1Discussion: a point A(x, y) is translated by the 2 × 1 matrix with entries a and b. What are the coordinates of the new point A', and how can A(x, y) be written as a matrix?
2Think about it: what are the 2 × 2 transformation matrices for reflection in the Y-axis (x = 0), in the line y = x, in the line y = -x, in the line x = h and in the line y = k?
3Think about it: what are the 2 × 2 transformation matrices for rotation through -90° (the negative direction) about the origin and for rotation through 180° about the origin?
4Think about it: what is the transformation matrix for an enlargement with centre (a, b) and scale factor k?
5Which transformation does the matrix \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \) represent: reflection in the X-axis, reflection in the Y-axis, reflection in the line y = x, or reflection in the line y = -x?
6Which matrix represents the unit square in the first quadrant: \( \begin{pmatrix} 0 & -1 & -1 & 0 \\ 0 & 1 & -1 & -1 \end{pmatrix} \), \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & -1 & -1 \end{pmatrix} \), \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \) or \( \begin{pmatrix} 0 & -1 & -1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \)?
7Which 2 × 2 matrix is related to reflection in the X-axis: \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \), \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \), \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \) or \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)?
8Which 2 × 2 matrix is related to a positive rotation of 90° about the origin: \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \), \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \), \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) or \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \)?
9Define the unit square and matrix transformation.
10Transform the point A(-4, 6) by the matrix \( \begin{pmatrix} 3 \\ 0 \end{pmatrix} \).
11Transform the point A(-4, 6) by the matrix \( \begin{pmatrix} 6 \\ 0 \end{pmatrix} \).
12Transform the point A(-4, 6) by the matrix \( \begin{pmatrix} 3 \\ 2 \end{pmatrix} \).
13Transform the point A(-4, 6) by the matrix \( \begin{pmatrix} \frac{1}{2} \\ \frac{1}{3} \end{pmatrix} \).
14Find the 2 × 2 matrix that transforms the point (x, y) to (2x + y, 3x + 4y).
15Find the matrix that transforms the point (x, y) to (x - 2y, 2x - 3y).
16The line segment PQ has end points P(3, 4) and Q(8, 4). Write the coordinates of the images when PQ is transformed by the matrix \( \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} \).
17A rectangle has vertices A(2, 3), B(2, 6), C(5, 2) and D(5, 6). It is transformed by the 2 × 2 matrix \( \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \) into the quadrilateral A'B'C'D'. Find the coordinates of A', B', C' and D'.
18The unit square \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \) is transformed by the 2 × 2 matrix \( \begin{pmatrix} 3 & 2 \\ 1 & 5 \end{pmatrix} \) into the quadrilateral O'A'B'C'. Find the coordinates of O', A', B' and C'.
19Find the 2 × 2 matrix that transforms the unit square \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \) into the parallelogram \( \begin{pmatrix} 0 & 3 & 4 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \).
20Find a 2 × 2 transformation matrix that transforms the unit square \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \) into the parallelogram \( \begin{pmatrix} 0 & 3 & 4 & 1 \\ 0 & 1 & 3 & 2 \end{pmatrix} \).
21Find a 2 × 2 transformation matrix that transforms the unit square into the parallelogram \( \begin{pmatrix} 0 & 2 & 5 & 3 \\ 0 & 3 & 4 & 1 \end{pmatrix} \).
22Prove by matrix transformation that a rotation through -90° about the origin followed by reflection in the line x = 0 is equivalent to reflection in the line y = -x.
23Using matrix transformation, prove that a positive rotation of +90° about the origin followed by reflection in the Y-axis is equivalent to reflection in the line y = x.
24A circle passes through the points (-2, 0) and (0, -2), and its centre lies on the line 2x - 3y + 1 = 0. Define a circle on the basis of conic sections.
25A circle passes through (-2, 0) and (0, -2), and its centre lies on the line 2x - 3y + 1 = 0. Find the coordinates of the centre.
26For the same circle, find the equation of the line through its centre that is perpendicular to the line 2x - 3y + 1 = 0.
27The line segment PQ passes through P(2, -3) and Q(-4, 9). Find the equation of the line PQ.
