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2 International GCSE MATHEMATICS FORMULAE SHEET HIGHER TIER Pythagoras Volume of cone = Theorem 1 3 r 2 h c Curved surface area of cone = b a l a 2 + b 2 = c 2 r Volume of sphere = rl Surface area of sphere = 4 r h r r 2 hyp opp adj = hyp opp = hyp opp = adj cos sin tan In any triangle ABC adj or sin opp hyp b C a cos tan adj hyp opp adj A a Sine rule: sin A b sin B c sin C 2 Cosine rule: a b c 2bc cos A c B cross section length Area of triangle 1 2 ab sin C Volume of prism = area of cross section length r Circumference of circle = 2 r Area of a trapezium = a 1 (a+b)h 2 Area of circle = r 2 h b r h Volume of cylinder = r 2 h Curved surface area of cylinder = 2 rh The Quadratic Equation 2 The solutions of ax + bx + c 0, where a 0, are given by 2 b+ b 4ac x 2a 2 *P44613A0224*

3 Answer ALL TWENTY TWO questions. Write your answers in the spaces provided. You must write down all the stages in your working. 1 1 euro = 120 yen 1 = 1.2 euros Change 50 to yen. yen (Total for Question 1 is 2 marks) 2 (a) Work out the value of (b) Work out the value of (Total for Question 2 is 4 marks) Do NOT write in this space. *P44613A0324* 3 Turn over

4 3 (a) Factorise 14x 35 (1) (b) Expand and simplify 3(2c 5) 2(c 4) (c) Simplify (4e 3 ) 2 (d) Expand and simplify (a + 5)(2a 1)... (Total for Question 3 is 7 marks) Do NOT write in this space. 4 *P44613A0424*

5 4 The diagram shows a shape with one line of symmetry. 10 cm Diagram NOT accurately drawn 12 cm 4 cm 15 cm Work out the area of the shape. cm 2 (Total for Question 4 is 4 marks) *P44613A0524* 5 Turn over

7 A second jar contains 100 coloured beads. 20 of these beads are brown. Ajit takes at random a bead from the jar. He records the colour of the bead and then returns the bead to the jar. He does this 60 times. (c) Work out an estimate for the number of times Ajit records a brown bead. (Total for Question 5 is 8 marks) 6 Solve 4(5y 1) = 3(6y + 7) Show clear algebraic working. y = (Total for Question 6 is 3 marks) *P44613A0724* 7 Turn over

8 7 Eloy s height was 125 cm when his age was 7 years. His height was 153 cm when his age was 12 years. (a) Work out the percentage increase in Eloy s height between the ages of 7 and 12 years. Eloy s height at the age of 12 years was 85% of his height at the age of 20 years. (b) Work out Eloy s height when his age was 20 years. % (3) cm (3) (Total for Question 7 is 6 marks) Do NOT write in this space. 8 *P44613A0824*

9 8 B Diagram NOT accurately drawn 15 cm 10 cm A D 12.5 cm C ABC is a triangle. The point D lies on AC. Angle BDC = 90 BD = 10 cm, AB = 15 cm and DC = 12.5 cm. (a) Calculate the length of AD. Give your answer correct to 3 significant figures. cm (3) (b) Calculate the size of angle BCD. Give your answer correct to 1 decimal place. (3) (Total for Question 8 is 6 marks) *P44613A0924* 9 Turn over

10 9 (a) Find the sum of the interior angles of a polygon with 7 sides. Diagram NOT accurately drawn x The diagram shows a regular polygon with 7 sides. (b) Work out the value of x. Give your answer correct to 1 decimal place. (Total for Question 9 is 4 marks) 10 *P44613A01024*

11 10 (a) Find the gradient of the line with equation 3y 2x = 6 (b) Find an equation of the line with gradient 3 that passes through the point (2, 5). (Total for Question 10 is 4 marks) *P44613A01124* 11 Turn over

12 = = (a) Find the highest common factor (HCF) of 3780 and 3240 Give your answer as a product of prime factors. (b) Find the lowest common multiple (LCM) of 3780 and 3240 Give your answer as a product of prime factors. (Total for Question 11 is 4 marks) 12 *P44613A01224*

