References only, not question text. Open Official paper ↗ where Cambridge publishes the paper,
or see the official Cambridge past-papers page; for other papers, ask your
school. “Paper 42” means Paper 4, variant 2.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Transformations · Core · Paper 1 style · no calculator · 2 marks
(a)Triangle A has vertices (1, 1), (4, 1), (1, 3).Triangle A is translated by the vector (3, −5), that is 3 units to the right and 5 units down.Write down the coordinates of the vertices of the image.[2]
Show answer
(a) (4, −4), (7, −4), (4, −2)
B2 all three correct (B1 for two correct)
Q2Transformations · Core · Paper 1 style · no calculator · 3 marks
(a)Triangle P has vertices (2, 1), (5, 1), (5, 3).Reflect triangle P in the y-axis. Write down the coordinates of the vertices of the image.[2]
(b)The image is then reflected in the x-axis. Write down the coordinates of the vertices of the final image.[1]
Show answer
(a) (−2, 1), (−5, 1), (−5, 3)
B2 (B1 for two correct)
(b) (−2, −1), (−5, −1), (−5, −3)
B1 follow through
Q3Transformations · Core · Paper 3 style · calculator · 2 marks
(a)Triangle B has vertices (1, 3), (4, 3), (4, 5).Reflect triangle B in the line y = 1. Write down the coordinates of the vertices of the image.[2]
Show answer
(a) (1, −1), (4, −1), (4, −3)
B2 (B1 for two correct)
Q4Transformations · Core · Paper 3 style · calculator · 2 marks
(a)Triangle C has vertices (1, 2), (3, 2), (3, 5).Rotate triangle C 90° clockwise about the origin. Write down the coordinates of the vertices of the image.[2]
Show answer
(a) (2, −1), (2, −3), (5, −3)
B2 (B1 for two correct, or a correct rotation anticlockwise)
Q5Transformations · Core · Paper 1 style · no calculator · 3 marks
(a)Shape D is a triangle with vertices (2, 1), (4, 1), (4, 4).Rotate shape D through 180° about (0, 0). Write down the coordinates of the vertices of the image.[2]
(b)The same rotation maps the point (k, −3) onto the point (−5, 3). Find the value of k.[1]
Show answer
(a) (−2, −1), (−4, −1), (−4, −4)
B2
(b) k = 5
B1 a 180° rotation about the origin changes the sign of both coordinates
Q6Transformations · Core · Paper 3 style · calculator · 4 marks
(a)Triangle E has vertices (1, 1), (3, 1), (1, 2).Enlarge triangle E by scale factor 2 with centre (0, 0). Write down the coordinates of the vertices of the image.[2]
(b)Work out the area of triangle E and the area of its image, in square units.[2]
Show answer
(a) (2, 2), (6, 2), (2, 4)
B2
(b) 1 square unit and 4 square units
B1 ½ × 2 × 1 = 1B1 1 × 2² = 4
Q7Transformations · Core · Paper 3 style · calculator · 2 marks
(a)Triangle W has vertices (1, 4), (3, 4), (3, 6).Triangle X has vertices (1, −2), (3, −2), (3, −4), with the vertices in corresponding order.Describe fully the single transformation that maps triangle W onto triangle X.[2]
Show answer
(a) Reflection in the line y = 1
B1 reflectionB1 y = 1
Q8Transformations · Standard · Paper 3 style · calculator · 3 marks
(a)Triangle F has vertices (2, 3), (3, 3), (2, 5).Enlarge triangle F by scale factor 3 with centre (1, 2). Write down the coordinates of the vertices of the image.[3]
Show answer
(a) (4, 5), (7, 5), (4, 11)
M1 using vectors from the centre, e.g. (2, 3) is (1, 1) from (1, 2), so the image is (1, 2) + 3(1, 1)A2 all three correct (A1 for two)
Q9Transformations · Standard · Paper 2 style · no calculator · 3 marks
(a)Triangle A has vertices (1, 2), (3, 2), (1, 5).Triangle B has vertices (−2, 1), (−2, 3), (−5, 1), where each vertex of B corresponds to the vertex of A in the same position in the list.Describe fully the single transformation that maps triangle A onto triangle B.[3]
