Anglican High 4048 02 AS

NAME: _______________________________ ( ) CLASS: 4 ( ) 9o MATHEMATICS 4048/02 Paper 2 31 August 2022 2 hours 30 minutes Candidates answer on the Question Paper and Graph Paper READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . 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 100. ---------------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Questions 1 2 3 4 5 6 7 Marks Questions 8 9 10 11 12 13 14 Marks Table of Penalties Units 100 Clarity/Logic Accuracy/Precision Parent’s Name and Signature: Date: This document consists of 23 printed pages and 1 blank page. SOLUTIONS S4

2 1 (a) Below are the first five terms of a sequence. 1 17 4 25 9 33 16 41 25 49 (i) Find the sixth term of the sequence. (ai) 36 12 57 19= B1 award mark even if student give 36 57 (ii) nT is the nth term of the sequence. Find an expression, in terms of n, for nT . (aii) 2 89 n nT n= + B2: 1 mark for each series. (iii) The kth term is 1. Find the value of k. (aiii) ( )( ) 2 2 89 8 9 0 9 1 0 9 or 1 (N.A.) kk kk kk kk =+ − − = − + = = =− M1 A1 (b) Ali invested $7500 in a saving account for 3 years. The rate of compound interest was fixed at r% per annum. At the end of 3 years, there was $8436.48 in his account. Find the value of r. b ( ) 3 3 3 $8436.48 $7500 1 100 $8436.481 100 $7500 $8436.481 100 $7500 1 1.04100 1.04 1 100 4 r r r r r r =+  += += += = −  = M1 M1 A1

3 2 The diagram shows a rhombus ABCD, with sides 5 units, and coordinates ( )2,0A − and ( )1, 4D . (a) Show that the equation of line BC is 3 4 12yx=− . (a) ( )( 2 5,0) 3,0BB− + = 40Gradient 1 ( 2) 4 3 −= −− = ( ) 4 3 403 3 4 4Equation of : 4 3 3 4 12 y x c c c BC y x yx =+ =+ =− =− =− M1: coordinates of B or C M1 M1 E is a point on BC, such that DE is perpendicular to AD. The gradient of the line DE is 3 4− . (b) Find the coordinates of E. B C x y O

4 (b) ( ) 3 4 341 4 19 4 3 19Equation of : 44 3 19 4 44 4 3 21 5 8 5 y x c c c DE y x xx x y =− + =− + = =− + − + = − = = Coordinates of E: 21 8 1 3, or 4 ,15 5 5 5            M1 M1 A1 (c) Given that the length of BE = 2 units, hence, find the area of ABED. (c) 22 21 81455 16 4 units DE    = − + −       = = ( )( ) 2 1Area of 2 5 4 2 14 units ABED=+ = M1, accept shoelace method A1 3 (a) A group of 24 tourists visited the National Museum. One of the 24 tourists is selected at random. The probability that it is a Korean male tourist is 1 3 . By showing clear workings, complete the table of information about the 24 tourists. Male Female Korean 5 Japanese 2 (a) Number of Korean male tourists 1 243 8 = = M1

5 Male Female Korean 8 5 Japanese 7 2 A1 (b) Two of the 24 tourists are selected at random. Find the probability that they are both male tourists. (b) ( )P both tourists are males 8 7 8 7 1 24 24 1 35 92 + + −  =   −   = M1 A1 4 Terry pays $15 a month for a season parking of his motorcycle in his workplace. James, also a motorist, uses a different scheme in which he pays a deposit of $50 and then a monthly payment of $8. Let the number of months that both Terry and James have been paying for their motorcycle be x. By forming an inequality, find the minimum number of months James will have to pay to ensure that his scheme is cheaper than Terry. 4 50 + 8x < 15x 50 < 7x 17 7x ⸫ James will have to pay for a minimum of 8 months for his scheme to be cheaper than Terry. M1 M1 A1 B1 5 In the diagram, A, B, C and D are four points on a circle with centre O. Angle OAB = 2x, angle OCB = 3x, and angle ADC = 2.5x. (a) Express angle ABC in terms of x. D C B A O

6 (a) Angle 2 (isoceles triangle, = same radii) Angle 3 (isoceles triangle, = same radii) Angle 2 3 5 OR 180 (angles in opposite segments are supp lementary) 2.5 180 1 ABO x OA OB OBC x OB OC ABC x x x ABC ADC ABC x ABC = = = +  =  + =   + =  = ( )80 2.5 x− M1 A1 M1 A1 (b) Find the value of x. (b) 180 (angles in opposite segments are supp lementary) 5 2.5 180 7.5 180 24 ABC ADC xx x x  + =  + =  = = M1 A1 6 (a) Given that 5 3 3 4 186 x a x a−− −= , express x in terms of a. (a) 5 3 3 4 186 3(5 3 ) 4(3 4 ) 124 15 9 12 16 24 3 24 7 24 7 3 x a x a x a x a x a x a xa ax −− −= − − − = − − + = =− −= M1 for combining into a single algebraic fraction A1 for expressing x in terms of a (b) Find the largest positive integer value of a if 29x − . (a) Given that x ≥ −29, 24 7 293 24 7 87 7 87 24 111 7 615 7 a a a a a − − − − − − − − −  ⸫The largest integer value of a = 15 M1 for forming this inequality M1 for solving this inequality A1

7 7 (a) ABCD is a major segment of a circle, centre O. BD is 32 cm, AC is 48 cm and angle DBA is 2  . (i) Show that the radius of the circle is 25 cm. (ai) Let the radius of the circle be r cm. ( ) ( ) ( )( ) ( ) 2 22 222 48 322 32 24 32 32 576 2 32 32 576 5762 32 32 2 32 18 2 50 25 rr rr r r r r r r r r r = + − − − = − + + − = −= −= −= = = The radius of the circle is 25 cm. (shown) M1 for forming an equation to solve for radius. M1 for solving the equation to find the radius A1 (ii) Calculate the area of the major segment ABCD. (aii) 24sin 25 1.2870 1.2870 2 2.574 AOB AOB AOC = =  =   M1 for finding angle AOC. O A B C D 32−r 24 cm O A B C D r

8 ( ) ( ) ( ) ( ) ( ) ( ) 22 22 2 Area of major segment = Area of major sector + Area of triang le 11 reflex sin acute 22 1125 2 2.574 25 sin 2.57422 1327.12 1330 cm AODC OADC OABC r AOC r AOC  =  +  = − +   M2 (one mark for each part) A1 (b) The sector OADC was cut from the above to form a cone where the OA is glued to OC. (i) Calculate the circumference of the base circle of the cone. (bi) ( ) Circumference of the base of the cone = arc length of the major sector = reflex 25(2 2.574) 92.730 92.7 cm OADC r AOC   =−   M1 A1 (ii) Hence, calculate the vertical height of the cone. (bii) m Radius of base circle 92.730= 14.758 c 2  ( ) ( ) ( ) 222 222 22 Let the vertical height be cm. 25 14.758 25 14.758 25 14.758 20.179 20.2 h h h h h h =+ =− =−   The vertical height is about 20.2 cm M1 M1 A1 8 (a) The heights of a group of 30 students were measured and the results are shown in the stem-and-leave diagram. 14 4 5 5 7 7 7 8 8 8 9 9 9 9