Shape A has vertices (1, 1), (3, 1), (3, 2) and (1, 2). Shape A is translated by the vector (4, -3) to give shape B. Find the coordinates of the vertices of shape B.
(Total for Question 1 is 2 marks)
2
Point P has coordinates (3, 5). Write down the coordinates of the image of P after a reflection in the x-axis.
(Total for Question 2 is 1 mark)
3
Point Q has coordinates (-2, 4). Write down the coordinates of the image of Q after a reflection in the y-axis.
(Total for Question 3 is 1 mark)
4
Point R has coordinates (6, -1). Write down the coordinates of the image of R after a reflection in the line y = x.
(Total for Question 4 is 1 mark)
5
Point S has coordinates (2, 5). Write down the coordinates of the image of S after a reflection in the line y = -x.
(Total for Question 5 is 1 mark)
6
Point T has coordinates (4, 2). Write down the coordinates of the image of T after a rotation of 90 degrees clockwise about the origin.
(Total for Question 6 is 1 mark)
7
Point U has coordinates (-3, 5). Write down the coordinates of the image of U after a rotation of 180 degrees about the origin.
(Total for Question 7 is 1 mark)
8
Point V has coordinates (1, -4). Write down the coordinates of the image of V after a rotation of 90 degrees anticlockwise about the origin.
(Total for Question 8 is 1 mark)
9
Point W has coordinates (5, 3). Find the coordinates of the image of W after a rotation of 180 degrees about the point (2, 2).
(Total for Question 9 is 2 marks)
10
Point X has coordinates (3, 6). Find the coordinates of the image of X after a rotation of 90 degrees clockwise about the point (1, 2).
(Total for Question 10 is 2 marks)
11
Triangle M has vertices (1, 1), (1, 3) and (2, 1). Find the coordinates of the vertices of the image of triangle M after an enlargement with scale factor 3, centre the origin.
(Total for Question 11 is 2 marks)
12
Point N has coordinates (6, 4). Write down the coordinates of the image of N after an enlargement with scale factor 1/2, centre the origin.
(Total for Question 12 is 1 mark)
13
Point K has coordinates (4, 2). Write down the coordinates of the image of K after an enlargement with scale factor -1, centre the origin.
(Total for Question 13 is 1 mark)
14
Point L has coordinates (5, 3). Find the coordinates of the image of L after an enlargement with scale factor -2, centre (1, 1).
(Total for Question 14 is 2 marks)
15
A shape is translated so that the point (2, 3) on the object maps to the point (5, 1) on the image. Find the vector that describes this translation.
(Total for Question 15 is 1 mark)
16
A shape is reflected in the line x = 4. State which points of the plane are invariant (unchanged) under this reflection.
(Total for Question 16 is 1 mark)
17
Shape A has vertices (1, 1), (1, 4) and (3, 1). Shape B has vertices (-1, 1), (-4, 1) and (-1, 3). Describe fully the single transformation that maps shape A onto shape B.
(Total for Question 17 is 3 marks)
18
Shape C has vertices (2, 2), (2, 5) and (4, 2). Shape C is reflected in the line y = 1 to give shape D. Find the coordinates of the vertices of shape D.
(Total for Question 18 is 3 marks)
19
Rectangle has vertices (0, 0), (4, 0), (4, 2) and (0, 2). The rectangle is enlarged with centre the origin so that the image has an area of 72 square units. Find the scale factor of the enlargement, and hence find the coordinates of the vertices of the image.
(Total for Question 19 is 4 marks)
20
Triangle P has vertices (1, 1), (1, 3) and (3, 1). Triangle P is rotated 90 degrees clockwise about the origin, and the image is then translated by the vector (2, -1), to form triangle Q.
(a)Find the coordinates of the vertices of triangle Q.(3)
(b)A student says: 'The single transformation that maps triangle Q back onto triangle P is a rotation of 90 degrees anticlockwise about the origin.' Show, using at least one vertex, that the student is incorrect.(2)
(Total for Question 20 is 5 marks)
21
Shape E has vertices (1, 1), (1, 3), (2, 3) and (2, 1). Shape F has vertices (2, 2), (2, 6), (4, 6) and (4, 2). Describe fully the single transformation that maps shape E onto shape F.
(Total for Question 21 is 3 marks)
22
Shape G has vertices (-3, 1), (-3, 4) and (-1, 1). Shape G is enlarged with scale factor -2, centre (-1, 1), to give shape H. Find the coordinates of the vertices of shape H, and state the ratio of the area of H to the area of G.
(Total for Question 22 is 4 marks)
23
Triangle J has vertices (0, 0), (2, 0) and (0, 3). Triangle J is enlarged with scale factor k, centre the origin, and the image is then reflected in the y-axis, to form triangle K with vertices (0, 0), (-6, 0) and (0, 9).
(a)Find the value of k.(2)
(b)M is the midpoint of the side of triangle J from (2, 0) to (0, 3). Find the coordinates of the image of M after the same combined transformation (the enlargement followed by the reflection).(2)
(Total for Question 23 is 4 marks)
24
Triangle A has vertices (1, 1), (1, 3) and (4, 1). Triangle A is rotated 90 degrees clockwise about the point (p, q) to form triangle B, with vertices (3, -3), (5, -3) and (3, -6). Find the values of p and q.
