Pythagoras’ theorem — Paper 2 · IGCSE Mathematics 0580

11 NOT TO SCALE O Q C 6 cm 4 cm P T Two circles, centres O and C, of radius 6 cm and 4 cm respectively, touch at Q. PT is a tangent to both circles. (a) Write down the distance OC. Answer(a) OC = cm [1] (b) Calculate the distance PT. Answer(b) PT = cm [3]

4 marks

Mark scheme: 11 (a) 10(.0..) 1 (b) 9.80 3 M2 √((a)2 – 22) or M1 PT2 + 22 = (a)2 h

This question in 0580/22 May/June 2010

19 For P Examiner's Use 8 cm NOT TO SCALE C B 10 cm M D 10 cm A The diagram represents a pyramid with a square base of side 10 cm. The diagonals AC and BD meet at M. P is vertically above M and PB = 8cm. (a) Calculate the length of BD. Answer(a) BD = cm [2] (b) Calculate MP, the height of the pyramid. Answer(b) MP = cm [3]

5 marks

Mark scheme: 19 (a) 14.1 2 M1 (BD2) = 102 + 102 or sin45 = 10/CD (b) 3.74 or 3.78 3 M1 (a)/2 M1 (their (a)/2)2 + PM2 = 82

This question in 0580/22 Oct/Nov 2010

16 For y Examiner's Use C (9,7) NOT TO SCALE A (1,3) x O B (3,0) The co-ordinates of A, B and C are shown on the diagram, which is not to scale. (a) Find the length of the line AB. Answer(a) AB = [3] (b) Find the equation of the line AC. Answer(b) [3]

6 marks

Mark scheme: 16 (a) 3.61 3 M1 (3 – 1)2 + (0 – 3)2 oe M1 2 2 + 3 2 1 1 1 1 (b) y = x + 2 oe 3 B2 y = x + k or y = kx + 2 2 2 2 2 1 1 or B1 kx + 2 or x + k 2 2 1 If 0 scored B1 m = 2 1 B1 c = 2 clearly identified in working 2 IGCSE – October/November 2011 0580 21 1

This question in 0580/21 Oct/Nov 2011

21 For P Examiner's Use NOT TO SCALE 5 cm D C 8 cm M A 8 cm B The diagram shows a pyramid on a square base ABCD. The diagonals of the base, AC and BD, intersect at M. The sides of the square are 8 cm and the vertical height of the pyramid, PM, is 5 cm. Calculate (a) the length of the edge PB, Answer(a) PB = cm [3] (b) the angle between PB and the base ABCD. Answer(b) [3]

6 marks

Mark scheme: 1 21 (a) 7.55 www 3 M2 ( √(82 + 82))2 + 52 or 42 + 52 + 42 seen 2 or M1 82 + 82 or 52 + 42 or 42 + 42 or 52 + (their MB)2 seen 5 5 (b) 41.5 www 3 M2 sin(B) = or tan(B) = or a( ) their MB their MB cos(B) = (a) or M1 recognition of angle PBM

This question in 0580/23 May/June 2012

24 For Q P Examiner's Use 6 cm NOT TO SCALE C B 5 cm D 10 cm A The diagram shows a triangular prism. ABCD is a horizontal rectangle with DA = 10 cm and AB = 5 cm. BCQP is a vertical rectangle and BP = 6 cm. Calculate (a) the length of DP, Answer(a) DP = cm [3] (b) the angle between DP and the horizontal rectangle ABCD. Answer(b) [3]

6 marks

Mark scheme: 24 (a) 12.7 3 M2 for 102 + 52 + 62 or M1 for one of 102 + 52 or 62 + 52 or 102 + 62 (b) 28.2 3 M2 for sin x = 6/(a) or M1 for identifying angle PDB 70

This question in 0580/23 Oct/Nov 2012

21 P 6 cm NOT TO SCALE D C 4 cm M 4 cm A B The diagram shows a pyramid on a square base ABCD with diagonals, AC and BD, of length 8 cm. AC and BD meet at M and the vertex, P, of the pyramid is vertically above M. The sloping edges of the pyramid are of length 6 cm. Calculate (a) the perpendicular height, PM, of the pyramid, Answer(a) PM = … cm [3] (b) the angle between a sloping edge and the base of the pyramid. Answer(b) … [3] __________________________________________________________________________________________ Question 22 is printed on the next page.