28Find the equation of the line parallel to PQ (through P(2, -3) and Q(-4, 9)) and passing through M(-1, 1).
29Find the equation of the line perpendicular to PQ (through P(2, -3) and Q(-4, 9)) and passing through N(1, 3).
30Find the images P' and Q' when the points P(2, -3) and Q(-4, 9) are translated by the displacement vector PQ.
31The vertical angle of a cone is 70°. A plane cuts the cone and the section is a parabola. Define an ellipse.
32The vertical angle of a cone is 70° and a plane cuts it in a parabola. Find the angle between the axis of the cone and the plane.
33The equation of a circle is \( 4x^2 + 4y^2 - 24x - 20y - 3 = 0 \). Which conic is formed when a plane cuts a cone parallel to its base?
34Find the centre and radius of the circle \( 4x^2 + 4y^2 - 24x - 20y - 3 = 0 \).
35One end of a diameter of the circle \( 4x^2 + 4y^2 - 24x - 20y - 3 = 0 \) is (1, 2). Find the equation of the diameter.
36The equation of a circle is \( x^2 + y^2 - 2x + 4y - 4 = 0 \). Define a circle on the basis of conic sections.
37Find the centre of the circle \( x^2 + y^2 - 2x + 4y - 4 = 0 \).
38One end of a diameter of the circle \( x^2 + y^2 - 2x + 4y - 4 = 0 \) is (1, 2). Find the equation of the line perpendicular to this diameter and passing through B(-3, 1).
39R1 is reflection in the X-axis and R2 is reflection in the Y-axis. Triangle ABC has vertices A(1, 5), B(5, 1) and C(8, 3). Find the equations of the sides AB and BC.
40R1 is reflection in the X-axis and R2 is reflection in the Y-axis. Which single transformation does the combined transformation R1oR2 represent?
41Using that single transformation (rotation through 180° about the origin), find the image of triangle ABC with A(1, 5), B(5, 1) and C(8, 3).
42Show triangle ABC with A(1, 5), B(5, 1), C(8, 3) and its image A'B'C' on the same graph.
43Triangle PQR has vertices P(1, 2), Q(5, 2) and R(1, 5). Write the transformation matrix for reflection in the X-axis.
44Find the images of P(1, 2), Q(5, 2) and R(1, 5) under the matrix \( \begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix} \).
45In triangle PQR with P(1, 2), Q(5, 2) and R(1, 5), find the equation of the perpendicular PS drawn from P to QR.
46Triangle MUN with M(-2, 2), U(2, 2), N(2, 8) is transformed by the translation vectors \( T_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and \( T_2 = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \). Find the combined transformation T1oT2.
47Using T1oT2 = (4, 3), find the image of triangle MUN with M(-2, 2), U(2, 2) and N(2, 8).
48Show triangle MUN with M(-2, 2), U(2, 2), N(2, 8) and its image M'U'N' on the same graph.
49R1 is reflection in the line y = x and R2 is reflection in the line x = 0. Which single transformation does R1oR2 represent?
50Using that single transformation, find the image of triangle ABC with A(3, 2), B(1, 2) and C(2, 3).
51Show triangle ABC with A(3, 2), B(1, 2), C(2, 3) and its image A'B'C' on the same graph.
52In triangle ABC with A(3, 2), B(1, 2) and C(2, 3), find the equation of the altitude AD drawn from A to BC.
53Triangle BUT has vertices B(3, 2), U(1, -1) and T(5, -5). If the slopes of two straight lines are m1 and m2 and the angle between them is θ, write the relation between m1, m2 and θ.
54For triangle BUT with B(3, 2), U(1, -1) and T(5, -5), find the equation of the line through the centroid G parallel to the side UT.
55Find the image of triangle BUT with B(3, 2), U(1, -1) and T(5, -5) under the matrix \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \).
56Why can a translation never be done by multiplying by a 2 × 2 matrix?
57Find the image of the unit square under the enlargement E[(0, 0), 2] and compare its area with the unit square.
58Find the image of the point (5, -2) under rotation through -90° about the origin, using its matrix.