Show answer
(a) Rotation, 90° anticlockwise, centre (0, 0)
B1 rotationB1 90° anticlockwiseB1 centre (0, 0)
Q10Transformations · Standard · Paper 2 style · no calculator · 3 marks
(a)Triangle P has vertices (2, 1), (4, 1), (4, 2).Triangle Q has vertices (6, 3), (12, 3), (12, 6), with the vertices in corresponding order.Describe fully the single transformation that maps triangle P onto triangle Q.[3]
Show answer
(a) Enlargement, scale factor 3, centre (0, 0)
B1 enlargementB1 scale factor 3B1 centre (0, 0)
Q11Transformations · Standard · Paper 2 style · no calculator · 3 marks
(a)Triangle R has vertices (1, 1), (3, 1), (3, 2).Triangle S has vertices (1, 1), (1, 3), (2, 3), with the vertices in corresponding order.Describe fully the single transformation that maps triangle R onto triangle S.[2]
(b)Triangle S is reflected in the line y = x. Describe where the image lies.[1]
Show answer
(a) Reflection in the line y = x
B1 reflectionB1 y = x
(b) It is back on triangle R (a reflection undoes itself)
B1
Q12Transformations · Standard · Paper 3 style · calculator · 3 marks
(a)Shape U has vertices (−3, 2), (−1, 2), (−1, 5).Shape V has vertices (2, −1), (4, −1), (4, 2), with the vertices in corresponding order.Describe fully the single transformation that maps shape U onto shape V.[2]
(b)Write down the vector of the translation that maps shape V back onto shape U.[1]
Show answer
(a) Translation by the vector (5, −3)
B1 translationB1 vector (5, −3) written as a column vector
(b) (−5, 3)
B1
Q13Transformations · Standard · Paper 4 style · calculator · 5 marks
(a)Triangle T has vertices (1, 2), (4, 2), (4, 3).Triangle T is reflected in the x-axis to give triangle T₁. Write down the coordinates of the vertices of T₁.[1]
(b)Triangle T₁ is rotated 90° clockwise about the origin to give triangle T₂. Write down the coordinates of the vertices of T₂.[2]
(c)Describe fully the single transformation that maps triangle T onto triangle T₂.[2]
Show answer
(a) (1, −2), (4, −2), (4, −3)
B1
(b) (−2, −1), (−2, −4), (−3, −4)
B2 follow through (B1 for two correct)
(c) Reflection in the line y = −x
B1 reflectionB1 y = −x
Q14Transformations · Standard · Paper 4 style · calculator · 4 marks
(a)Triangle Y has vertices (1, 1), (4, 1), (1, 3).Enlarge triangle Y by scale factor 2.5 with centre (1, 1). Write down the coordinates of the vertices of the image.[2]
(b)Calculate the area of triangle Y and the area of its image, in square units.[2]
Show answer
(a) (1, 1), (8.5, 1), (1, 6)
B2 (B1 for two correct)
(b) 3 and 18.75 square units
B1 ½ × 3 × 2 = 3B1 3 × 2.5² = 18.75
Q15Transformations · Standard · Paper 1 style · no calculator · 3 marks
(a)Triangle Z has vertices (2, 1), (4, 1), (2, 3).Rotate triangle Z 90° anticlockwise about the point (1, 0). Write down the coordinates of the vertices of the image.[3]
Show answer
(a) (0, 1), (0, 3), (−2, 1)
M1 using the vector from (1, 0) to each vertex, e.g. (1, 1) → (−1, 1)A2 all three correct (A1 for two)
Q16Transformations · Challenge · Paper 4 style · calculator · 4 marks
(a)Triangle G has vertices (5, 3), (9, 3), (5, 9).Enlarge triangle G by scale factor −½ with centre (1, 1). Write down the coordinates of the vertices of the image.[3]
(b)The area of triangle G is 12 square units. Find the area of the image.[1]
Show answer
(a) (−1, 0), (−3, 0), (−1, −3)
M1 vector from centre to each vertex multiplied by −½, e.g. (4, 2) → (−2, −1)A2 all three correct (A1 for two)
(b) 3 square units
B1 12 × (½)²
Q17Transformations · Challenge · Paper 2 style · no calculator · 3 marks
