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Date	November 2020	Marks available	1	Reference code	20N.1.AHL.TZ0.F_7
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Determine	Question number	F_7	Adapted from	N/A

Question

Points in the plane are subjected to a transformation $T$ in which the point $(x, y)$ is transformed to the point $(x', y')$ where

$[\begin{matrix} x' \\ y' \end{matrix}] = [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$ .

Describe, in words, the effect of the transformation $T$ .

[1]

Show that the points $A (1, 4), B (4, 8), C (8, 5), D (5, 1)$ form a square.

[3]

b.i.

Determine the area of this square.

[1]

b.ii.

Find the coordinates of $A', B', C', D'$ , the points to which $A, B, C, D$ are transformed under $T$ .

[2]

b.iii.

Show that $A' B' C' D'$ is a parallelogram.

[3]

b.iv.

Determine the area of this parallelogram.

[2]

b.v.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

a stretch of scale factor $3$ in the $x$ direction

and a stretch of scale factor $2$ in the $y$ direction A1

[1 mark]

the four sides are equal in length $(5)$ A1

$Grad AB = \frac{4}{3}, Grad BC = - \frac{3}{4}$ A1

so product of gradients $= - 1$ , therefore $AB$ is perpendicular to $BC$ A1

therefore $ABCD$ is a square AG

[3 marks]

b.i.

area of square $= 25$ A1

[1 mark]

b.ii.

the transformed points are

$A' = (3, 8)$

$B' = (12, 16)$

$C' = (24, 10)$

$D' = (15, 2)$ A2

Note: Award A1 if one point is incorrect.

[2 marks]

b.iii.

$Grad A'B' = \frac{8}{9}; Grad C'D' = \frac{8}{9}$ A1

therefore $A'B'$ is parallel to $C'D'$ R1

$Grad A'D' = - \frac{6}{12}; Grad B'C' = - \frac{6}{12}$ A1

therefore $A'D'$ is parallel to $B'C'$

therefore $A' B' C' D'$ is a parallelogram AG

[3 marks]

b.iv.

$area of parallelogram = |determinant| \times area of square$

$= 6 \times 25$ (M1)

$= 150$ A1

[2 marks]

b.v.

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

b.iv.

[N/A]

b.v.

Syllabus sections

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

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22M.2.AHL.TZ1.6c.i:
Find an expression for $h (x)$ .
22M.2.AHL.TZ1.6c.ii:
Find the value of $a$ .
22M.2.AHL.TZ1.7a.i:
By considering the image of $(0, 0)$ , find $q$ .
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.TZ0.16a:
Describe fully the geometrical transformation represented by B.
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.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:
Given that $T : (\begin{matrix} p \\ q \end{matrix}) \mapsto 2 (\begin{matrix} p \\ q \end{matrix})$ , find the value of $p$ and the value of $q$ .
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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Topic 3—Geometry and trigonometry

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