1.30 pm 2.30 pm. Mathematics Module M4 Paper 1 (Non-calculator) Higher Tier [GMM41] 1 hour.

Transcription

1 Centre Number 71 Candidate Number General Certificate of Secondary Education 2006 Mathematics Module M4 Paper 1 (Non-calculator) Higher Tier [GMM41] GMM41 MONDAY 5 JUNE 1.30 pm 2.30 pm TIME 1 hour. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number in the spaces provided at the top of this page. Write your answers in the spaces provided in this question paper. Answer all ten questions. Any working should be clearly shown in the spaces provided since marks may be awarded for partially correct solutions. You must not use a calculator for this paper INFORMATION FOR CANDIDATES The total mark for this paper is 49. Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. You should have a ruler, compasses, set-square and protractor. The Formula Sheet is on page 2. For Examiner s use only Question Number BP GMM41S Total

2 Formula Sheet Area of trapezium = 1 2 (a + b)h a h b Volume of prism = area of cross section length In any triangle ABC Cross section A length Area of triangle = 1 2 ab sin C a b c Sine rule : = = sin A sin B sin C Cosine rule: a 2 = b 2 + c 2 2bc cos A B c a b C Volume of sphere = 4 3 πr 3 Surface area of sphere = 4πr 2 r Volume of cone = 1 3 πr 2 h Curved surface area of cone = πrl l r h Quadratic equation: The solutions of ax 2 + bx + c = 0, where a 0, are given by BP b± b 4ac x = 2a 2 GMM41S [Turn over

3 1 y B C z O Diagram not drawn accurately D 100 E x A O is the centre of the circle. Find the sizes of the angles (a) x Answer [1] (b) y Answer [1] (c) z Answer [1] BP GMM41S [Turn over

4 2 In a closing down sale, all prices were reduced by 70%. The sale price of a camera was 48. What was the original price before the sale? Answer [3] 3 Evaluate (a) 10 3 Answer [1] (b) 6 0 Answer [1] BP GMM41S [Turn over

5 4 (a) Solve x x 15 = 0 Answer x = [3] (b) On the grid below, illustrate the inequalities y 8, y 3x + 4, y x + 1, x 0 Mark with R the region satisfying all four inequalities. [4] y x BP GMM41S [Turn over

6 5 Evaluate (a) Answer [1] (b) Answer [1] (c) Answer [2] (d) Answer [2] BP GMM41S [Turn over

7 6 The two triangles shown are similar. 9cm 7.5 cm xcm Diagram not drawn accurately 12cm (a) Calculate the length of the side marked x. Answer cm [2] (b) Given that the area of the small triangle is 22.5 cm 2, calculate the shaded area between the two triangles. Answer cm 2 [3] BP GMM41S [Turn over

8 7 The table gives information about the heights of plants in a crop test. Height (h cm) 0 h 6 6 h 8 8 h h h 20 Frequency (a) Show this information on a histogram. [3] BP GMM41S [Turn over

9 (b) The plants are arranged in order of height and every tenth one is selected to form a sample. Explain why this is not a suitable way to select a sample. [2] (c) In a stratified sample of all plants over 9 cm in height, 20 up to 10 cm were selected and 8 over 15 cm were selected. How many plants in the original crop were over 9 cm in height? Answer [3] BP GMM41S [Turn over

10 8 Factorise fully (a) 6cd 9c 2 Answer [2] (b) 3a 2 27d 2 Answer [3] 9 Solve the equation = 3 (x 3) (2x 1) Show your working BP Answer [7] GMM41S [Turn over

11 10 Find the values of c and d for which y 2 12y + c (y + d) 2 Answer c = d = [3] BP GMM41S [Turn over

