Excel / Education GCSE Mathematics Paper 3B (Calculator) Higher Tier Time: 2 hours 3B Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used. Items included with question papers Formulae sheet Instructions to Candidates Answer ALL the questions in the spaces provided in this question paper. Supplementary answer sheets may be used. Information for Candidates The total mark for this paper is 100. The marks for questions and parts of questions are shown in round brackets: e.g.. Calculators may be used. Advice to Candidates Show all your working.
1. (a) Expand (i) 3 (4x 2) (ii) -2x(x + 7) (b) Factorise 8ab 12ac 2. y A x (a) On the diagram above, draw the image of shape A when it is rotated 90 o anticlockwise about the origin. Label this shape B. (b) On the diagram above, draw the image of shape A when it is enlarged by scale factor - 2, centre the origin. Label this shape C.
3. (a) (i) Write the number 64 as a product of its prime factors (ii) Write the number 36 as a product of its prime factors (b) Find the highest common factor of 64 and 36 (c) Simplify the fraction 36 64 (1) 4. A puppy is born weighing 900g. Each month its weight increases by 15% of what it was the previous month. (a) How much does the puppy weigh when it is one month old?
(b) How much does the puppy weigh when it is three months old? 5. A 9cm B C 12cm D 20 cm E In the diagram above, BC is parallel to DE (a) Explain why triangle ABC is SIMILAR to triangle ADE AB is 9cm, BD is 12cm, DE is 20cm. (b) Calculate the length of BC cm
6 From a position 200m from a vertical cliff C the angle of elevation of the top of the cliff from a boat B is 25 o. 25 o B 200m C h (a) Calculate the height of the cliff (marked h on the diagram). (b) The boat moves to 80m from the cliff. What is the angle of elevation of the top of the cliff from the boat now?
7. (a) Simplify (2y 4 ) 2 (b) Simplify 18x 9 3x 7.... 8. Maths textbooks cost 9.75 each and a teacher needs to buy 33 copies for her class. Show how the teacher could ESTIMATE the total cost of the books and state what her estimated total would be. 9. A 12.5 m long rope is tied to the top of a flag pole and the other end is staked into the ground as shown in the diagram. The flag pole is 5.1 m tall. 5.1m 12.5m x (a) Calculate how far from the base of the pole the rope is staked to the ground. This is the distance marked x in the diagram. Give your answer to a sensible degree of accuracy..m (4)
Regulations say that the rope must be staked exactly 15m from the foot of the pole. (b) How much longer does the rope have to be?.m 10. The equation of a line is y = 2x + 3 (a) Complete the following table of values for this equation. x -1 0 1 2 3 y 5 (b) Use the grid on the next page to plot the graph of y = 2x + 3 y 10 8 6 4 2-4 -3-2 -1 1 2 3 4 x -2-4 -6-8
(c) Does the point (25, 53) lie on the line? Give a reason for your answer. (d) Write down the equation of a line which is parallel to the line y = 2x + 3. 11. A biologist is studying rock pools. He measures the distance of each pool from the sea and counts the number of shells in each pool. pool A B C D E F G distance (m) 20 23 15 7 10 12 9 shells 4 2 10 18 12 12 15 (a) On the grid (over), draw a scatter diagram to illustrate this data.
shells 21 19 17 15 13 11 9 7 5 3 1 1 3 5 7 9 11 13 15 17 19 21 23 dist (m) (b) Draw a line of best fit on the graph (1) (c) Describe the relationship between the distance from the ocean and the number of shells.........
12. Solve these simultaneous equations; 5x 3y = 14 4x 5y = 6 x =. y =. (4) 13. The graph (over page) shows Saima s journey on a hiking trip to a café 8km from home. (a) What is her average speed on her journey to the café? Give your answer in km / h. Saima sets off home at 3pm, walking at 6 km / h. km / h (4) (b) On the graph, draw the line representing her journey home.
Dist. (m) 8500 7500 6500 5500 4500 3500 2500 1500 500 12pm 12.30 1.00 1.30 2.00 2.30 3.00 3.30 4.00 4.30 Time 14. The diagram shows a slice of birthday cake cut from a circular cake. The angle at the centre, AOB, is 30 o. Length OA is 10cm and length OD is 6cm. A B 10cm 30 o O 6 cm D
(i) Find the area of the top of the cake, that is sector AOB....cm 2 (ii) Hence find the volume of the piece of cake, giving your answer to a suitable degree of accuracy....cm 3 15. (a) A cylinder has height 4cm and radius (x - 2) cm. Given that its volume is 16π cm 3, find the value of x. (b) Hence find the total surface area of the cylinder. (4) (4)
16. C 15 o A 40 o 10 m B The diagram shows a triangle ABC. Angle BAC is 40 o, Angle ACB is 15 o and AB is 10m. Calculate the length of side BC. Give your answer to an appropriate degree of accuracy...m (4)
17. A straight line passes through the points A(x, x + 2) and B(x 2 + 1, x 2 + 4) (a) Show that the gradient of the line is given by x 2! x + 2 x 2! x +1 (b) Given that the gradient of the line is 2, find the possible values of x... (4) End of Questions (Total Marks: 100)