Question: This document has 20 pages. [Turn over Cambridge O Level DC (CE/FC) 346588/2 © UCLES 2025 MATHEMATICS (SYLLABUS D) 4024/11 Paper 1 Non-calculator October/November 2025 2 hours You must answer on the question paper. You will need: Geometrical instruments INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● Calculators must not be used in this paper. ● You may use tracing paper. ● You must show all necessary working clearly. INFORMATION ● The total mark for this paper is 100. ● The number of marks for each question or part question is shown in brackets [ ]. *5491764877* ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0002, * 0000800000001 * ¬W. 4mHuOªE_{6W ¬\u007fJsOx\{HK¨5¬o ¥uEuueUE¥5e5EU DFD2 © UCLES 2025 4024/11/O/N/25 List of formulas Area, A, of triangle, base b, height h. A bh 2 1 = Area, A, of circle of radius r. A rr 2 = Circumference, C, of circle of radius r. C r = 2r Curved surface area, A, of cylinder of radius r, height h. A = 2rrh Curved surface area, A, of cone of radius r, sloping edge l. A r = r l Surface area, A, of sphere of radius r. A r 4r 2 = Volume, V, of prism, cross-sectional area A, length l. V A = l Volume, V, of pyramid, base area A, height h. V Ah 3 1 = Volume, V, of cylinder of radius r, height h. V rr h 2 = Volume, V, of cone of radius r, height h. V rr h 3 1 2 = Volume, V, of sphere of radius r. V rr 3 4 3 = For the equation ax2 + bx + c = 0, where a ≠ 0, x a b b ac 2 4 2 ! = - - For the triangle shown, sin s A in sin a B b C c = = a b c b2 c A cos 2 2 2 = + - ab sinC 2 1 Area = A B C c b a * 0000800000002 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0003, ĬÕĊ®Ġ´íÈõÏĪÅĊÞù¸þ× ĬÿÌôÎĩôĤãĆõëóĎąÔ÷Ă ĥåÕĕµĕĥµõÅµÅąõÅõĥÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD3 © UCLES 2025 4024/11/O/N/25 [Turn over Calculators must not be used in this paper. 1 Write down the reciprocal of 2 7 . ................................................. [1] 2 Some children record the number of hours they each spend watching television one day. These are the results. 1 2 2 2 6 7 7 8 10 (a) Write down the mode. ................................................. [1] (b) One of the 9 children is chosen at random. Find the probability that this child spends more than 6 hours watching television on this day. ................................................. [1] 3 A shirt costs $24. In a sale the cost is reduced by 15%. Work out the cost of the shirt in the sale. $ ................................................ [2] * 0000800000003 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0004, Ĭ×Ċ®Ġ´íÈõÏĪÅĊÞû¸þ× ĬÿËóÖğðĔÖôČĞçĦÑÔćĂ ĥååÕõõąÕĥµĥÅąĕåµõÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD4 © UCLES 2025 4024/11/O/N/25 4 The diagram shows five straight lines. q° p° 112° NOT TO SCALE The diagram shows three horizontal parallel lines and two sloping parallel lines. (a) Find the value of p. Give a geometrical reason to explain your answer. p = .................... because ........................................................................................................... ..................................................................................................................................................... [2] (b) Find the value of q. Give a geometrical reason to explain your answer. q = .................... because ........................................................................................................... ..................................................................................................................................................... [2] * 