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Date	May Example question	Marks available	2	Reference code	EXM.2.AHL.TZ0.16
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Describe	Question number	16	Adapted from	N/A

Question

The matrices A and B are defined by $A = \left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&4 \end{array}} \right)$ and $B = \left( {\begin{array}{*{20}{c}} { - 1}&0 \\ 0&1 \end{array}} \right)$ .

Triangle X is mapped onto triangle Y by the transformation represented by AB. The coordinates of triangle Y are (0, 0), (−30, −20) and (−16, 32).

Describe fully the geometrical transformation represented by B.

[2]

Find the coordinates of triangle X.

[5]

Find the area of triangle X.

[2]

c.i.

Hence find the area of triangle Y.

[3]

c.ii.

Matrix A represents a combination of transformations:

A stretch, with scale factor 3 and y-axis invariant;
Followed by a stretch, with scale factor 4 and x-axis invariant;
Followed by a transformation represented by matrix C.

Find matrix C.

[4]

Markscheme

reflection in the y-axis A1A1

[2 marks]

$X = {\left( {AB} \right)^{ - 1}}Y$ M1

EITHER

$AB = \left( {\begin{array}{*{20}{c}} { - 3}&{ - 2} \\ { - 2}&4 \end{array}} \right)$ , so ${\left( {AB} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} { - \frac{1}{4}}&{ - \frac{1}{8}} \\ { - \frac{1}{8}}&{\frac{3}{{16}}} \end{array}} \right)$ M1A1

$X = {B^{ - 1}}{A^{ - 1}}Y$ M1A1

THEN

$X = \left( {\begin{array}{*{20}{c}} 0&{10}&0 \\ 0&0&8 \end{array}} \right)$ (A1)

So the coordinates are (0, 0), (10, 0) and (0, 8). A1

[5 marks]

$\frac{{10 \times 8}}{2} = 40$ units² M1A1

[2 marks]

c.i.

$\det \left( {AB} \right) = - 16$ M1A1

Area $= 40 \times 16 = 640$ units² A1

[3 marks]

c.ii.

A stretch, with scale factor 3 and y-axis invariant is given by $\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)$ A1

A stretch, with scale factor 4 and x-axis invariant is given by $\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)$ A1

So $C = A{\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)^{ - 1}}{\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&{ - \frac{1}{2}} \\ {\frac{2}{3}}&1 \end{array}} \right)$ M1A1

[4 marks]

Examiners report

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.9—Matrix transformations

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22M.2.AHL.TZ1.7a.i:
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22M.2.AHL.TZ1.7c.ii:
Hence find the angle and direction of the rotation represented by $R$ .
22M.2.AHL.TZ1.7b:
Write down the matrix $S$ .
22M.1.AHL.TZ2.15c:
Show that the equation of the resulting line does not depend on $m$ or $c$ .
EXM.2.AHL.TZ0.15d:
Describe fully the geometrical transformation represented by X.
EXM.2.AHL.TZ0.16c.i:
Find the area of triangle X.
22M.2.AHL.TZ1.6c.iii:
Find the value of $b$ .
22M.2.AHL.TZ1.7a.ii:
By considering the image of $(1, 0)$ and $(0, 1)$ , show that

$P = (\begin{matrix} \frac{\sqrt{3}}{4} & \frac{1}{4} \\ - \frac{1}{4} & \frac{\sqrt{3}}{4} \end{matrix})$ .
22M.2.AHL.TZ1.7d.ii:
Find the value of $a$ and the value of $b$ .
21N.2.AHL.TZ0.4b:
Three drones are initially positioned at the points $A$ , $B$ and $C$ . After performing the movements listed above, the drones are positioned at points $A'$ , $B'$ and $C'$ respectively.

Show that the area of triangle $ABC$ is equal to the area of triangle $A' B' C'$ .
21N.2.AHL.TZ0.4a.iii:
Find $P^{2}$ .
21N.2.AHL.TZ0.4c:
Find a single transformation that is equivalent to the three transformations represented by matrix $P$ .
22M.2.AHL.TZ1.7d.i:
Write down an equation satisfied by $(\begin{matrix} a \\ b \end{matrix})$ .
22M.2.AHL.TZ1.7c.i:
Use $P = R S$ to find the matrix $R$ .
21N.2.AHL.TZ0.4a.i:
Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.
21N.2.AHL.TZ0.4a.ii:
Find a single matrix $P$ that defines a transformation that represents the overall change in position.
21N.2.AHL.TZ0.4a.iv:
Hence state what the value of $P^{2}$ indicates for the possible movement of the drone.
EXM.2.AHL.TZ0.16b:
Find the coordinates of triangle X.
EXM.2.AHL.TZ0.15a:
Describe fully the geometrical transformation represented by A.
EXM.2.AHL.TZ0.15c:
Find the matrix X.
20N.1.AHL.TZ0.F_7b.i:
Show that the points $A (1, 4), B (4, 8), C (8, 5), D (5, 1)$ form a square.
EXM.2.AHL.TZ0.15b:
Find the area of the image of P.
EXM.2.AHL.TZ0.16d:
Matrix A represents a combination of transformations:

A stretch, with scale factor 3 and y-axis invariant;
Followed by a stretch, with scale factor 4 and x-axis invariant;
Followed by a transformation represented by matrix C.

Find matrix C.
EXN.3.AHL.TZ0.2c.ii:
Write down $A_{n}$ , in terms of $n$ .
EXN.3.AHL.TZ0.2c.i:
Write down $A_{1}$ .
20N.1.AHL.TZ0.F_7a:
Describe, in words, the effect of the transformation $T$ .
20N.1.AHL.TZ0.F_7b.v:
Determine the area of this parallelogram.
20N.1.AHL.TZ0.F_7b.ii:
Determine the area of this square.
20N.1.AHL.TZ0.F_7b.iv:
Show that $A' B' C' D'$ is a parallelogram.
EXM.2.AHL.TZ0.16c.ii:
Hence find the area of triangle Y.
20N.1.AHL.TZ0.F_7b.iii:
Find the coordinates of $A', B', C', D'$ , the points to which $A, B, C, D$ are transformed under $T$ .
21M.1.AHL.TZ2.14a:
Find the coordinates of the image of the point $(6, - 2)$ .
21M.1.AHL.TZ2.14b:
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21M.1.AHL.TZ2.14c:
A triangle $L$ with vertices lying on the $x y$ plane is transformed by $T$ .

Explain why both $L$ and its image will have exactly the same area.

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