frac{2 \sqrt{2}}{3}}$ $\left(S_{\infty}=\right) 157(157.367 \ldots, 81+54 \sqrt{2})$

Note: No FT if their $|r| \geq 1$ .

[Maximum number: 7]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

Write down $z_{1}$ and $z_{2}$ in exponential form, with a constant modulus.

[ 3 ]

at least one expression in the form $r \mathrm{e}^{g(t) \mathrm{i}}$ $z_{1}=1.5 \mathrm{e}^{(0.00939 t+2.83) \mathrm{i}}, z_{2}=1.6 \mathrm{e}^{(0.00939 t) \mathrm{i}}$

(b)

Hence or otherwise find an equation for L in the form $L(t)=p \sin (q t+r)+d$ , where $p, q, r, d \in \mathbb{R}$ .

[ 4 ]

EITHER

\begin{aligned} & z_{1}-z_{2}=1.5 \mathrm{e}^{(0.00939 t+2.83) \mathrm{i}}-1.6 \mathrm{e}^{(0.00939 t) \mathrm{i}} \\ & =\mathrm{e}^{0.00939 t \mathrm{i}}\left(1.5 \mathrm{e}^{2.83 \mathrm{i}}-1.6\right) \\ & =\mathrm{e}^{0.00939 t \mathrm{i}}\left(3.06249 \ldots \mathrm{e}^{2.99086 \ldots \mathrm{i}}\right) \end{aligned}

OR
graph of L or f
p=3.06249...
$r=-0.150729 \ldots$ OR $r=2.99086 \ldots$
Note: The p and r variables (or equivalent) must be seen.

THEN
$L(t)=3.06 \sin (0.00939 t+2.99)+12.5$ $(L(t)=3.06248 \ldots \sin (0.00939 t+2.99086 \ldots)+12.4833 \ldots)$
Note: Accept equivalent forms, e.g. $L(t)=3.06 \sin (0.00939 t-0.151)+12.5$ . Follow through on their answer to part (g) replacing 12.5.

IB Maths AI HL 1.2 Number and Algebra Ahl Content Topic Practice

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IB Maths AI HL 1.2 Number and Algebra Ahl Content Topic Practice

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Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)