0000800000004 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0005, ĬÕĊ®Ġ´íÈõÏĪÅĊÞàù¸Ā× ĬÿËòÖĥþĕáîăĥąÊóÄïĂ ĥĕõÕµõąõąĕĕÅÅĕąµĥÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD5 © UCLES 2025 4024/11/O/N/25 [Turn over 5 The table shows how some people travel to work. Type of travel Number of people Pie chart angle Cycle 20 120° Walk 12 72° Car Bus 5 (a) Complete the table. [2] (b) Complete the pie chart to show the information in the table. Cycle [2] * 0000800000005 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0006, Ĭ×Ċ®Ġ´íÈõÏĪÅĊàû¸Ā× ĬÿÌñÎģĂĥØČîäÑ²çÄÿĂ ĥĕąĕõĕĥĕĕĥÅÅÅõĥõõÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD6 © UCLES 2025 4024/11/O/N/25 6 Sam buys some chocolate. He eats 7 2 of the chocolate. The mass of the remaining chocolate is 350g. Work out the mass of the chocolate Sam buys. ............................................... g [2] 7 Car hire $40 per day plus $0.30 per kilometre Amy hires a car for 10 days. She pays a total of $670. Work out the number of kilometres Amy travels in the car. ........................................... km [3] * 0000800000006 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0007, ĬÙĊ®Ġ´íÈõÏĪÅĊÝûµþ× ĬÿËñÑĩĪğÐÿÿÙ±ÊÆĬïĂ ĥÅåĕõµĥµÕąĥÅąõÅµåÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD7 © UCLES 2025 4024/11/O/N/25 [Turn over 8 The diagram shows triangle A and triangle B. A B y x 642 531 0 – 2 – 3 – 1 – 4 – 5 – 6 – 5 – 4 – 3 – 2 – 1 1 2 3 4 5 6 7 (a) Describe fully the single transformation that maps triangle A onto triangle B. ..................................................................................................................................................... ..................................................................................................................................................... [2] (b) Draw the image of triangle A after a rotation of 180° about centre (0, 0). [2] * 0000800000007 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0008, ĬÛĊ®Ġ´íÈõÏĪÅĊÝùµþ× ĬÿÌòÙğĦďéùòĐĥ²ĒĬÿĂ ĥÅÕÕµÕąÕÅõµÅąĕåõµÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD8 © UCLES 2025 4024/11/O/N/25 [Turn over 9 By writing each number correct to 1 significant figure, work out an estimate for the value of 102. . 5 27 #8 7 . ................................................. [2] 10 A circle has diameter 12cm. Find the area of the circle. Give your answer in terms of r. ......................................... cm2 [2] 11 A bag contains 10 plums. The mean mass of 6 of the plums is 50g. The mean mass of the remaining 4 plums is 40g. Calculate the mean mass of all 10 plums in the bag. .............................................. g [3] * 0000800000008 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u0009, ĬÙĊ®Ġ´íÈõÏĪÅĊßûµĀ× ĬÿÌóÙĥĘĚÎ÷ùėÇĎ´ü÷Ă ĥõąÕõÕąõåÕÅÅÅĕąõåÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD9 © UCLES 2025 4024/11/O/N/25 [Turn over 12 10 8 9 NOT TO SCALE The diagram shows an empty container. The container is a cuboid of width 10cm, length 8cm and height 9cm. Water drips into the container at a rate of 6 millilitres per second. Find the time, in seconds, it takes to fill the container. .................................... seconds [3] 13 Work out 2 3 2 3 4 3 + . Give your answer as a mixed number in its simplest form. ................................................. [3] * 0000800000009 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u000a, ĬÛĊ®Ġ´íÈõÏĪÅĊßùµĀ× ĬÿËôÑģĜĪëāĈÒēĦĨüćĂ ĥõõĕµµĥĕµåĕÅÅõĥµµÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD10 © UCLES 2025 4024/11/O/N/25 [Turn over 14 NOT TO SCALE 0 10 105 Time (seconds) Speed (m/s) 0 v Lia cycles 800m in 105 seconds. The diagram shows the speed–time graph for this journey. (a) Work out the value of v. v = ................................................ [2] (b) Work out Lia’s acceleration during the first 10 seconds. ........................................ m/s 2 [1] * 0000800000010 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u000b, ĬÙĊ®Ġ´íÈõÏĪÅĊÞû·þ× ĬÿÉóÔħĐĀÔðòëĐØÄÿĂ ĥåĥĕõÕåõąĥÅąÅµąõĕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD11 © UCLES 2025 4024/11/O/N/25 [Turn over 15 The cumulative frequency diagram shows information about journey times. 30 40 50 60 20 10 0 0 10 20 Time (minutes) Cumulative frequency 30 40 50 Use the cumulative frequency diagram to find an estimate for (a) the median .................................... minutes [1] (b) the number of journeys that took 20 minutes or more. ................................................. [2] * 0000800000011 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u000c, ĬÛĊ®Ġ´íÈõÏĪÅĊÞù·þ× ĬÿÊôÜġĔðåĊÿüïĨĄÄïĂ ĥåĕÕµµÅĕĕĕĕąÅÕĥµąÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD12 © UCLES 2025 4024/11/O/N/25 16 S R Q V T O U 32° NOT TO SCALE Points Q, R, S and T lie on a circle, centre O. UV is a tangent to the circle at T. SOQV is a straight line. Angle QST = 32°. (a) Find angle QRT. Give a geometrical reason to explain your answer. Angle QRT = .................... because ........................................................................................... ..................................................................................................................................................... [2] (b) Find angle TQS. Angle TQS = ................................................ [2] (c) Find angle TVQ. Angle TVQ = ................................................ [2] * 0000800000012 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\r, ĬÙĊ®Ġ´íÈõÏĪÅĊàû·Ā× ĬÿÊñÜīĢùÒĈĈóÍÌâÔćĂ ĥĕÅÕõµÅµõµĥąąÕÅµĕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD13 © UCLES 2025 4024/11/O/N/25 [Turn over 17 NOT TO SCALE 10 13 3 The diagram shows a prism. The cross-section of the prism is an isosceles triangle with sides 13cm, 13 cm and base 10cm. The length of the prism is 3cm. (a) Show that the volume of the prism is 180 cm . 3 [4] (b) A mathematically similar prism has a volume of 1440 cm . 3 The cross-section of this prism is an isosceles triangle with base x cm. Calculate the value of x. x = ................................................ [3] * 0000800000013 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u000e, ĬÛĊ®Ġ´íÈõÏĪÅĊàù·Ā× ĬÿÉòÔĝĞĉçòù¶ĉ´öÔ÷Ă ĥĕµĕµÕåÕĥÅµąąµåõąÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD14 © UCLES 2025 4024/11/O/N/25 18 Write 0 1. 4 o as a fraction. ................................................. [2] 19 ( )x x g 2 = +4 h( )x x = - 3 1 Solve hg( )x = 5. x = ................................................ [3] * 0000800000014 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0001\u000f, ĬÕĊ®Ġ´íÈõÏĪÅĊßú¶Ă× ĬÿÊôÑģđñÕøîęåĩòĤïĂ ĥµµÕõµĥõąÕµÅÅõąµÕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD15 © UCLES 2025 4024/11/O/N/25 [Turn over 20 Write as a single fraction in its simplest form. 2x x 5 7 4 - ................................................. [2] 21 (a) Evaluate. 27 3 2 - ................................................. [2] (b) Simplify. x 100 1 100 2 1 e o ................................................. [2] * 0000800000015 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0010, Ĭ×Ċ®Ġ´íÈõÏĪÅĊßü¶Ă× ĬÿÉóÙĥčāäĂăÐñđæĤÿĂ ĥµÅĕµÕąĕĕåĥÅÅĕĥõÅÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD16 © UCLES 2025 4024/11/O/N/25 22 The table shows some values for y x 3 2 x 3 = - + . x -3 -2 -1 0 1 2 3 y -16 4 2 0 20 (a) Complete the table. [2] (b) Draw the graph of y x 3 2 x 3 = - + for -3 3 G Gx . x –3 –2 –1 0 1 2 3 y –5 5 10 15 20 –10 –15 –20 [4] * 0000800000016 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0011, ĬÕĊ®Ġ´íÈõÏĪÅĊÝú¶Ą× ĬÿÉòÙğğĈ×ĀČ×ÓĈô÷Ă ĥąĕĕõÕąµõąĕÅąĕÅõÕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD17 © UCLES 2025 4024/11/O/N/25 [Turn over (c) By drawing a suitable line on the grid, find the solutions of the equation x x7 1 0 3 - + = . x = .................... or x = .................... or x = .................... [4] 23 (a) Simplify. 8 6 # + 3 ................................................. [2] (b) Rationalise the denominator and simplify. 2 5 3 - ................................................. [3] * 0000800000017 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0012, Ĭ×Ċ®Ġ´íÈõÏĪÅĊÝü¶Ą× ĬÿÊñÑĩģøâúõĒćÅÔôćĂ ĥąĥÕµµĥÕĥõÅÅąõåµÅÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD18 © UCLES 2025 4024/11/O/N/25 24 The diagram shows a rectangle and a triangle. x NOT TO SCALE x -1 2 3 x - x + 2 The length of the rectangle is x cm and the area of the rectangle is 30 cm2 . The triangle has sides of length ( ) x x - + 1 2 cm, ( )cm and ( ) 2 3 x- cm. (a) Write down an expression, in terms of x, for the width of the rectangle. ............................................ cm [1] (b) The perimeter of the rectangle is equal to the perimeter of the triangle. Form an equation in x and show that it simplifies to x x 30 0 2 - - = . [4] * 0000800000018 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0013, ĬÕĊ®Ġ´íÈõÏĪÅĊàú¸Ă× ĬÿÌòÔĝħĒÙćăí¯¯ĤÌÿĂ ĥÕõÕõÕåµÕµĕąąµÅõĥÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD19 © UCLES 2025 4024/11/O/N/25 [Turn over (c) Solve the equation x x 30 0. 2 - - = x = .................... or x = .................... [3] (d) Work out the perimeter of the rectangle. ............................................ cm [2] 25 Factorise. 654 xxx 3 2 + - ................................................. [3] * 0000800000019 * ,\u0001\u0001\u0001\u0001 \u0001\u0001\u0001\u0001\u0001\u0014, Ĭ×Ċ®Ġ´íÈõÏĪÅĊàü¸Ă× ĬÿËñÜīīĢàñî¼īÇ¸ÌïĂ ĥÕąĕµµÅÕÅÅÅąąÕåµõÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD20 © UCLES 2025 4024/11/O/N/25 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. 26 O B 4b a A T NOT TO SCALE The diagram shows triangle OAB. OA = a and OB = 4b. T is a point on AB such that AT : TB = 3 : 2. (a) Find AB in terms of a and b. AB = ................................................ [1] (b) Show that the position vector of T is ( ) a b . 5 2 +6 [3] (c) Q is a point on OB and QT ( ) a b . 5 1 = - 2 3 Find OQ : QB. ....................... : ....................... [3] * 0000800000020 * ,\u0001\u0001\u0001\u0001 \u0001, ĬÕĊ®Ġ´íÈõÏĪÅĊÞú¸Ą× ĬÿËôÜġęħÛïõ³ÉīĖÜćĂ ĥĥÕĕõµÅõåĥµąÅÕąµĥÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DFD

1. **State the problem:** Find the answers to the questions shown in the exam paper. 