6 marks

Mark scheme: 21 (a) 4.47 or 4.472[…] 3 M2 for 6 2 − 4 2 or M1 for [PM ]2 + 4 2 = 6 2 or 6 2 − 4 2 4 (b) 48.2 or 48.18 to 48.19 3 M2 for cos[correct angle] = oe 6 or M1 for recognising a correct angle Page 5 MMarrk SSccheemee Sylllabbuss Papeer IGC SEE – M ay//Juunee 2014 055800 222

This question in 0580/22 May/June 2014

11 NOT TO 8 cm SCALE 5 cm x cm Calculate the value of x. Answer x = … [3]

3 marks

Mark scheme: 11 6.24 or 6.244 to 6.245 3 M2 for 8 2 − 5 2 or M1 for 8 2 = 5 2 + x 2 or better 3 15 27 9× 3

This question in 0580/22 Oct/Nov 2015

23 7 cm E F 5 cm B NOT TO A SCALE H G 3 cm D C The diagram shows a cuboid. HD = 3 cm, EH = 5 cm and EF = 7 cm. Calculate (a) the length CE, CE = … cm [4] (b) the angle between CE and the base CDHG. … [3]

7 marks

Mark scheme: 23 (a) 9.11 or 9.110… 4 M3 for 5 2 + 32 + 7 2 or M2 for 5 2 + 32 or 32 + 7 2 or 5 2 + 7 2 or M1 for 5² + 3² or 3² + 7² or 5² + 7² 5 (b) 33.3 or 33.28 to 33.29 3 M2 for sin = oe their ( a ) or B1 for identifying angle ECH

This question in 0580/23 May/June 2016

24 S R NOT TO P Q 8 cm SCALE D C 8 cm A 8 cm B The diagram shows a cube of side length 8 cm. (a) Calculate the length of the diagonal BS. BS = … cm [3] (b) Calculate angle SBD. Angle SBD = … [2]

5 marks

Mark scheme: 24 (a) 13.9 or 13.85 to 13.86 3 M2 for 8 2 + 8 2 + 8 2 oe or M1 for 82 + 82 or better for one face 8 8 2 + 8 2 (b) 35.1 to 35.5[4…] 2 M1 for sin = or cos = their(a) their(a) 8 or tan = oe 8 2 + 8 2

This question in 0580/21 Oct/Nov 2016

9 The diagram shows a pyramid with a square base ABCD. All the sloping edges of the pyramid are 20 cm long and AC = 17 cm. V NOT TO SCALE 20 cm D C 17 cm A B Calculate the height of the pyramid. … cm [3]

3 marks

Mark scheme: 2  1  oe9 18.1 or 18.10…. 3 M2 for 20 − (17 )   2  2  1  2 2 20 or M1 for h +  ( 17 )  =  2 

This question in 0580/22 Feb/March 2017

7 A NOT TO 2.5 cm SCALE C B 4.1 cm Calculate the length of AC. AC = … cm [2]

2 marks

Mark scheme: 7 4.8[0] or 4.802… 2 M1 for [ AC 2 = ] 2.5 2 + 4.12

This question in 0580/21 May/June 2018

16 A M NOT TO SCALE O B The diagram shows a circle, centre O. AB is a chord of length 12 cm. M is the mid-point of AB and OM = 4.5 cm. Calculate the radius of the circle. … cm [3]

3 marks

Mark scheme: 16 7.5 nfww 3 2 2  12 2 4.5 oe M2 for [OB =]  +  2  or B1 for recognition of right angle

