ACSI 2012 Y3 Core Mathematics Paper 2

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 1 Anglo - Chinese School (Independent) FINAL YEAR EXAMINATION 2012 YEAR 3 INTEGRATED PROGRAMME CORE MATHEMATICS PAPER 2 Monday 8 October 2012 1 hour 30 minutes Additional Materials Answer Paper ( 6 sheets) Graph Paper ( 1 sheet) INSTRUCTIONS TO CANDIDATES • Write your index number in the boxes on the cover page. • Do not open this examination paper until instructed to do so. • A calculator is required for this paper. • Answer all the questions on the answer sheets provided. • At the end of the examination, attach the answer sheets to the cover page provided. • Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. • The maximum mark for this paper is 80.                 _____________________________________________________________________  This paper consists of 6 printed pages. [Turn over

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 2 Full marks are not necessarily awarded for a co rrect answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.  Answer all the questions on the answer sheets provided. Please start each question on a new page. 1. [Maximum mark: 7] A triangular prism is shown in the diagram above. BCDE , ACDF and ABEF are rectangles. It is given that M is the midpoint of BC , 5 cmAB AC= = , 8 cmBC= and 12 cmCD= . (Volume of prism = Area of cross-section × Length) i) Find ADM∠ . [4 marks] ii) Find the volume of the prism. [3 marks] 2. [Maximum mark: 7] The two curves 33yx= and 2 5y x= intersect at the point A. i) Find the coordinates of the point A. [3 marks] ii) Hence sketch the graphs of 33yx= and 2 5y x= on the same diagram, indicating clearly the coordinates of Aand where the graphs cut the axes (if any). [4 marks] M 12 cm 5 cm F E D C B A 8 cm

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 3 3. [Maximum mark: 8] The diagram above shows the cross-section of a ci rcular tube used in a wind instrument, with point Oas the centre. The regular octagon ABCDEFGH is a hollow section which fits exactly in the tube with all 8 vertices on the ci rcumference of the circle. It is given that the radius mmOB x= and the length of the chord 7 mmAB= . i) Find AOB∠ . [2 marks] ii) Show that 9.146 mmx= , correct to 3 decimal places. [3 marks] iii) Find the area of the octagon ABCDEFGH . [3 marks] B F G H E D C A O x mm 7 mm

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 4 4. [Maximum mark: 10] Diagram I shows a spherical steel ball of radius 3 cm. (Volume of sphere 34 3 rπ= ; Volume of cylinder 2rhπ= ; Volume of cone 21 3 rhπ= ) i) Find the volume of one such spherical steel ball. [2 marks] Diagram II shows the cross sectional view of an open cylindrical container filled with 7 such identical spherical steel balls touching one another and the side of the container. The height of the container is 6 cm. ii) Find the volume of the empty space in the container. [4 marks] iii) These 7 spherical steel balls are melted and recast into the shape as shown in Diagram III. The new steel structure consists of a cylinder and a cone of same height and radius, r cm. Find the value of r . [4 marks] 5. [Maximum mark: 13] a) Find the range of values of m for which the line 2yx m= − intersects the curve ( ) 22 8yx m++ = at two distinct points. [5 marks] b) The roots of the quadratic equation 229 5 0xx+ += are αand β. i) State the values of α β+ and αβ . [2 marks] ii) Find the quadratic equation whose roots are 213 α− and 213 β− . [ 6 m a r k s ] 3 cm r cm r cm r cm Diagram I Diagram II Diagram III

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 5 6. [Maximum mark: 10] Solution to this question by accurate drawing will not be accepted. The diagram above, not drawn to scale, shows a parallelogram ABCD where the coordinates of the points A , B and C are (3 ,8 )− − , (3, 4)− and (6 , 5 )− respectively. E is a point with coordinates (1 0 , 9 )− and BCE is a straight line. i) Find the coordinates of the point D . [1 mark] ii) Find the gradient of AB . [1 mark] iii) Find the equation of the perpendicular bisector of AB and show that the point E lies o n i t . [5 marks] iv) Find the area of ABE+ . [3 marks] O y x B (3, െ4) E (–10, 9) D A (െ3, െ8) C (െ6, 5)

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 6 7. [Maximum mark: 12] Solve the following equations a) 225 3 0xxee−− −− = ; [4 marks] b) 2 23log log 12yy+= ; [4 marks] c) 73 1 2 1 75 zz e−+ = . [4 marks] 8. [Maximum mark: 13] The table shows experimental values of two variables x and y , which are connected by an equation of the form nyA x= , where A and n are constants. x 2 3 4 5 6 y 9.9 18.2 28.0 39.1 51.4 i) Using a scale of 2cm to 0.1 unit on the lg x –axis and 2cm to 0.2 units on the lg y – axis, plot lg y against lgx and draw a straight line graph. [4 marks] ii) Use your graph to estimate the value of A and of n . [4 marks] iii) On the same diagram, draw the line representing the equation 10yx= . [2 marks] iv) Hence find the range of values of x for which 10.nAxx≤ [3 marks] End of Paper 2

ACS(Independent)/Y3IPCoreMathsP2/2012/Endyear 7 ANSWERS: 1(i) 13.3 (3sf)ADM∠= D 5(a) 44 33 m− << 1(ii) 3144 cm 5(b)(i) 95 and 22αβ α β+ =− = 2(i) (1.11, 4.08)A= 5(b)(ii) 2 175 23 0 or42xx++ = 24 175 46 0xx+ += 2(ii) 6(i) (1 2 , 1 )D= − 6(ii) 2 3 6(iii) 3 62yx=−− 6(iv) 265 units 3(i) 45AOB∠= D 7(a) ln 3 1.10x=−= − 3(iii) 2237 mm 7(b) 12 525 . 2 8y== 4(i) 3113 cm 7(c) 0.341z =− 4(ii) 3735 cm 8(ii) 3.47A= Accept 3.24 3.80A≤≤ 1.51n= Accept 1.30 1.70n≤≤ 4(iii) 5.74 cm 8(iv) 1.16x≥