IB Maths AI HL 1.2 Number and Algebra Ahl Content Questions

Practise IB Mathematics AI HL 1.2 by extending number-and-algebra models with advanced sequences, financial mathematics, technology and parameter analysis.

Exam points

Calculate and interpret infinite geometric sums, checking convergence and deriving effective ratios in repeated-stage models.
Operate with Cartesian, polar and exponential complex forms, using modulus, argument, powers and geometric interpretation.
Use matrices for transformations, systems, coding/decoding and repeated processes, including inverses and matrix powers.
Find and use eigenvalues and eigenvectors to diagonalise 2×2 matrices and model long-run behaviour.

Covered syllabus points

AHL 1.9 (HL)—Logarithm laws AHL 1.10 (HL)—Rational exponents AHL 1.11 (HL)—Infinite geometric sequences AHL 1.12 (HL)—Complex numbers AHL 1.13 (HL)—Complex forms and applications AHL 1.14 (HL)—Matrices AHL 1.15 (HL)—Eigenvalues and eigenvectors

[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.

[Maximum number: 12]

François is a video game designer. He designs his games to take place in two dimensions, relative to an origin O . In one of his games, an object travels on a straight line $L_{1}$ with vector equation

r=\binom{-1}{1}+\lambda\binom{2}{-1}

(a)

Given that T=R X,

[ 12 ]

(i)

use your answers to part (c) to find matrix X.

[ 6 ]

attempt to multiply matrices from part (c) (in any order)

\boldsymbol{X}=\begin{pmatrix}5\sqrt{2}&0\\0&5\sqrt{2}\end{pmatrix}\begin{pmatrix}\cos\left(\frac{\pi}{4}\right)&-\sin\left(\frac{\pi}{4}\right)\\\sin\left(\frac{\pi}{4}\right)&\cos\left(\frac{\pi}{4}\right)\end{pmatrix}

\boldsymbol{X}=\begin{pmatrix}5&-5\\5&5\end{pmatrix}

(ii)

hence, find the value of $\alpha$ .

[ 6 ]

substituting $\boldsymbol{T}$ , $\boldsymbol{R}$ and $\boldsymbol{X}$

\begin{pmatrix}1&7\\7&-1\end{pmatrix}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}\begin{pmatrix}5&-5\\5&5\end{pmatrix}

multiplying by inverse (in any order)

\begin{pmatrix}1&7\\7&-1\end{pmatrix}\begin{pmatrix}5&-5\\5&5\end{pmatrix}^{-1}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}

\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}=\begin{pmatrix}-\frac{3}{5}&\frac{4}{5}\\\frac{4}{5}&\frac{3}{5}\end{pmatrix}

\cos 2\alpha=-\frac{3}{5}\quad\text{and}\quad\sin 2\alpha=\frac{4}{5}

\alpha=1.11(=1.107148\ldots)\quad\text{OR}\quad63.4^{\circ}(63.4349\ldots^{\circ})

[Maximum number: 2]

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

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

[Maximum number: 4]

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

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

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}

[Maximum number: 6]

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.
The model is of the form

\binom{X_{n+1}}{Z_{n+1}}=\boldsymbol{M}\binom{X_n}{Z_n}

where $X_n$ is the probability of the gene being in its normal state after dividing for the nth time, and $Z_n$ is the probability of it being in another state after dividing for the nth time, where $n\in\mathbb{N}$ .
Matrix $\boldsymbol{M}$ is found to be $\left(\begin{smallmatrix}0.94&b\\0.06&0.98\end{smallmatrix}\right)$ .

(a)

Find the eigenvalues of M.

[ 3 ]

$\det\left(\begin{smallmatrix}0.94-\lambda&0.02\\0.06&0.98-\lambda\end{smallmatrix}\right)=0$
(M1)
$(0.94-\lambda)(0.98-\lambda)-0.0012=0$ OR $\lambda^2-1.92\lambda+0.92=0$ $\lambda=1,\ 0.92\ (\frac{23}{25})$
Note: Award M1 for an attempt to find eigenvalues. Any indication that $\det(\boldsymbol{M}-\lambda\boldsymbol{I})=0$ has been used is sufficient.

(b)

Find the eigenvectors of M.

[ 3 ]

$\left(\begin{smallmatrix}0.94&0.02\\0.06&0.98\end{smallmatrix}\right)\binom{x}{y}=\binom{x}{y}$ OR $\left(\begin{smallmatrix}0.94&0.02\\0.06&0.98\end{smallmatrix}\right)\binom{x}{y}=0.92\binom{x}{y}$
Note: This M1 can be awarded for attempting to find either eigenvector.
0.02y-0.06x=0 OR 0.02y+0.02x=0
Eigenvectors $\binom13$ and $\binom1{-1}$ .
Note: Accept any multiple of the given eigenvectors.

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

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Question (b)

Question 3

Question (a)

Question (i)

Question (ii)

Question 4

Question 5

Question 6

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Question (b)