IB Mathematics Applications & Interpretation AHL 1.12—complex Numbers Introduction Questionbank

Lumen Games stores every two-dimensional transform used by its animation tool as a single number written in the form $a + bi$, where $a$ and $b$ are real. The tool holds each of the two components $a$ and $b$ correct to two decimal places, and to no greater accuracy than that. Assume the tool evaluates all formulas exactly using the stored decimal values; only the stated storage rounding is applied.

The documentation for the tool prints one test and nothing else. For a pair of stored transforms $z$ and $w$ it defines an alignment number $B$ and a twist number $D$ by

$B = zw^{*} + z^{*}w \text{ and } D = zw^{*} - z^{*}w,$

where $z^{*}$ denotes the conjugate of $z$. A scene built from the pair $z$ and $w$ renders aligned exactly when $B = 0$.

Before committing a scene to an overnight render, an artist at the studio has to decide from the stored numbers alone whether it will render aligned, because a scene that comes back unaligned in the morning has cost a night of machine time.

The first scene in the artist's file is built from the stored pair $z = 3 + 5i$ and $w = 4 - 2i$.

(a) Write down $w^{*}$.

(d) Show that $B = 2\operatorname{Re}\left(zw^{*}\right)$ for every pair of transforms $z$ and $w$.

Meridian Reviews runs an online survey platform. Every survey question on the platform has two numbers attached to it, called its index values, and the platform's documentation states that these are the two solutions of

$x^{2}+px+q=0,$

where $p$ and $q$ are real numbers written into an audit log. The documentation also states that the two index values of a survey question are never real.

The archive record of a survey question keeps only two fields: the score $s$, which is the real part of either index value, and the confidence $c$, which is the modulus of either index value. Score and confidence are the platform's own names for these two fields, and the definitions just given are the only ones needed here.

A database migration has destroyed every audit log, so no value of $p$ or $q$ survives anywhere. The archive itself is untouched, and the platform's data manager must decide whether the audit logs can be rebuilt from the archive alone.

For parts (a) to (f), treat the displayed archive values as exact. Parts (g) and (h) consider their actual stored accuracy. The archive record for survey question $14$ gives a score of $5.6$ and a confidence of $6.5$.

(b) Show that, for every survey question, $p=-2s$ and $q=c^{2}$.

The archive records of three further survey questions are shown in the table. The $p$ and $q$ columns are empty because the audit logs that held them were destroyed.

(c) Find the six missing entries of the table.

The archive record for survey question $52$ gives a score of $4.8$ and a confidence of $4.1$.

An engineering simulation stores a pair of complex signal coefficients as the roots of

$x^2-2kx+k^2+9=0,$

where $k$ is a real design parameter. The real and imaginary parts are the two components of each coefficient.

(b) Calculate $zw^{*}$, giving your answer in the form $a + bi$ where $a$ and $b$ are real.

(c) Hence calculate the alignment number $B$ and the twist number $D$ for this scene.

(a) Calculate the two index values of survey question $14$.

(d) Explain why the platform, as documented, cannot have produced the archive record for survey question $52$.

The platform defines a further quantity, the spread. The spread $d$ of a survey question is the imaginary part of whichever of its two index values has a positive imaginary part.

(a) Show that the equation has no real roots for every real $k$, and determine its two complex roots.

(b) It is known that each root has modulus $5$. Determine all possible values of $k$.

(c) One root lies in quadrant II and has positive imaginary part. Determine $k$, write the corresponding quadratic, and find the quotient of this root by the other root in Cartesian form.