(a)Triangle J has vertices (2, 2), (4, 2), (2, 6).Triangle K has vertices (−1, −1), (−2, −1), (−1, −3), with the vertices in corresponding order.Describe fully the single transformation that maps triangle J onto triangle K.[3]
Show answer
(a) Enlargement, scale factor −½, centre (0, 0)
B1 enlargementB1 scale factor −½B1 centre (0, 0)
Q18Transformations · Challenge · Paper 4 style · calculator · 3 marks
(a)Triangle L has vertices (1, 1), (3, 1), (1, 2).Triangle M has vertices (0, −2), (0, 0), (−1, −2), with the vertices in corresponding order.Triangle M is the image of triangle L after a rotation of 90° anticlockwise.Find the coordinates of the centre of rotation.[3]
Show answer
(a) (2, −1)
M1 the centre is equidistant from corresponding points, e.g. on the perpendicular bisectors of (1, 1)–(0, −2) and (3, 1)–(0, 0)M1 solving / checking a candidate centreA1 (2, −1)
Q19Transformations · Challenge · Paper 2 style · no calculator · 5 marks
(a)Triangle N has vertices (0, 1), (2, 1), (0, 3).Triangle N is reflected in the line x = 1 and then the image is reflected in the line x = 4.Find the coordinates of the vertices of the final image.[3]
(b)Describe fully the single transformation that maps triangle N onto the final image.[2]
Show answer
(a) (6, 1), (8, 1), (6, 3)
M1 first image (2, 1), (0, 1), (2, 3)A2 final image correct (A1 for two vertices)
(b) Translation by the vector (6, 0)
B1 translationB1 (6, 0)
Q20Vectors (column, magnitude) · Core · Paper 4 style · calculator · 5 marks
(a)a = (3, −2) and b = (−1, 4) are column vectors (top number first).Find a + b.[1]
(b)Find 2a − 3b.[2]
(c)Find |a|, the magnitude of a. Give your answer correct to 3 significant figures.[2]
Show answer
(a) (2, 2)
B1
(b) (9, −16)
M1 (6, −4) − (−3, 12)A1
(c) 3.61
M1 √(3² + (−2)²) = √13A1 3.61
Q21Vectors (column, magnitude) · Core · Paper 4 style · calculator · 4 marks
(a)p = (5, 12). Find |p|.[2]
(b)q = (−4, k) and |q| = 5. Find the two possible values of k.[2]
Show answer
(a) 13
M1 √(25 + 144)A1 13
(b) k = 3 or k = −3
M1 16 + k² = 25A1 ±3
Q22Vectors (column, magnitude) · Standard · Paper 2 style · no calculator · 3 marks
(a)A is the point (2, −1) and B is the point (7, 3).Write AB as a column vector.[1]
(b)AC = (−3, 6). Find the coordinates of C.[1]
(c)Write BC as a column vector.[1]
Show answer
(a) (5, 4)
B1
(b) (−1, 5)
B1
(c) (−8, 2)
B1 follow through
Q23Vectors (column, magnitude) · Standard · Paper 4 style · calculator · 4 marks
(a)u = (6, −8) and v = (x, 3).2u + v = (15, −13). Find x.[2]
(b)Find |u + v|. Give your answer correct to 3 significant figures.[2]
Show answer
(a) x = 3
M1 12 + x = 15A1 3
(b) 10.3
M1 u + v = (9, −5), √(81 + 25)A1 10.3
Q24Vectors (column, magnitude) · Challenge · Paper 2 style · no calculator · 4 marks
(a)m(2, 3) + n(−1, 4) = (8, 1), where (2, 3), (−1, 4) and (8, 1) are column vectors.Find the values of m and n.[4]
Show answer
(a) m = 3, n = −2
M1 2m − n = 8M1 3m + 4n = 1M1 eliminating one variableA1 m = 3, n = −2
Q25Vectors (column, magnitude) · Challenge · Paper 4 style · calculator · 5 marks
(a)Positions are measured in kilometres from a harbour at the origin.A boat starts at the point (3, −2) and moves with constant velocity (12, 5) km/h.Find the position of the boat after 2.5 hours.[2]
(b)Find the speed of the boat.[1]
(c)Find the distance of the boat from the harbour after 2.5 hours. Give your answer correct to 3 significant figures.[2]
Show answer
(a) (33, 10.5)
M1 (3, −2) + 2.5 × (12, 5)A1
(b) 13 km/h
B1 √(144 + 25)
(c) 34.6 km
M1 √(33² + 10.5²)A1 34.6
Q26Vector geometry · Standard · Paper 2 style · no calculator · 4 marks
OABC is a parallelogram. OA = a and OC = c (vectors). M is the midpoint of AB.
