IGCSE Mathematics 0580 · Topic practice
Practise describing and performing transformations precisely, then work with vector notation, magnitude and geometric proofs. A full description names every detail: centre, angle, direction, scale factor or vector.
Appears inPaper 1 19Paper 2 118Paper 3 30Paper 4 83
You’re practising 2016–2018 papers. The 2019–2026 papers, including 2026, need a plan.
Unlock from $6/mo5 The diagram shows a regular pentagon and a kite. Complete the following statements. (a) The regular pentagon has ................................... lines of symmetry. [1] (b) The kite has rotational symmetry of order ................................... . [1]
7 A B C D E F p q The diagram shows a regular hexagon ABCDEF. p CD = and q CB = . Find CA, in terms of p and q, giving your answer in its simplest form. CA = ........................................ [2]
14 q p O Q T P NOT TO SCALE O is the origin, p OP = and q OQ = . QT : TP = 2 : 1 Find the position vector of T. Give your answer in terms of p and q, in its simplest form. ................................................. [2]
14 Describe fully the single transformation represented by the matrix 0 1 1 0 c m. ...................................................................................................................................................................... ...................................................................................................................................................................... [2]
15 5 1 3 2 M = - - e o (a) Find 3M. 3M = f p [1] (b) Find M 1 - . M 1 - = f p [2]
16 y x R 7 6 5 4 3 2 1 0 –1 –2 –3 –1 –2 –3 1 2 3 4 5 6 7 On the grid, draw the image of shape R after the transformation represented by the matrix 0 1 1 0 - c m. [3]
20 A 1 9 1 9 = c m B 9 1 0 8 = c m C 1 1 3 3 = c m I 1 0 0 1 = c m (a) Here are four matrix calculations. AI IA C2 B + I Work out which matrix calculation does not give the answer 1 9 1 9 c m. ................................................. [2] (b) Find B . ................................................. [1] (c) Explain why matrix A has no inverse. .............................................................................................................................................................. [1]
20 A C B 9 y x 8 7 6 5 4 3 2 1 –1 0 –2 –3 –4 –5 –6 –7 –1 1 2 3 5 6 7 9 10 11 –2 –3 –4 –5 –6 –7 –8 4 8 Describe fully the single transformation that maps (a) shape A onto shape B, .............................................................................................................................................................. .............................................................................................................................................................. [3] (b) shape A onto shape C. .............................................................................................................................................................. .............................................................................................................................................................. [3]
20 M 8 7 2 3 = c m N 4 3 1 5 = - - c m (a) Find MN. MN = f p [2] (b) Find M–1. M–1 = f p [2]
22 NOT TO SCALE C D O E –2a + 3b 4a + b a – 2b In the diagram, O is the origin, a b OC 2 3 =- + and a b OD 4 = + . (a) Find CD, in terms of a and b, in its simplest form. CD = ................................................ [2] (b) a b DE 2 = - Find the position vector of E, in terms of a and b, in its simplest form. ................................................. [2]
24 P 3 2 1 3 = c m Q 1 2 4 1 = - c m Find (a) 3P, 3P = f p [1] (b) PQ, PQ = f p [2] (c) Q–1. Q–1 = f p [2]
26 10 9 8 7 6 5 4 3 2 1 –1 –1 –2 –3 –4 1 0 2 3 4 5 6 7 8 x y U T (a) Describe fully the single transformation that maps triangle T onto triangle U. .............................................................................................................................................................. .............................................................................................................................................................. [3] (b) On the grid, draw the image of triangle T after a rotation through 90° clockwise about the point (7, 3). [3] 12
26 NOT TO SCALE C P B O X a c A In the diagram, OABC is a parallelogram. OP and CA intersect at X and CP : PB = 2 : 1. OA OC and a c = = . (a) Find OP, in terms of a and c, in its simplest form. OP = ................................................ [2] (b) CX : XA = 2 : 3 (i) Find OX , in terms of a and c, in its simplest form. OX = ................................................ [2] (ii) Find OX : XP. OX : XP = ................... : ................... [2] 12
1 (a) 0 y x –5 –4 –7 –6 –3 –2 –1 1 2 3 4 5 6 7 –6 –5 –4 –7 –3 –2 –1 1 2 3 4 5 6 7 T P (i) Describe fully the single transformation that maps triangle T onto triangle P. ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) Translate triangle T by the vector 2 5 - - e o. [2] (iii) Rotate triangle T through 90° anticlockwise about (0, 0). [2] (iv) Enlarge triangle T by scale factor 2 1 - with centre (0, 0). [2] 3 (b) O x y A (3, 2) B (5, 6) NOT TO SCALE (i) Find the column vector AB. AB = f p [1] (ii) Find AB . AB = ................................................. [2] (iii) B is the mid-point of the line AC. Find the co-ordinates of C. ( ........................ , ....................... ) [2] (iv) Find the equation of the straight line that passes through A and B. .................................................. [3] (v) The straight line that passes through A and B cuts the y-axis at D. Write down the co-ordinates of D. ( ........................ , ....................... ) [1]