This question in 0580/22 May/June 2018

2 B M NOT TO SCALE A O D N E The diagram shows a circle, centre O. AB and DE are chords of the circle. M is the mid-point of AB and N is the mid-point of DE. AB = DE = 9 cm and OM = 5 cm. Find ON. ON = … cm [1]

1 marks

Mark scheme: 2 5 1

This question in 0580/22 Oct/Nov 2018

13 The length of one side of a rectangle is 12 cm. The length of the diagonal of the rectangle is 13 cm. Calculate the area of the rectangle. … cm2 [3]

3 marks

Mark scheme: 13 60 3 2 2 M2 for 12 × 13 − 12 or M1 for 132 – 122 or for 12 × their 5 from Pythagoras or trig

This question in 0580/22 Oct/Nov 2020

9 A is the point (5, - 5 ) and B is the point (9, 3). (a) Find the coordinates of the midpoint of AB. ( … , … ) [2] (b) Find the length of AB. … [3]

5 marks

Mark scheme: 9(a) (7, − 1) 2 B1 for each 9(b) 8.94 or 8.944… 3 2 2 M2 for ( 9 − 5 ) + ( 3 −−5 ) oe 2 2 or M1 for ( 9 − 5 ) + ( 3 −−5 ) oe

This question in 0580/21 May/June 2021

16 A is the point (5, 7) and B is the point (9, - 1). (a) Find the length AB. … [3] (b) Find the equation of the line AB. … [3]

6 marks

Mark scheme: 16(a) 8.94 or 8.944… 3 2 2 M2 for ( 9 − 5 ) + ( −−1 7 ) oe 2 2 or M1 for ( 9 − 5 ) + ( −−1 7 ) oe 16(b) y = –2x + 17 oe final answer 3 B2 for answer –2x + 17 OR −−1 7 M1 for oe 9 − 5 M1 for correct substitution of (5, 7) or (9, –1) into y = their mx + c oe

This question in 0580/22 May/June 2021

23 D C 4 cm Q P 5 cm NOT TO SCALE A B 12 cm The diagram shows a triangular prism. Angle BPC = 90°. (a) Calculate AC. AC = … cm [3] (b) Calculate the angle between AC and the base ABPQ. … [3]

6 marks

Mark scheme: 23(a) 13.6 or 13.60… 3 M2 for 12 2 + 5 2 + 4 2 or M1 for 5 2 + 4 2 or 12 2 + 4 2 or 12 2 + 5 2 23(b) 17.1 or 17.08 to 17.10… 3 4 M2 for sin = oe or their (a) 4 their AP tan = or cos = their AP their (a) or M1 for recognising angle CAP.

This question in 0580/23 Oct/Nov 2021

24 A cuboid measures 24 cm by 12 cm by 8 cm. Calculate the length of a diagonal of the cuboid. … cm [3]

3 marks

Mark scheme: 24 28 3 M2 for 242 + 122 + 82 or M1 for 242 + 122 or 242 + 82 or 122 + 82

This question in 0580/21 May/June 2022

10 NOT TO 14 cm SCALE h cm 10 cm The diagram shows a right-angled triangle. (a) Calculate the value of h. h = … [3] (b) Find the perimeter of this triangle. … cm [1]

4 marks

Mark scheme: 10(a) 9.8[0] or 9.797 to 9.798 3 M2 for 142 –102 oe or better or M1 for 102 + h2 = 142 oe or better 10(b) 33.8 or 33.79 to 33.80 1 FT 24 + their (a)

This question in 0580/23 Oct/Nov 2022

21 H M G NOT TO E F SCALE 8 cm D C 5 cm A B 14 cm The diagram shows a cuboid ABCDEFGH. AB = 14 cm, BC = 5 cm and CG = 8 cm. M is the midpoint of HG. (a) Calculate BM. … cm [3] (b) Calculate the angle that BM makes with the base ABCD. … [3] Question 22 is printed on the next page.

