Cambridge International Eaminations Cambridge International General Certificate of Secondar Education * 1 4 9 5 4 7 5 6 * CAMBRIDGE INTERNATIONAL MATHEMATICS 67/ Paper (Etended) Ma/June 15 45 minutes Candidates answer on the Question Paper. Additional Materials: Geometrical Instruments READ THESE INSTRUCTIONS FIRST Write our Centre number, candidate number and name on all the work ou hand in. Write in dark blue or black pen. Do not use staples, paper clips, glue or correction fluid. You ma use an HB pencil for an diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. CALCULATORS MUST NOT BE USED IN THIS PAPER. All answers should be given in their simplest form. You must show all the relevant working to gain full marks and ou will be given marks for correct methods even if our answer is incorrect. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 4. This document consists of 8 printed pages. DC (AC/SG) 9983/6 [Turn over
Formula List For the equation a + b + c = b! b 4ac = - - a Curved surface area, A, of clinder of radius r, height h. A = rrh Curved surface area, A, of cone of radius r, sloping edge l. A = rrl Curved surface area, A, of sphere of radius r. A = 4rr 1 Volume, V, of pramid, base area A, height h. V = Ah 3 Volume, V, of clinder of radius r, height h. V = rr h Volume, V, of cone of radius r, height h. 1 V = rr h 3 Volume, V, of sphere of radius r. A 4 3 V = rr 3 a b c = = sin A sin B sin C c b a = b + c - bc cos A 1 Area = bc sin A B a C
3 Answer all the questions. 1 (a) Work out (. 3). Answer(a)... [1] (b) Find n when 5 n =. 6 4 Answer(b) n =... [1] (a) Find the value of (i) 5, Answer(a)(i)... [1] (ii) 1 3. Answer(a)(ii)... [1] (b) Write as a single power of 5. 1 5 5 # 5 3 Answer(b)... [1] J - 6 N 3 Find the magnitude of K O. L 4 P Write our answer in surd form as simpl as possible. Answer... [3] [Turn over
4 Anneke, Babar, Céline, and Dieter each throw the same biased die. The want to find the probabilit of throwing a si with this die. The each throw the die a different number of times. These are their results. 4 Anneke Babar Céline Dieter Number of throws 4 1 5 Number of sies 46 1 15 1 (a) Complete the table below to show the relative frequencies of their results. Write our answers as decimals. Anneke Babar Céline Dieter Relative frequenc of throwing a si [] (b) Whose result gives the best estimate of the probabilit of throwing a si with the biased die? Give a reason for our answer. Answer(b)... because...... [1] (c) The probabilit of throwing a si with a different biased die is.41. Find the epected number of sies when this die is thrown 6 times. Answer(c)... [1]
5 5 A is the point (, 8) and B is the point (6, ). (a) Find the co-ordinates of the midpoint of AB. Answer(a) (...,...) [1] (b) Find the gradient of AB. Answer(b)... [] 6 Simplif ( 5 + 3). Answer... [] 7 Solve. + 3 G 4( - ) Answer... [3] [Turn over
6 8 3 m 6 m NOT TO SCALE 1 m The diagram shows a shape made from a clinder and a cone. The clinder and cone have a common radius of 6 m. The height of the clinder is 1 m and the height of the cone is 3 m. Calculate the total volume of the shape. Leave our answer as a multiple of π. Answer... m 3 [3] 9 Solve these simultaneous equations. 5 + = 11 4-3 = 18 Answer =... =... [4]
7 1 Solve the following equations. (a) log + log 3 = log 1 Answer(a) =... [1] (b) log = 3 Answer(b) =... [1] (c) log log 5 = log Answer(c) =... [3] 11 A, B, C and D are points on the circle, centre O. PQ is a tangent to the circle at the point C. Angle PCD = 55 and angle ADO = 4. A B Q NOT TO SCALE O 55 C 4 Find (a) angle COD, D P Answer(a)... [] (b) angle DAC, Answer(b)... [1] (c) angle ABC. Answer(c)... [1] Question 1 is printed on the net page. [Turn over
8 1 These are sketches of the graphs of si functions. A B C D E F In the table below are four functions. Write the correct letter in the table to match each function with its graph. Function Graph f ( ) = - 3 f( ) = ( - ) f( ) = 4-3 f ( ) = 5 - [4] Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (UCLES) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. To avoid the issue of disclosure of answer-related information to candidates, all copright acknowledgements are reproduced online in the Cambridge International Eaminations Copright Acknowledgements Booklet. This is produced for each series of eaminations and is freel available to download at www.cie.org.uk after the live eamination series. Cambridge International Eaminations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge.