184/09 MATHEMATICS HIGHER TIER PAPER 1. P.M. MONDAY, 4 June (2 Hours) CALCULATORS ARE NOT TO BE USED FOR THIS PAPER

Transcription

1 Candidate Name Centre Number Candidate Number WELSH JOINT EDUCATION COMMITTEE General Certificate of Secondary Education CYD-BWYLLGOR ADDYSG CYMRU Tystysgrif Gyffredinol Addysg Uwchradd 184/09 MATHEMATICS INSTRUCTIONS TO CANDIDATES JD* HIGHER TIER PAPER 1 P.M. MONDAY, 4 June 2007 (2 Hours) Write your name, centre number and candidate number in the spaces at the top of this page. Answer all the questions in the spaces provided. Calculators are not allowed for this paper. Take πas INFORMATION FOR CANDIDATES CALCULATORS ARE NOT TO BE USED FOR THIS PAPER You should give details of your method of solution when appropriate. Unless stated, diagrams are not drawn to scale. Scale drawing solutions will not be acceptable where you are asked to calculate. The number of marks is given in brackets at the end of each question or part-question. No certificate will be awarded to a candidate detected in any unfair practice during the examination. For Examiner s use Question Maximum Mark TOTAL MARK Mark Awarded

2 2 Formula List Volume of prism = area of cross-section length crosssection length Volume of sphere = 4 πr 3 3 Surface area of sphere = 4πr 2 r Volume of cone = 1 3 πr 2 h l h Curved surface area of cone = πrl r In any triangle ABC C Sine rule a b c = = sin A sin B sin C b a Cosine rule a 2 = b 2 + c 2 2bc cos A Area of triangle = 1 2 ab sin C A c B The Quadratic Equation The solutions of ax 2 + bx + c = 0 where a 0 are given by x b b ac = ± ( 2 4 ) 2a Standard Deviation Standard deviation for a set of numbers x 1, x 2,..., x n, having a mean of x is given by s = ( x x )2 or s n 2 2 x x = n n

3 3 Examiner 1. (a) Express 180 as a product of prime numbers. (b) Given that 105 = 3 5 7, find the highest common factor (HCF) of 105 and 180. Turn over.

4 4 Examiner 2. The table shows some of the values of y = 7 + 2x x 2 for values of x from 3 to 4. (a) Complete the table by finding the value of y for x = 2. x y = 7 + 2x x [1] (b) (c) On the graph paper opposite, draw the graph of y = 7 + 2x x 2 for values of x between 3 and 4. Draw the line y = 3 on your graph paper and write down the x-values of the points where your two graphs intersect.

5 5 Examiner For use with question 2 10 y x Turn over.

6 6 Examiner 3. (a) Draw the image of the triangle A after reflection in the dotted line y = x. Label the image B. y A x y = x 5

7 7 Examiner (b) Rotate the shape C through 90 clockwise about the point ( 2, 1). Label the image D. y C x Turn over.

8 8 Examiner 4. Find and shade the region of points inside triangle XYZ that satisfy both of the following conditions. (i) The points are nearer to XY than to XZ. (ii) The points are more than 6cm from Y. [3] Z X Y

9 9 Examiner 5. Explain clearly why a regular polygon cannot have exterior angles of Solve the following simultaneous equations by an algebraic (not graphical) method. Show all your working. 3x + 4y = 7 2x + 3y = 6 [4] Turn over.

10 10 Examiner 7. Square paving stones are manufactured so that the length of each of them is 51 cm, measured to the nearest centimetre. (a) Write down the least and greatest length of a square paving stone. Least length cm Greatest length cm (b) The length of a path is 30 0 m, measured correct to the nearest 0 1 of a metre. Starting at one end of the path, 60 of the paving stones are laid end to end in a straight line along the path. Explain, showing all your calculations, why the paving stones will always be longer than the path. [5] 8. Each of the following quantities has a particular number of dimensions. Give the number of dimensions of each quantity. The first one has been done for you. Quantity Number of dimensions The capacity of a jug. 3 The distance between Cardiff and Mold. The area of a field. The perimeter of a hexagon. The volume of a cuboid.