6 marks

Mark scheme: 21(a) 11.7 or 11.74 to 11.75 3 2  14  M2 for   + 52 + 82 oe  2   14  2  14  2 or M1 for   + 52, 52 + 82 or   + 82  2   2  21(b) 42.9 to 43.14 3 8 M2 for sin [….] = oe their (a) or M1 for recognising angle MBX where X is the midpoint of DC

This question in 0580/23 Oct/Nov 2023

18 NOT TO E SCALE D 18 cm 6 cm 60° A B C 17 cm The quadrilateral ACDE is formed by two right-angled triangles ABE and BCD. AC = 17 cm, AE = 18 cm and BD = 6 cm. (a) Show that CD = 10 cm. [5] (b) Find the perimeter of the quadrilateral ACDE. Give your answer in the form p + k q . … cm [4]

9 marks

Mark scheme: 18(a) AB M1 = cos60 or [AB = ] 18cos 60 18 1 A1 cos 60 = and [AB =] 9 2 Correct use of Pythagoras’ theorem M1 i.e. [CD2 =] 6 2 + (17 −theirAB ) 2 oe Correct evaluation for their AB M1 [CD2 =] 36 + 64 or CD = 36 + 64 100 = 10 A1 Dep on M1A1M1M1 18(b) 39 + 9 3 4 B3 for [BE =] 9 3 or for answer k + 9 3 or for answer equivalent to 39 + 9 3 but not in required form OR BE 3 M2 for = oe or better 18 2 BE or M1 for = sin60 oe or better 18 M1 for 18 + 17 + 10 + their BE – 6 oe OR M2 for 182 −their 92 oe or M1 for BE2 + (their 9)2 = 182 oe M1 for 18 + 17 + 10 + their BE – 6 oe

This question in 0580/22 Feb/March 2025

17 B is the point (-3, 1) and D is the point (-5, 9). BD is a diagonal of the kite ABCD. (a) The ratio of the lengths of the diagonals BD : AC = 2 : 3. Work out the length of AC. Give your answer as a surd in its simplest form. … [5] (b) Find the coordinates of the midpoint of BD. ( … , … ) [2] (c) The diagonal AC of the kite passes through the midpoint of BD. Find an equation of AC. Give your answer in the form y = mx + c . y = … [4]

11 marks

Mark scheme: 17(a) 3 17 cao 5 B4 for answer equivalent to 3 17 but 3 68 not in the correct form e.g. , 2 6 17 , 153 2 OR B3 for 68 or 2 17 or M2 for (9 – 1)2 + (–5 – – 3)2 oe or M1 for (9 – 1) or (–5 – – 3) oe and 3 M1 for their 68  oe 2 17(b) (– 4, 5) 2 B1 for each coordinate 17(c) 1 4 − 1 y = x + 6 final answer M1 for  grad BD = 9 oe 4 −−−5 3 −1 M1 for  grad AC =  their grad BD M1 for their (– 4, 5) substituted into y = their mx + c oe

This question in 0580/22 Oct/Nov 2025

19 7 cm NOT TO SCALE 5 cm 5 cm The diagram shows a box in the shape of a cuboid. Mala has a straight rod of length 10 cm. Show that this rod does not fit completely inside the box. [3]

3 marks

Mark scheme: 19 99 and states this is less than 10 oe 3 B2 for 99 or M2 for 52 + 52 + 72 oe or M1 for 52 + 52 or 52 + 72 oe

This question in 0580/22 Oct/Nov 2025

10 C 11 cm NOT TO SCALE x° B A 8 cm The diagram shows a right-angled triangle ABC. (a) Work out the exact length of AC. … cm [3] (b) cos x = k Write down the value of k. k = … [1]

4 marks

Mark scheme: 10(a) 57 3 M2 for 112 – 82 or M1 for 112 = AC2 + 82 10(b) 8 1 11

This question in 0580/23 Oct/Nov 2025