Vectors & TransformationsCambridge IGCSE Maths: Topic test

20 questions, 52 marks

Cambridge IGCSE Maths

Vectors & Transformations topic test

Total 52 marks

Name

Class

Date

1
Vector $\mathbf{u} = \binom{5}{-12}$ , vector $\mathbf{v} = \binom{-2}{3}$ , and the point W(4, -6).
(a)
What is the magnitude of vector u?
[1 mark]
A13
B-7
C17
D169
(b)
What is u + v?
[1 mark]
A(7, -15)
B(3, -15)
C(3, -9)
D(3, 9)
(c)
A rotation of 180 degrees about the origin maps point W to which point?
[1 mark]
A(4, 6)
B(4, -6)
C(-4, -6)
D(-4, 6)
Total for question 1: 3 marks
2
A boat travels from port C to buoy D, a displacement of $\binom{8}{-6}$ km. From D, it continues to lighthouse E, a displacement of $\binom{-3}{10}$ km.
(a)
Find the displacement vector from C to E.
[2 marks]
(b)
Find the distance from C to E, giving your answer to 1 decimal place.
[2 marks]
Total for question 2: 4 marks
3
Point F has coordinates (3, 5).
(a)
F is reflected in the line y = x, and the image is then enlarged with centre the origin and scale factor -2. Find the coordinates of the final image.
[3 marks]
(b)
F is instead translated by vector (-4, 2), and the image is then rotated 90 degrees clockwise about the origin. Find the coordinates of the final image.
[3 marks]
Total for question 3: 6 marks
4
Quadrilateral GHIJ has position vectors $\overrightarrow{OG} = \binom{2}{1}$ , $\overrightarrow{OH} = \binom{6}{1}$ and $\overrightarrow{OJ} = \binom{2}{5}$ relative to origin O. K is the point such that $\overrightarrow{GK} = 2\overrightarrow{GH}$ .
(a)
Find the vector GH, and hence find the position vector of K.
[4 marks]
(b)
Find the vector HJ, and determine whether HJ is parallel to GK (found in part (a)).
[4 marks]
(c)
The point K found in part (a) is enlarged with centre the origin and scale factor 0.5, and the image is then reflected in the x-axis. Find the coordinates of the final image, and find the translation vector that would map G directly to this final image.
[5 marks]
Total for question 4: 13 marks
5
Vector $\mathbf{s} = \binom{-6}{8}$ , vector $\mathbf{t} = \binom{3}{3}$ , and point Y(-2, 7).
(a)
What is the magnitude of vector s?
[1 mark]
A14
B10
C2
D100
(b)
What is 3t - s?
[1 mark]
A(15, 1)
B(9, -5)
C(3, 17)
D(15, 9)
(c)
A reflection in the y-axis maps point Y to which point?
[1 mark]
A(-2, -7)
B(-2, 7)
C(2, 7)
D(2, -7)
Total for question 5: 3 marks
6
A cyclist rides from checkpoint L to checkpoint M, a displacement of $\binom{-9}{12}$ km. From M, the cyclist rides to checkpoint N, a displacement of $\binom{5}{-4}$ km.
(a)
Find the displacement vector from L to N.
[2 marks]
(b)
Find the distance from L to N, giving your answer to 1 decimal place.
[2 marks]
Total for question 6: 4 marks
7
Point Z has coordinates (-3, 4).
(a)
Z is rotated 90 degrees anticlockwise about the origin, and the image is then translated by vector (2, -5). Find the coordinates of the final image.
[3 marks]
(b)
Z is instead enlarged with centre the origin and scale factor 3, and the image is then reflected in the line y = -x. Find the coordinates of the final image.
[3 marks]
Total for question 7: 6 marks
8
Triangle ABC has position vectors $\overrightarrow{OA} = \binom{4}{3}$ , $\overrightarrow{OB} = \binom{10}{3}$ and $\overrightarrow{OC} = \binom{4}{9}$ relative to origin O. The area of triangle ABC is 18 square units.
(a)
Find the vector AB and the vector AC, and hence find the position vector of the midpoint N of BC.
[4 marks]
(b)
Triangle ABC is enlarged with centre the origin and scale factor 2, and the image is then translated by vector (-3, 1). Find the coordinates of the image of vertex A after both transformations.
[4 marks]
(c)
Using the scale factor 2 enlargement from part (b) (before the translation) and the given area of triangle ABC, calculate the area of the enlarged triangle A'B'C', and state the ratio of the area of ABC to the area of A'B'C'.
[5 marks]
Total for question 8: 13 marks

End of questions