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Date	May Example question	Marks available	2	Reference code	EXM.2.AHL.TZ0.15
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	15	Adapted from	N/A

Question

The matrix A is defined by $A = (\begin{matrix} 3 & 0 0 & 2 \end{matrix})$ $A = \left( {\begin{array}{*{20}{c}} 3&0 \\ 0&2 \end{array}} \right)$ .

Pentagon, P, which has an area of 7 cm², is transformed by A.

The matrix B is defined by $B = \frac{1}{2} (\begin{matrix} 3 \sqrt{3} & 3 - 2 & 2 \sqrt{3} \end{matrix})$ $B = \frac{1}{2}\left( {\begin{array}{*{20}{c}} {3\sqrt 3 }&3 \\ { - 2}&{2\sqrt 3 } \end{array}} \right)$ .

B represents the combined effect of the transformation represented by a matrix X, followed by the transformation represented by A.

Describe fully the geometrical transformation represented by A.

[5]

Find the area of the image of P.

[2]

Find the matrix X.

[3]

Describe fully the geometrical transformation represented by X.

[2]

Markscheme

stretch A1

scale factor 3, A1

y-axis invariant (condone parallel to the x-axis) A1

and

stretch, scale factor 2, A1

x-axis invariant (condone parallel to the y-axis) A1

[5 marks]

$det (A) = 6$ ${\text{det}}\left( A \right) = 6$ A1

$7 \times 6 = 42 c m^{2}$ $7 \times 6 = 42\,{\text{c}}{{\text{m}}^2}$ A1

[2 marks]

$B = A X$ $B = AX$ (A1)

$X = A^{- 1} B$ $X = {A^{ - 1}}B$ (M1)

$X = (\begin{matrix} 0.866 & 0.5 - 0.5 & 0.866 \end{matrix}) ⎛ ⎝ = ⎛ ⎝ \begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix} ⎞ ⎠ ⎞ ⎠$ $X = \left( {\begin{array}{*{20}{c}} {0.866}&{0.5} \\ { - 0.5}&{0.866} \end{array}} \right)\left( { = \left( {\begin{array}{*{20}{c}} {\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}} \\ { - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}} \end{array}} \right)} \right)$ A1

[3 marks]

Rotation A1

clockwise by 30° about the origin A1

[2 marks]

Examiners report

[N/A]

Syllabus sections

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

Show 36 related questions

22M.2.AHL.TZ1.6c.i:
Find an expression for $h (x)$ $h (x)$ .
22M.2.AHL.TZ1.6c.ii:
Find the value of $a$ $a$ .
22M.2.AHL.TZ1.7a.i:
By considering the image of $(0, 0)$ $(0, 0)$ , 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.TZ0.16d:
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:
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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