2 –6 –6 –4 –10 –8 –2 0 2 4 6 8 y x –4 –8 –2 2 4 6 8 10 D C B A (a) Describe fully the single transformation that maps (i) flag A onto flag B, ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) flag A onto flag C, ..................................................................................................................................................... ..................................................................................................................................................... [3] (iii) flag A onto flag D. ..................................................................................................................................................... ..................................................................................................................................................... [3] (b) Draw the reflection of flag A in the line y 1 =- . [2]
3 –6 –4 –2 –5 –7 –3 –1 2 1 3 5 7 4 6 8 x y –2 0 –4 –6 –1 –3 –5 2 4 6 1 3 5 B A (a) (i) Draw the image of triangle A after a reflection in the line x = 2. [2] (ii) Draw the image of triangle A after a translation by the vector 4 2 - c m. [2] (iii) Draw the image of triangle A after an enlargement by scale factor 2 1 - , centre (3, 1). [3] (b) Describe fully the single transformation that maps triangle A onto triangle B. .............................................................................................................................................................. .............................................................................................................................................................. [3] (c) Describe fully the single transformation represented by the matrix 0 1 1 0 - - c m. .............................................................................................................................................................. .............................................................................................................................................................. [2]
3 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 y x –5 –4 –7 –3 –2 –1 1 2 3 4 5 6 A B (a) Describe fully the single transformation that maps triangle A onto triangle B. .............................................................................................................................................................. .............................................................................................................................................................. [3] (b) On the grid, draw the image of (i) triangle A after a reflection in the x-axis, [1] (ii) triangle A after a translation by the vector 7 5 - e o, [2] (iii) triangle A after the transformation represented by the matrix . . 0 5 0 0 0 5 e o . [3]
4 B A C D –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 6 5 8 y x 7 4 3 2 1 1 0 2 3 4 5 6 7 8 (a) Describe fully the single transformation that maps (i) triangle A onto triangle B, ..................................................................................................................................................... ..................................................................................................................................................... [2] (ii) triangle A onto triangle C, ..................................................................................................................................................... ..................................................................................................................................................... [3] (iii) triangle A onto triangle D. ..................................................................................................................................................... ..................................................................................................................................................... [3] (b) On the grid, draw the image of triangle A after an enlargement by scale factor 2, centre ,7 3 ^ h. [2]
8 (a) M 2 4 1 3 = c m N 1 2 = ^ h P 4 1 = f p (i) For the following calculations, put a tick (ü) if it is possible or put a cross (û) if it is not possible. There is no need to carry out any of the calculations. Calculation ü or û N + P NP M2 N2 MN NM [4] (ii) Work out P 1 2 + f p . ................................................ [1] (iii) Work out PN. ................................................ [2] (iv) Work out M 1 - . ................................................ [2] (b) Describe fully the single transformation represented by the matrix 0 1 1 0 - f p. ............................................................................................................................................................. ............................................................................................................................................................. [3]
9 (a) Find the magnitude of the vector 1 7 - c m. ................................................. [2] (b) The determinant of the matrix m m 6 5 2 c m is 24. Find the value of m. m = ................................................ [2] (c) L = 2 3 5 9 c m M = 4 2 - c m N = 1 7 ^ h Work out the following. (i) NM ................................................ [2] (ii) LM ................................................ [2] (iii) L2 ................................................ [2] (iv) L 1 - ................................................ [2]