2. **Question 1** The reciprocal of $\frac{2}{7}$ is found by flipping the fraction. $$\frac{2}{7} \rightarrow \frac{7}{2}$$ **Answer:** $\frac{7}{2}$ 3. **Question 2** The data are $1, 2, 2, 2, 6, 7, 7, 8, 10$. - The mode is the number that appears most often. - More than $6$ hours means $7, 7, 8, 10$. - There are $4$ such children out of $9$. $$P(\text{more than } 6) = \frac{4}{9}$$ **Answers:** - (a) $2$ - (b) $\frac{4}{9}$ 4. **Question 3** A shirt costs $24$ and is reduced by $15\%$. First find $15\%$ of $24$. $$15\% \text{ of } 24 = \frac{15}{100} \times 24 = 3.6$$ Subtract the reduction. $$24 - 3.6 = 20.4$$ **Answer:** $20.4$ 5. **Question 4** The full diagram is not visible here, so the values of $p$ and $q$ cannot be determined reliably from the text alone. **Answer:** Need the exact diagram to find $p$ and $q$ 6. **Question 5** Use the fact that pie chart angles are proportional to the number of people. From cycle, $20$ people corresponds to $120^\circ$. So $1$ person corresponds to $\frac{120}{20} = 6^\circ$. - Walk: $12 \times 6 = 72^\circ$ - Bus: $5 \times 6 = 30^\circ$ - Car angle is the rest. $$360 - 120 - 72 - 30 = 138$$ The total number of people is $$20 + 12 + 5 + 23 = 60$$ So the car number is $23$. **Answers:** - (a) Car $= 23$, Bus angle $= 30^\circ$ - (b) Complete the pie chart with the remaining $138^\circ$ for car 7. **Question 6** Sam eats $\frac{2}{7}$ of the chocolate, so the remaining part is $$1 - \frac{2}{7} = \frac{5}{7}$$ If $\frac{5}{7}$ is $350$ g, then the whole mass is $$350 \div \frac{5}{7} = 350 \times \frac{7}{5} = 490$$ **Answer:** $490$ g 8. **Question 7** Cost for $10$ days: $$40 \times 10 = 400$$ The rest is the mileage charge. $$670 - 400 = 270$$ Each kilometre costs $0.30$, so $$270 \div 0.30 = 900$$ **Answer:** $900$ km 9. **Question 8** The exact coordinates of triangles $A$ and $B$ are not clear enough from the text image to identify the transformation accurately. **Answer:** Need the diagram details to determine the transformation and rotation image 10. **Question 9** Estimate by rounding each number to $1$ significant figure. The expression is unclear in the text extraction, so the estimate cannot be calculated reliably. **Answer:** Need the exact expression 11. **Question 10** Diameter $= 12$ cm, so radius $r = 6$ cm. Area of a circle is $$A = \pi r^2$$ Substitute $r = 6$: $$A = \pi \times 6^2 = 36\pi$$ **Answer:** $36\pi$ cm$^2$ 12. **Question 11** Total mass of the first $6$ plums: $$6 \times 50 = 300$$ Total mass of the remaining $4$ plums: $$4 \times 40 = 160$$ Total mass of all $10$ plums: $$300 + 160 = 460$$ Mean mass: $$460 \div 10 = 46$$ **Answer:** $46$ g 13. **Question 12** Volume of the cuboid: $$10 \times 8 \times 9 = 720$$ Since $1$ cm$^3$ = $1$ ml, the container holds $720$ ml. Water fills at $6$ ml per second. Time taken: $$720 \div 6 = 120$$ **Answer:** $120$ seconds 14. **Question 13** The fraction expression is unclear in the text extraction, so it cannot be evaluated reliably. **Answer:** Need the exact fraction expression 15. **Question 14** Lia cycles $800$ m in $105$ s. The area under the speed-time graph is the distance. The graph shape is not readable enough to calculate $v$ exactly from the text alone. Acceleration during the first $10$ s is $$a = \frac{v - 0}{10} = \frac{v}{10}$$ **Answer:** Need the exact graph details to find $v$ and the acceleration 16. **Question 15** The cumulative frequency graph is not clear enough from the text extraction to read the median and frequency values accurately. **Answer:** Need the graph to estimate the median and the number of journeys $20$ minutes or more 17. **Question 16** - (a) Angle in the same segment gives $$\angle QRT = \angle QST = 32^\circ$$ - (b) In cyclic quadrilateral $QRST$, opposite angles add to $180^\circ$: $$\angle TQS = 180 - (32 + 32) = 116^\circ$$ - (c) Radius is perpendicular to tangent, and the angle at the tangent can be found from the triangle formed. Since $QT$ subtends the same angle, the angle between tangent $TV$ and chord $TQ$ equals $32^\circ$. **Answers:** - (a) $32^\circ$ - (b) $116^\circ$ - (c) $32^\circ$ 18. **Question 17** Cross-section is an isosceles triangle with sides $13$, $13$, and base $10$. Half the base is $5$. Height by Pythagoras: $$h^2 = 13^2 - 5^2 = 169 - 25 = 144$$ $$h = 12$$ Area of triangle: $$\frac{1}{2} \times 10 \times 12 = 60$$ Volume of prism: $$60 \times 3 = 180$$ For the similar prism, volume scale factor is $$\frac{1440}{180} = 8$$ Linear scale factor: $$\sqrt[3]{8} = 2$$ So the base doubles: $$x = 10 \times 2 = 20$$ **Answers:** - (a) $180$ cm$^3$ - (b) $20$ cm 19. **Question 18** The expression appears to be $0.14$ written as a recurring decimal or possibly $0.1\dot{4}$, but the text is unclear. If it is $0.14$, then as a fraction: $$0.14 = \frac{14}{100} = \frac{7}{50}$$ **Answer:** $\frac{7}{50}$ if the decimal is $0.14$ 20. **Question 19** The function notation is partially unclear, but if the composition is $h(g(x))=5$ with $$g(x)=x+4$$ and $$h(x)=x-3+1$$ then more detail is needed to solve it exactly. **Answer:** Need the exact function definitions 21. **Question 20** The expression is unclear in the text extraction, so it cannot be simplified reliably. **Answer:** Need the exact fraction expression 22. **Question 21** - (a) The expression appears to be $27^{\frac{3}{2}} - 2$ or similar, but the text is unclear. - (b) The simplification expression is also unclear. **Answer:** Need the exact expressions 23. **Question 22** The table is for $y = x^3 - 2x + 3$. Calculate the missing values: - For $x = -1$: $$y = (-1)^3 - 2(-1) + 3 = -1 + 2 + 3 = 4$$ - For $x = 1$: $$y = 1^3 - 2(1) + 3 = 1 - 2 + 3 = 2$$ - For $x = 2$: $$y = 8 - 4 + 3 = 7$$ - For $x = 3$: $$y = 27 - 6 + 3 = 24$$ **(a) Answers:** $4, 2, 7, 24$ **(b)** Plot the points for $-3 \le x \le 3$. **(c)** To solve $x^3 - 2x + 3 = 0$, a suitable line would be $y = 0$. The exact roots need the graph; from calculation one root is $x = -\frac{3}{2}$ if the equation is different, so the text is unclear. 24. **Question 23** - (a) The expression is unclear, but if it is $8x^6 + 3$, it is already simplified. - (b) Rationalising $\frac{2}{\sqrt{5} - 3}$ would use the conjugate. Multiply top and bottom by $\sqrt{5} + 3$. $$\frac{2}{\sqrt{5} - 3} \times \frac{\sqrt{5} + 3}{\sqrt{5} + 3} = \frac{2(\sqrt{5} + 3)}{5 - 9} = \frac{2(\sqrt{5} + 3)}{-4} = -\frac{\sqrt{5} + 3}{2}$$ **Answer:** $-\frac{\sqrt{5} + 3}{2}$ 25. **Question 24** - (a) Area of rectangle $= 30$ and length $= x$, so width is $$\frac{30}{x}$$ - (b) Perimeter of rectangle: $$2\left(x + \frac{30}{x}\right)$$ Perimeter of triangle: $$(x-1) + (x+2) + (2x-3)$$ Set equal: $$2\left(x + \frac{30}{x}\right) = (x-1) + (x+2) + (2x-3)$$ Simplifying gives $$x^2 - x - 30 = 0$$ - (c) Solve: $$x^2 - x - 30 = 0$$ $$$(x-6)(x+5)=0$$ So $x = 6$ or $x = -5$ - (d) Perimeter of rectangle: $$2\left(6 + \frac{30}{6} \right) = 2(6+5)=22$$ **Answers:** - (a) $\frac{30}{x}$ cm - (c) $6$ or $-5$ - (d) $22$ cm 26. **Question 25** Factorise the expression as written in the scan; the text is unclear, but if it is $3x^2 + 6x - 45$, then $$3(x^2 + 2x - 15)=3(x+5)(x-3)$$ **Answer:** Need the exact expression 27. **Question 26** - (a) $AB = 4b - a$ - (b) Since $AT:TB = 3:2$, point $T$ divides $AB$ in the ratio $3:2$. Using the section formula, $$\vec{OT} = \frac{2\vec{OA} + 3\vec{OB}}{5} = \frac{2a + 3(4b)}{5} = \frac{2a + 12b}{5}$$ - (c) If $Q$ is on $OB$, then $\vec{OQ} = \lambda(4b)$. From $\vec{QT} = \frac{1}{5}a - 2b$, more exact text is needed to determine $OQ:QB$. **Answers:** - (a) $4b-a$ - (b) $\frac{2a+12b}{5}$ - (c) Need the exact vector equation