11 11 Examiner 9. (a) Explain clearly why triangles ABC and PQR are similar. R C 8 cm 12 cm 12 cm 18 cm A 14 cm B P 21 cm Diagrams not drawn to scale. Q (b) Triangle XYZ, in which XZ = 3 cm and ZY = 4 5 cm, is similar to triangle ABC in part (a). Calculate the length of XY. Z 3 cm 4 5 cm X Diagram not drawn to scale. Y Turn over.

12 12 Examiner 10. (a) (i) Solve the inequality n n (ii) Write down the greatest whole number that satisfies this inequality [1] (b) Simplify 4a 3 b 4 3a 3 b. (c) Make u the subject in the following. 5(t + u) = 6 + 3t [3]

13 11. The probability that Linda wins a game of darts is 1. 4 (a) 13 Examiner Linda plays two games of darts. Complete the following tree diagram to show the probabilities of what can happen when Linda plays two games of darts. Game 1 Game Linda wins Linda loses (b) Calculate the probability that Linda loses both games. Turn over.

14 14 Examiner 12. On the graph paper provided, draw the region which satisfies all of the following inequalities. x + y X 10 y x 2 x + 4 x x 1 Make sure that you clearly indicate the region that represents your answer. [3] y x 4 6

15 15 Examiner 13. Make f the subject of the following formula. 4ef + 7e = 3f + 5d [3] 14. Express 450 in the form a b, where a is a whole number and b is a prime number. Turn over.

16 16 Examiner 15. Four points A, B, C and D lie on the circumference of the circle centre O. The tangent TP touches the circle at C and the radius OB is parallel to DC. A O D B P 58 C T Diagram not drawn to scale. $ Given that DCT = 58, find each of the following angles, giving reasons for your answers. $ (a) BOC $ (b) BAC

17 17 Examiner 16. Given that y is inversely proportional to x 2, and that y = 4 when x = 10, (a) find an expression for y in terms of x, [3] (b) calculate (i) the value of y when x = 20, [1] (ii) a value of x when y = Turn over.

18 18 Examiner 17. A survey was carried out to find the distribution of the heights of trees in an orchard. The histogram below illustrates the results of the survey. Frequency density Height of tree (m) (a) Use the histogram to calculate the number of trees in the orchard measured. (b) Estimate the height exceeded by 44% of the trees. [1]

19 19 Examiner 18. A bag contains 5 red balls, 4 blue balls and one yellow ball. Two balls are drawn at random without replacement from the bag. (a) Calculate the probability that the two balls are of the same colour. [3] (b) Calculate the probability that at least one blue ball is drawn. [3] Turn over.

20 20 Examiner 19. (a) Factorise 6x x (b) Simplify 2 6x + 19x x + 25.

21 21 BLANK PAGE Turn over.

22 22 Examiner 20. The table shows some of the values of y = x 3 + 6x 2 x 27 for values of x from 6 to 3. x y (a) (b) On the graph paper opposite, draw the graph of y = x 3 + 6x 2 x 27 for values of x between 6 and 3. On the graph paper opposite, draw an appropriate straight line and solve the equation x 3 + 6x 2 x 17 = 0. [3]

23 23 Examiner For use with question y x Turn over.

24 24 Examiner 21. Express the following as a single fraction in its simplest form. 3x x 5 4x x + 7 [3]

25 25 Examiner 22. In the following expressions n is any positive integer. State whether each expression is always odd, always even, or sometimes even and sometimes odd. Give reasons for your answer. (a) 2n + 1 (b) (n + 1) (n + 2) Turn over.

26 26 Examiner 23. During an experiment the value of a variable y was found for different values of x. The results are recorded in the following table. x y (a) On the graph paper plot the values of y against the values of x y x 2 (b) It is known that y is approximately equal to ax 2 + b. Use your graph to estimate the values of a and b. [3]

27 27 Examiner 24. The prisms shown below are similar. Diagrams not drawn to scale. The area of the uniform cross-section of the smaller prism is 5 cm 2 and its length is 3 cm. The area of the cross-section of the larger prism is 80 cm 2. Calculate the volume of the larger prism. [5]