13 12 Solve the simultaneous equations 5y 4x = 8 y + x = 7 Show clear algebraic working. x = y = (Total for Question 12 is 3 marks) *P44613A01324* 13 Turn over

14 13 The cumulative frequency graph gives information about the intelligence quotients (IQ) of a random sample of 100 adults. Cumulative frequency IQ (a) Use the cumulative frequency graph to find an estimate for the number of adults in the sample who have an IQ between 85 and 115 (b) Find an estimate for the upper quartile of the IQ of adults in the sample. (Total for Question 13 is 4 marks) 14 *P44613A01424*

15 14 S Diagram NOT accurately drawn 5 cm O R T 6 cm P 3 cm Q P, Q, R and S are points on a circle, centre O. QS is a diameter of the circle. QS and PR intersect at the point T. OS = 5 cm, QT = 3 cm and TR = 6 cm. Work out the length of PT. cm (Total for Question 14 is 3 marks) *P44613A01524* 15 Turn over

16 15 Diagram NOT accurately drawn 32 cm Zane buys mineral water in large bottles and in small bottles. The large bottles are mathematically similar to the small bottles. Large bottles have a height of 32 cm and a volume of 2000 cm 3 Small bottles have a volume of 500 cm 3 Work out the height of a small bottle. Give your answer correct to 3 significant figures cm cm 3 cm (Total for Question 15 is 3 marks) 16 *P44613A01624*

17 16 Gemma has 9 counters. Each counter has a number on it Gemma puts the 9 counters into a bag. She takes at random a counter from the bag and does not replace the counter. She then takes at random a second counter from the bag. (a) Work out the probability that the number on each counter is an even number. (b) Work out the probability that the number on the first counter added to the number on the second counter gives an odd number. (3) (Total for Question 16 is 5 marks) *P44613A01724* 17 Turn over

18 17 P is directly proportional to q 3 P = 270 when q = 7.5 (a) Find a formula for P in terms of q (3) (b) Work out the positive value of q when P = q q = (Total for Question 17 is 5 marks) 18 *P44613A01824*

19 18 y = x 3 4x 2 + 4x + 3 (a) Find d y dx y P O x The diagram shows a sketch of the curve with equation y = x 3 4x 2 + 4x + 3 The point P is a turning point on the curve. (b) Work out the coordinates of P. Show clear algebraic working. (c) Write down the range of values of x for which the curve has a negative gradient. (...,...) (4) (Total for Question 18 is 8 marks) *P44613A01924* 19 Turn over

20 19 E A B The Venn diagram shows all of the elements in sets A, B and E. (a) Write down the elements in A (1) (b) Find n(a B) (1) (c) Find the elements in (A B) (A B) (1) A C = Ø B C = {5, 6, 7, 8, 9} n(c) = 3 (d) Write down the elements in C. (1) (Total for Question 19 is 4 marks) Do NOT write in this space. 20 *P44613A02024*

21 2x 20 f:x 2x g:x where x 1 x 1 (a) Express the composite function gf in the form gf:x... Give your answer as simply as possible. gf:x (b) Express the inverse function g 1 in the form g 1 :x... g 1 :x (3) (Total for Question 20 is 5 marks) *P44613A02124* 21 Turn over

22 21 Diagram NOT accurately drawn 15 cm A solid cone has a height of 15 cm. The volume of the cone is 320 cm 3 Work out the curved surface area of the cone. Give your answer correct to 3 significant figures. cm 2 (Total for Question 21 is 5 marks) 22 *P44613A02224*

23 22 C (2x 1) cm 2 7 cm Diagram NOT accurately drawn A 60 (2x + 1) cm B The diagram shows a triangle ABC. AB = (2x + 1) cm, AC = (2x 1) cm and BC = 2 Angle BAC = 60 7 cm. Work out the value of x. Show clear algebraic working. x =... (Total for Question 22 is 3 marks) TOTAL FOR PAPER IS 100 MARKS *P44613A02324* 23

24 BLANK PAGE Do NOT write in this space. 24 *P44613A02424*

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