S78230A ©2023 Pearson Education Ltd. mel justmaths.co.uk Worked Solutions there may be 0 the valid methods to those shown here S78230A ©2023 Pearson Education Ltd. Answer ALL questions. Write your answers in the spaces provided. You must write down all the stages in your working. 1 ( a ) Write 508 200 000 in standard form. ...................................................... ( 1 ) ( b ) Write 8.06 × 10 – 4 as an ordinary number. ...................................................... ( 1 ) ( Total for Question 1 is 2 marks ) ___________________________________________________________________________ 2 ( a ) Find the highest common factor ( HCF ) of 72 and 126 ......... ............................................. ( 2 ) A = 2 × 5 3 × 7 B = 2 2 × 5 2 × 7 2 ( b ) Find the lowest common multiple ( LCM ) of A and B ......... ............................................. ( 1 ) ( Total for Question 2 is 3 marks ) ___________________________________________________________________________ 5 082 108 O 00080 72 2x 2 2 3 3 126 2 3 3 7 MCF 2 3 3 18 18 Hof 21 52 7 350 Lcm 350 2 5 7 24500 24500 S78230A ©2023 Pearson Education Ltd. 3 The table shows information about the weights of 100 dogs. Weight ( w kg ) Frequency 0 ≤ w < 10 17 10 ≤ w < 20 25 20 ≤ w < 30 38 30 ≤ w < 40 11 40 ≤ w < 50 9 Draw a frequency polygon for this information. ( Total for Question 3 is 2 marks ) ___________________________________________________________________________ 5 IS 25 35 45 x X x X x S78230A ©2023 Pearson Education Ltd. 4 A prism has a volume of 78 cm 3 The length of the prism is 12.5 cm. Work out the area of the cross section of the prism. ...................................................... cm 2 ( Total for Question 4 is 2 marks ) ___________________________________________________________________________ 5 Here is a Venn diagram. ( a ) Write down the numbers that are in set ( i ) A ∩ B ..................................... ............................................. ( 1 ) ( ii ) B ′ ..................................... ............................................. ( 1 ) A number is chosen at random from the universal set E ( b ) Find the probability that the number is in the set A ∪ B ......... ............................................. ( 2 ) ( Total for Question 5 is 4 marks ) ___________________________________________________________________________ hey AA 12.5 12.8 A 6.24 6 24 2410 1,315,719,1413 14 a 14 S78230A ©2023 Pearson Education Ltd. 6 The diagram shows a plan of Rico’s garden. ABCD is a trapezium. BC is the diameter of a semicircle. Rico is going to cover his garden with stones. Each bag of stones covers 11 m 2 of garden. Each bag of stones costs £229.25 Work out how much it will cost Rico to buy all the bags of stones he needs. £......... .............................. ............... ( Total for Question 6 is 5 marks ) ___________________________________________________________________________ diameter 6 radius 3M Area xarde trapezium x it 32 e Iz b 18 7 x 4 14.137 t 294 43.53716 m2 needs 43.53716 ill 3 95 so 4 bags co SE 4 229 25 917 917 S78230A ©2023 Pearson Education Ltd. 7 ( a ) Expand and simplify ( x – 5 )( x – 1 ) ..................................... ............................................. ( 2 ) ( b ) Solve 3 y + 6 > 41 – 2 y ......... ............................................. ( 3 ) ( Total for Question 7 is 5 marks ) ___________________________________________________________________________ I X Soc 15 2 62C 15 3y 6 7 41 2y 2g 2g Sy 6 7 41 G G y 7 5g 7 35 3 I S78230A ©2023 Pearson Education Ltd. 8 Tabitha has some gold with a total mass of 9.65 g She is going to melt the gold and use all the gold to make a cube. The density of gold is 19.3 g/cm s Tabitha thinks the side length of the cube will be greater than 8 mm. Is Tabitha correct? You must show how you get your answer. ( Total for Question 8 is 3 marks ) ___________________________________________________________________________ Golde mass 9.65g density 19.3g Ian's D M v Volume im 9 6 5 D 19.3 0.5cm's Cobeofside length 8mm Volume 0.8 0 8 0.8 0 512cm's Tabitha is wrong 0.540.512 S78230A ©2023 Pearson Education Ltd. 9 The length of a running track is 125 metres, correct to the nearest 5 metres. Complete the error interval for the length of the running track. ................................................ ...... m ≤ length < ................................................ ...... m ( Total for Question 9 is 2 marks ) ___________________________________________________________________________ 1 0 Use your calculator to work out √ 12 6 2 - tan 75 o 3 12 1 1 2 Give your answer correct to 3 significant figures. .................................................................................. ( Total for Question 10 is 2 marks ) ___________________________________________________________________________ 1 1 The value of an object depreciates at a constant rate of 7.3% each year. At the end of 2022, the value of the object is £3150 At the end of which year will the object first have a value of less than £2000? ...................................................... ( Total for Question 11 is 2 marks ) ___________________________________________________________________________ 120 125 I 30 p p 122.5 127 S I 544343776 3s f 1.54 100 7 3 92.7 SO O 927 3150 0.9275 2156 30 2022 6 2156 0.927 1998.89 SO 6 years 2028 S78230A ©2023 Pearson Education Ltd. 1 2 Beth is asked to enlarge shape A by a scale factor of 2 1 2 with centre of enlargement ( 5, 6 ) She draws triangle B Write down a mistake Beth has made. ............................................................................................................................. ......................... ............................................................................................................................. ......................... ............................................................................................................................. ......................... ( Total for Question 12 is 1 mark ) ___________________________________________________________________________ X 2x 2.5 3 25 5 I 7 S t The base length is too big it should be 5 units long S78230A ©2023 Pearson Education Ltd. 1 3 The diagram shows triangle PQR S is the midpoint of TR T is the midpoint of PS Calculate the size of angle QPT Give your answer correct to 1 decimal place. ...................................................... ° ( Total for Question 13 is 4 marks ) ___________________________________________________________________________ 4.93 O M oof e l G 30 6.30 A C 2 6.30 12.6 DSIR son 38 0 5 8 QS 85m38 4.925291803 cos 38 51 8 SR 8 cos 38 6.304086029 TS SR PE 0 PAR tank 4.925 12.608 X tan 0.390 21.33774 21.3 Idp S78230A ©2023 Pearson Education Ltd. 1 4 Make t the subject of the formula u = 4 + √ 5 𝑡 ..................................... .............................. ............... ( Total for Question 14 is 3 marks ) ___________________________________________________________________________ 1 5 Wyatt wants to work out an estimate for the total number of wild ponies in a forest. In July, Wyatt catches 42 ponies in the forest. He puts a tag on each of these ponies and releases them. In August, Wyatt catches 60 ponies in the forest. He finds that 5 of the 60 ponies are tagged. Work out an estimate for the total number of ponies in the forest. ......... .............................. ............... ( Total for Question 15 is 3 marks ) ___________________________________________________________________________ U K i TE is i ee 4 St r Cu 4 2 E Cu 4 2 t.ca 5 July August 42 5 60 4.7 i 5 GO 504 42 60 5 S78230A ©2023 Pearson Education Ltd. 1 6 The ratio of the price of a summer ferry ticket to the price of a winter ferry ticket is 5 : 3 The price of each ferry ticket increases by £3.50 The ratio of the price of a summer ticket to the price of a winter ticket is now 11 : 7 Work out the price of a summer ticket and the price of a winter ticket before the increase. summer £ ................................................ ...... winter £ ......... .............................. ............... ( Total for Question 16 is 4 marks ) ___________________________________________________________________________ S w 5 3 5 3 so 3 3.50 50C c 3.50 I 32C 3 SO 7 7 52 3 50 1113 3.50 352C 24.5 332C t 38.5 22C 38 5 24.5 2x 14 X 7 so 5 7 35 c 3.50 38.50 3 7 21 t 3.50 24.50 Check 35 21 S 3 3850 24.50 35 II i 7 21 S78230A ©2023 Pearson Education Ltd. 1 7 ( a ) Use the iteration formula x n + 1 = 3 3 5 n x + to find the values of x 1 , x 2 and x 3 Start with x 0 = 2 x 1 = ................................................ ...... x 2 = ................................................ ...... x 3 = ......... ............................................. ( 3 ) The values of x 1 , x 2 and x 3 found in part ( a ) are estimates of the solution of an equation of the form x 3 = ax + b where a and b are integers. ( b ) Find the value of a and the value of b a = ................................................ ...... b = ......... ............................................. ( 1 ) ( Total for Question 17 is 4 marks ) ___________________________________________________________________________ Xo 2 2C i z 3 2 5 2 2239 202 z 3 2.22 15 2 268 2cg z 3 2 268 es 2.276 2.22 2 27 2 28 203 3 15 3 5 S78230A ©2023 Pearson Education Ltd. 1 8 A and B are two boxes each in the shape of a cuboid. Box A and box B are similar. surface area of box A : surface area of box B = 324 : 25 Box A is full of sand. Box B is empty. Ryan completely fills box B with sand from box A He then empties box B Ryan repeats these two steps as many times as possible. Work out the number of times Ryan completely fills box B with sand. You must show all your working. ......... .............................. ............... ( Total for Question 18 is 4 marks ) ___________________________________________________________________________ full of sand area SF 12.96 length SF TE 3 6 Vol SF a 3 63 46 656 So 46 46 S78230A ©2023 Pearson Education Ltd. 1 9 ( a ) Solve 3 x 2 = 5 x + 7 Give your solutions correct to 3 significant figures. ..................................... ............................................. ( 3 ) ( b ) ( i ) Write x 2 + 14 x – 3 in the form ( x + a ) 2 + b where a and b are integers. ..................................... ............................................. ( 2 ) ( ii ) Hence write down the coordinates of the turning point of the graph of y = x 2 + 14 x – 3 ( ........................ .... , ........................ .... ) ( 1 ) ( Total for Question 19 is 6 marks ) ___________________________________________________________________________ 3 2 Soc 7 0 a 3b Scs 7 x C S 1 1 4 35 7 2 3 SIFFERT a Sto SO 6 2.57338 and O 9067 2 57 and O 907 x 1712 49 3 ace 7 2 52 7 2 52 7 52 S78230A ©2023 Pearson Education Ltd. 2 0 OAB is a triangle OA → = 3 a OB → = 5 b P is the point on AB such that AP : PB = 9 : 10 Show that OP → is parallel to the vector 2 a + 3 b ( Total for Question 20 is 4 marks ) ___________________________________________________________________________ 9 9 I 19 AT Sb Za AI i ga Sb 3A Ep 3 a 4,1g b Itza Eats a 2 a t b 3 a t Lida b is 12 a t 3 b so is parallel to La c 3b S78230A ©2023 Pearson Education Ltd. 2 1 The histogram shows information about the weights, in grams, of some eggs Milo collected. None of the eggs weighed less than 30 grams or more than 85 grams. Small - sized eggs weigh 53 grams or less. Work out an estimate for the ratio number of small - sized eggs : total number of eggs ..................................... .............................. ............... ( Total for Question 21 is 3 marks ) ___________________________________________________________________________ 2 10 20 I 2 3 6 I l 4 5 i f 7 I I I i 0 75 20 I 0.3 20 IS i G a 53 1St 6 1St 20 17 c 6 21 i 48 21 i 48 Oe S78230A ©2023 Pearson Education Ltd. 2 2 The diagram shows a sketch of part of the curve with equation y = tan x ° ( a ) Write down the coordinates of the point P ( ........................ .... , ........................ .... ) ( 2 ) Here is the graph of y = sin x ° for 0 x 360 ( b ) Find estimates for all four solutions of sin x ° = – 0.8 for – 360 ≤ x ≤ 360 Give your solutions correct to the nearest degree. ............................................................................................................................. ............... .......... ( 2 ) ( Total for Question 22 is 4 marks ) ___________________________________________________________________________ 90 45 I sur 0 8 53.139 180 153 180 153 360 53 180 0 127 53 233 307 S78230A ©2023 Pearson Education Ltd. 2 3 On a mountain, if it snows one day, the probability that it will snow the next day is 0.9 if it does not snow one day, the probability that it will snow the next day is 0.6 Given that it snows on the mountain on Wednesday, work out the probability that it will not snow on the mountain on Saturday. You must show all your working. ..................................... .............................. ............... ( Total for Question 23 is 4 marks ) ___________________________________________________________________________ 0.9 S 0.9 S O l NS 0.9 0.9 0 I 0.087 s 0 6 S 0.9 O INS 0 4 Ns 0.9 0.1 0 4 0.036 S 0.9 S O I O 6 S Ol NS 0.1 0.6 0 I 0.006 NS O Ng OG s 0.4 NS O l x 0 4 0.4 0 016 given that it snows so 0.081 0 036 on weds 0.006 0 016 NS l W T F Sae 0 139 S78230A ©2023 Pearson Education Ltd. 2 4 Show that ( ) 2 3 3 5 2 x x + − can be written as 6 19 15 18 50 x x x + + − ( Total for Question 24 is 4 marks ) ___________________________________________________________________________ TOTAL FOR PAPER IS 8 0 MARKS 2 3 x 1 3rd 5 2 3 2 bio 10 2rad 3 655C t 10 x GTE IO x GTE t 10 I 2x 2052C 18 Fx t 30 362C t 6055C 6052 100 12 385 30 362C 100 6 19 152 as required 182C 50