(a)Find OB in terms of a and c.[1]
(b)Find OM in terms of a and c.[1]
(c)Find CM in terms of a and c, in its simplest form.[2]
Show answer
(a) a + c
B1
(b) a + ½c
B1
(c) a − ½c
M1 CO + OM = −c + a + ½cA1
Q27Vector geometry · Standard · Paper 2 style · no calculator · 4 marks
OABC is a trapezium. OA = a, OC = c and CB = 2a (vectors).N is the midpoint of CB.
(a)Find AB in terms of a and c.[2]
(b)Find ON in terms of a and c.[1]
(c)What do your answers tell you about the lines AB and ON?[1]
Show answer
(a) a + c
M1 AO + OC + CB = −a + c + 2aA1
(b) a + c
B1 c + a
(c) AB and ON are parallel and equal in length (OABN is a parallelogram)
B1
Q28Vector geometry · Standard · Paper 4 style · calculator · 4 marks
(a)A is the point (3, 1) and B is the point (11, 7).Find |AB|, the length of the vector AB.[2]
(b)P lies on AB such that AP : PB = 3 : 1.Find the position vector of P.[2]
Show answer
(a) 10
M1 AB = (8, 6), √(64 + 36)A1 10
(b) (9, 5.5)
M1 (3, 1) + ¾(8, 6)A1
Q29Vector geometry · Challenge · Paper 2 style · no calculator · 7 marks
OAB is a triangle with OA = a and OB = b (vectors).P is the point on AB such that AP : PB = 1 : 2.Q is the midpoint of OB.R is the point on OA extended such that OR = 2a.
(a)Find OP in terms of a and b, in its simplest form.[2]
(b)Find QP and QR in terms of a and b.[2]
(c)Show that Q, P and R lie on a straight line and find the ratio QP : PR.[3]
Show answer
(a) ⅔a + ⅓b
M1 a + ⅓(b − a)A1
(b) QP = ⅔a − ⅙b, QR = 2a − ½b
B1 B1
(c) QR = 3QP, so they are collinear; QP : PR = 1 : 2
M1 QR = 3(⅔a − ⅙b)A1 QR is a multiple of QP and they share point QB1 1 : 2
Q30Vector geometry · Challenge · Paper 2 style · no calculator · 6 marks
OPQ is a triangle with OP = p and OQ = q (vectors).M is the point on PQ such that PM : MQ = 2 : 3.X is the point such that OX = (5/3)OM.
(a)Show that OM = (1/5)(3p + 2q).[2]
(b)Find PX in terms of q.[2]
(c)Explain what your answer to part (b) shows about PX and OQ, and write down the ratio PX : OQ.[2]
Show answer
(a) OM = p + ⅖(q − p) = ⅗p + ⅖q
M1 p + ⅖(q − p)A1 shown
(b) ⅔q
M1 OX = p + ⅔q, PX = OX − OPA1
(c) PX is parallel to OQ; PX : OQ = 2 : 3
B1 parallelB1 2 : 3
Similar questions
Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.
References only: find each question in its paper (official link where available, otherwise through your school).