(Total for Question 24 is 5 marks)
25
A shape is enlarged with scale factor k, centre the origin, and the image is then translated by the vector (h, h), where k and h are constants. A student claims that, for some values of k and h, this combined transformation maps every point (a, b) onto the point (b, a). Show that the student's claim is impossible: that is, show that there are no values of k and h for which this is true.
(Total for Question 25 is 4 marks)
Mark scheme · 3.17D Transformations: Fluency and Exam Drill
Question 1
M1 adds 4 to each x-coordinate and subtracts 3 from each y-coordinate oe, at least one correct vertex shown
A1 all four vertices correct: (5,-2), (7,-2), (7,-1), (5,-1) cao
Answer: (5, -2), (7, -2), (7, -1), (5, -1)
Question 2
B1 (3, -5) cao
Answer: (3, -5)
Question 3
B1 (2, 4) cao
Answer: (2, 4)
Question 4
B1 (-1, 6) cao
Answer: (-1, 6)
Question 5
B1 (-5, -2) cao
Answer: (-5, -2)
Question 6
B1 (2, -4) cao
Answer: (2, -4)
Question 7
B1 (3, -5) cao
Answer: (3, -5)
Question 8
B1 (4, 1) cao
Answer: (4, 1)
Question 9
M1 uses image = (2 x 2 - 5, 2 x 2 - 3) oe, or a correct diagram/vector method
A1 (-1, 1) cao
Answer: (-1, 1)
Question 10
M1 translates X by (-1,-2) to (2,4), applies (x,y) -> (y,-x) to get (4,-2), oe valid method
A1 (5, 0) cao
Answer: (5, 0)
Question 11
M1 multiplies each coordinate by 3 oe, at least one correct vertex shown
A1 all three vertices correct: (3,3), (3,9), (6,3) cao
Answer: (3, 3), (3, 9), (6, 3)
Question 12
B1 (3, 2) cao
Answer: (3, 2)
Question 13
B1 (-4, -2) cao
Answer: (-4, -2)
Question 14
M1 finds the vector from the centre to L as (4,2) oe
A1 (-7, -3) cao
Answer: (-7, -3)
Question 15
B1 (3, -2) oe, may be written as a column vector
Answer: (3, -2)
Question 16
B1 any point on the line x = 4 oe, e.g. (4, y) for any y
Answer: Every point on the line x = 4 (points of the form (4, y))
(a) M1 rotates at least one vertex of P by 90 degrees clockwise about the origin using (x,y) -> (y,-x)
(a) M1 dep translates the rotated vertices by the vector (2,-1)
(a) A1 all three vertices correct: (3,-2), (5,-2), (3,-4) cao
(a) Answer: (3, -2), (5, -2), (3, -4)
(b) M1 applies a 90 degree anticlockwise rotation about the origin to a vertex of Q, e.g. (3,-2) -> (2,3)
(b) A1 correct conclusion that this does not give the corresponding vertex of P (e.g. (2,3) is not equal to (1,1)), so the student is incorrect, because triangle P and Q are related by a rotation combined with a translation, not by a rotation alone
(b) Answer: Incorrect: rotating (3,-2) by 90 degrees anticlockwise about the origin gives (2,3), not (1,1), so a single rotation about the origin cannot map Q back onto P (a translation was also applied)
Question 21
B1 enlargement (oe 'enlarged')
B1 scale factor 2
B1 centre (0, 0)
Answer: Enlargement, scale factor 2, centre the origin (0, 0)
Question 22
M1 uses image = centre + (-2)(vertex - centre) for at least one vertex, oe valid method
A1 two vertices correct
A1 all three vertices correct: (3,1), (3,-5), (-1,1) cao
B1 area ratio 4 : 1 oe (H has 4 times the area of G), since area scale factor = (-2)2 = 4
Answer: H has vertices (3, 1), (3, -5), (-1, 1); area of H : area of G = 4 : 1
Question 23
(a) M1 forms a correct equation from corresponding coordinates, e.g. enlarging (2,0) by k then reflecting in the y-axis gives (-2k,0), so -2k = -6 oe (or using (0,3) -> (0,9): 3k = 9)
(a) A1 k = 3 cao
(a) Answer: k = 3
(b) M1 finds M = (1, 1.5) oe 3/2, and enlarges by k=3 to get (3, 4.5) oe 9/2, ft their k
(b) A1 reflects in the y-axis to get (-3, 4.5) cao, ft their k
(b) Answer: (-3, 4.5)
Question 24
M1 sets up the general image of a 90 degree clockwise rotation about (p,q): (x,y) -> (y-q+p, p-x+q)
M1 forms two equations in p and q using a vertex and its image, e.g. from (1,1) -> (3,-3): 1-q+p=3 and p-1+q=-3
M1 solves the equations simultaneously (or uses an equivalent method, e.g. locating the centre as the intersection of the perpendicular bisectors of two vertex-image pairs)
A1 p = 0
A1 q = -2
Answer: p = 0, q = -2
Question 25
M1 writes the image of the general point (a,b) under the combined transformation as (ka+h, kb+h)
M1 sets ka+h = b and kb+h = a, stating this must hold for every value of a and b
M1 compares coefficients of a (or b) in ka+h=b: since the right-hand side b does not depend on a, the coefficient of a on the left must be 0, so k=0
A1 cso: substituting k=0 gives h=b for every value of b, which is impossible since h is a fixed constant; therefore no values of k and h exist, so the student's claim is false
Answer: No such k and h exist (proof by contradiction, cso)