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Date November 2020 Marks available 2 Reference code 20N.1.AHL.TZ0.F_7
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Find 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

x'y'=3002xy.

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

[1]
a.

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]

a.

the four sides are equal in length 5         A1

Grad AB=43, Grad BC=-34         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'=89; Grad C'D'=89         A1

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

Grad A'D'=-612; Grad B'C'=-612         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×area of square

=6×25         (M1)

=150         A1


[2 marks]

b.v.

Examiners report

[N/A]
a.
[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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Topic 3—Geometry and trigonometry

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