J20

19. Using a scale of 1 cm to one unit in each case draw the axes taking values of x from
-4 to +6 and values of y from 0 to 12.
(a) Draw and label the quadrilateral OABC with O(0, 0), A(2, 0), B(4, 2), C(0, 2)
(b) Find and draw the image of OABC under the transformation whose matrix is
2.4
1.8
R, where R = .
1.8
2.4
(c) Calculate, in surd form, the lengths OB and O ′ B′
(d) Calculate the angle AO A′ .
(e) Given that the transformation R consists of a rotation about O followed by an
enlargement, state the angle of the rotation and the scale factor of the
enlargement.
cos
sin
20. The matrix R =
represents a positive rotation of θ 0 about
sin
cos
the origin. Find the matrix which represents a rotation of:
(a) 90 0 (b) 180 0
(c) 30 0 (d) -90 0
(e) 60 0 (f) 150 0
(g) 45 0 (h) 53.1 0
Confirm your results for parts (a), (e), (h) by applying the matrix to the quadrilateral
O90, 0), A(0, 2), B(4, 2), C(4, 0).
x ′,
y′
of a point (x, y) under a transformation is given by
x
3
0 x 2
= +
y
1
2
y
5
(a) Find the coordinates of the image of the point (4, 3).
(b) The image of the point (m, n) is the point (11, 7). Write down two equations
involving m and n and hence find the values of m and n.
(c) The image of the point (h, k) is the point (5, 10). Find the values of h and k.
21. The image ( )
22. Draw A(0, 2), B(2, 2), C(0, 4) and its image under an enlargement,
A′(2, 2), B′(6, 2), C′ (2, 6).
(a) What is the centre of enlargement?
(b) Find the image of ABC under a reflection in the line x = 0
(c) Find the translation which maps this image onto A′ B′ C′ .
(d) What is the matrix X and vector v which represents a reflection in the line x
= 2?
2
0
1
0
23. A = ; h and k are numbers so that A 2 = hA + kI, where I = .
1
2
0
1
Find the values of h and k.
a
1
1
0
24. M = . Find the values of a if: (a) M
2
= 17
1
a
0
1
(b) M = -10.