12 BP S 1/ (133) [Turn over

13 Centre Number 71 Candidate Number General Certificate of Secondary Education 2006 Mathematics Module M4 Paper 2 (With calculator) Higher Tier [GMM42] GMM42 MONDAY 5 JUNE 2.45 pm 3.45 pm TIME 1 hour. INSTRUCTIONS TO CANDIDAT ES Write your Centre Number and Candidate Number in the spaces provided at the top of this page. Write your answers in the spaces provided in this question paper. Answer all eleven questions. Any working should be clearly shown in the spaces provided since marks may be awarded for partially correct solutions. INFORMATION FOR CANDIDATES The total mark for this paper is 49. Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. You should have a calculator, ruler, compasses, set-square and protractor. The Formula Sheet is on page 2. GMM42S For Examiner s use only Question Number Total

14 Formula Sheet Area of trapezium = 1 2 (a + b)h a h b Volume of prism = area of cross section length In any triangle ABC Cross section A length Area of triangle = 1 2 ab sin C a b c Sine rule : = = sin A sin B sin C Cosine rule: a 2 = b 2 + c 2 2bc cos A B c a b C Volume of sphere = 4 3 πr 3 Surface area of sphere = 4πr 2 r Volume of cone = 1 3 πr 2 h Curved surface area of cone = πrl l r h Quadratic equation: The solutions of ax 2 + bx + c = 0, where a 0, are given by x = 2 b± b 4ac 2a GMM42S [Turn over

15 1 The cumulative frequency graph gives information about the percentage marks obtained by 300 candidates in an examination. Use the graph to estimate (a) the median percentage mark, Answer [1] (b) the interquartile range, Answer [2] (c) the percentage mark separating the top 20 candidates from the others. Answer [2] Cumulative frequency % marks (less than) GMM42S [Turn over

16 2 (a) A is (0, 4). B is (3, 13) Find the equation of the line AB. Answer [3] (b) Find the equation of the line parallel to the line AB and passing through (0, 8). Answer [2] (c) Find the equation of the line perpendicular to the line AB and passing through (9, 0). Answer [2] GMM42S [Turn over

17 3 Q 8cm Diagram not drawn accurately P 11cm R In the triangle PQR, angle PQR is 90, PR = 11 cm and PQ = 8 cm. Calculate the size of the angle QPR. Answer [3] 4 Calculate the volume of a sphere of radius 12 cm. Answer [3] 5 Solve the simultaneous equations 6x + 7y = 1 2x y = 7 Show your working. A solution by trial and improvement will not be accepted. Answer x = y = [3] GMM42S [Turn over

18 6 D G E C F 7cm B 8cm O 10cm A (a) Calculate the length of the space-diagonal OG in the cuboid OABCDEFG. Answer cm [2] (b) Calculate the size of the angle GOA between the lines OA and OG. Answer [2] 7 Prove that opposite angles of a cyclic quadrilateral add up to 180. [3] GMM42S [Turn over

19 8 The area of a rectangle is 6 cm 2 and the perimeter is 20 cm. (a) If x is the length of one side of the rectangle, show that x 2 10x + 6 = 0 [3] (b) Find the lengths of the sides of the rectangle. You may give your answers correct to two decimal places. Answer cm [3] GMM42S [Turn over

20 9 y x 1 y = cosx Use the graph to find, for 0 x 360, the solutions of (a) cos x = 0.45 Answer [2] (b) 8 cos x = 5 Answer [2] GMM42S [Turn over

21 10 Start 5km 3km 3km Bill Diagram not drawn accurately Ben Bill and Ben are hiking. They hike together 5 km due East. They then split up. Bill continues 3 km due East but Ben travels 3 km in a straight line in a different direction. When they have both stopped Ben is 1 km nearer the starting point than Bill. Calculate Ben s bearing from the starting point. A solution by scale drawing will not be accepted. Answer [4] 11 Solve simultaneously the equations x 2 + 4y 2 = 10 2y + 3x = 10 A solution by trial and improvement will not be accepted. Show your working. Answer x = y = [7] GMM42S [Turn over

22 THIS IS THE END OF THE QUESTION PAPER GMM42S [Turn over

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24 S 1/ (134) [Turn over