PHS 2022 S4NA Prelim MA P2 MS

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL MATHEMATICS 4045/02 PAPER TWO 26 July 2022 Tuesday 2 hours PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL MARK SCHEME

2 Presbyterian High School 4045/02/4NA Prelim/2022 Section A (52 marks) Answer all the questions in this section. 1 The stem-and-leaf diagram shows the weight, in kilogram, of 20 students. 4 0 1 5 7 9 9 5 0 1 1 1 4 5 9 9 6 0 3 6 6 7 1 2 Key: 4 | 0 means 40 Find (a) the number of students with weight more than 59, Answer 6 students [B1] [1] (b) the median weight. 51 54median 52.5 2 +== Answer 52.5 kg [B1] [1] 2 The amounts of flour, butter and sugar required to bake a cookie are in the ratio 3 : 2 : 1. The total weight of all three ingredients is 450 grams. Find the amount of each ingredient used. 3flour 450 2256 2butter 450 1506 1sugar 450 756 =  = =  = =  = Answer flour: 225 g [B1] butter: 150 g [B1] sugar: 75 g [B1] [3]

3 Presbyterian High School 4045/02/4NA Prelim/2022 [Turn over 3 (a) Calculate (i) the exact value of 9.3 4.62 5.7 2.1  − , Answer 11.935 [B1] [1] (ii) 23 15 2.57+ . Give your answer correct to 1 decimal place. 23 15 2.5 2.2507+= Answer 2.3 [B1] [1] (b) The table shows the number of coronavirus cases of four countries in Europe. Country Number of cases (millions) France 30.51 Germany 27.77 Russia 18.41 Italy 18.18 (Source: https://www.worldometers.info/coronavirus/) Calculate the total number of coronavirus cases of the four countries. Give your answer in standard form. 7 total number 30.51 27.77 18.41 18.18 [M 1] 94.87 million = 9.487 10 = + + + =  Answer 79.487 10 [A1] [2] 4 (a) Expand and simplify ( )( )4 1 3xx+− . ( )( ) 2 2 4 1 3 4 12 3 [M1 - 2 correct expansion] 4 11 3 x x x x x xx + − = + − − = − − Answer 24 11 3xx−− [A1] [2] (b) (i) Solve the inequality 3 74x . Answer 224 3x [B1] [1] (ii) Write down the largest integer value for x. Answer 24 [B1] [1]

4 Presbyterian High School 4045/02/4NA Prelim/2022 5 (a) Write 1008 as a product of its prime factors. Answer 422 3 7 [2] [B2 – 3 correct, B1 – 2 correct] (b) What is the largest perfect square that is a factor of 1008? 422 3 144= Answer 144 [B1] [1] (c) 233780 2 3 5 7=    Write down the lowest common multiple of (LCM) of 1008 and 3780. Give your answer as a product of its prime factors. 43LCM 2 3 5 7=    Answer 432 3 5 7   [B1] [1]

5 Presbyterian High School 4045/02/4NA Prelim/2022 [Turn over 6 The diagram shows a quadrilateral ABDE. ACD and BCE are straight lines and AB is parallel to ED. (a) Complete the sentence. The mathematical name of quadrilateral ABDE is trapezium [B1] . [1] (b) Given that triangle ABC is similar to triangle DEC, identify all pairs of angles in triangle ABC and triangle DEC that are equal. Answer ABC DEC BCA ECD BAC EDC  =  =  = [B2 – 3 correct pairs, B1 – 2 correct pairs] [2] (c) AB = 8 cm, BC = 6.5 cm and DE = 12 cm. (i) Find the scale factor of enlargement of triangle ABC to triangle DEC. Answer 3 2 [B1] [1] (ii) Calculate EC. 8 6.5 [M1]12 12 6.5 8 9.75 EC EC = = = Answer 9.75 cm [A1] [2] B A E D C 8 12 6.5 6.5

6 Presbyterian High School 4045/02/4NA Prelim/2022 7 (a) Factorise completely 3 6 2 4ax ay x y− + − . ( ) ( ) ( )( ) 3 6 2 4 3 2 2 2 [M1 - common factor of 2 or 3 ] 3 2 2 ax ay x y a x y x y a a x y − + − = − + − = + − Answer ( )( )3 2 2a x y+− [2] (b) Rearrange this equation xby bx= − to make b the subject. ( ) ( ) [M1] [M1] xby bx y b x xb yb yx xb yb xb yx b y x yx yxb yx = − −= −= −= −= = − ( ) ( ) [M1] [M1] xby bx y b x xb yb yx xb yx xb yb yx b x y yxb xy = − −= −= − = − − = − =− − Answer b = yx yx− [A1] [3]

7 Presbyterian High School 4045/02/4NA Prelim/2022 [Turn over 8 This table of values is for 32 3 6 8y x x x= − − + . x – 3 – 2 – 1 0 1 2 3 4 5 y p 0 10 8 0 – 8 q 0 28 (a) Calculate the value of p and the value of q. Answer p = –28 [B1] q = –10 [2] (b) On the grid provided, plot the graph of 32 3 6 8y x x x= − − + for 35 x−   . Answer on next page [3] [P2 – 9 points, P1 – 5 points] [C1 – smooth curve] (c) Find the value of x when y = 20. Answer x = 4.8 [B1] [1] (d) By drawing a tangent, find the gradient of the curve 32 3 6 8y x x x= − − + when x = 4. tangent at x = 4 [M1] ( )15 18 4.8 3 18.3 m −−= − = Answer 18.3 [B1] [2] (range : 17 – 19) Answer: 8(b)

8 Presbyterian High School 4045/02/4NA Prelim/2022 9 PRS is a right-angled triangle and PQR is a straight line.

9 Presbyterian High School 4045/02/4NA Prelim/2022 [Turn over PS = 10 cm, SR = 6 cm, angle 90QRS = and angle 32QSR= . (a) Find sin SPR . Give your answer as a fraction in its lowest terms. 63 10 5= Answer 3 5 [B1] [1] (b) Calculate the length of QS. 6cos 32 [M1] 6 cos 32 7.0750 QS QS = =  = Answer 7.08 cm [A1] [2] (c) Show that the angle PSQ is 21.1 . Answer 1 3cos [M1]5 3cos 5 53.130 53.130 32 21.1 [A1] PSR PSR PSQ − = =  =  = −  = [2] (d) Calculate the area of triangle QPS. ( )( ) ( )1 7.0750 10 sin 21.130 [M1]2 12.75216  = ( )( ) ( )1 7.0750 10 sin 21.1 [M1]2 12.7348  = Answer 12.8 cm2 or 12.7 cm2 [A1] [2] S P R Q 6 10 32o

10 Presbyterian High School 4045/02/4NA Prelim/2022 10 Mr Lo owns a car. He usually drives 18 km from home to work for 30 minutes daily. (a) A car wheel is made up of steel rim and rubber tyre. The outer diameter of the car wheel is 66 cm. The diameter of the steel rim is 41 cm. (i) Find the section height of the tyre. 66 41 12.52 − = Answer 12.5 cm [B1] [1] (ii) One revolution of a wheel will make the car move a distance equal to its circumference of the wheel. How many complete revolutions does the car wheel need to make from home to work? ( )circumference 66 [M1] 1800000number of complete rotations 66 8681.17   = = = Answer 8682 revolutions [A1] [2] (b) On a particular day, he drove a further 6 km in 15 min. Find the average speed, in km/h for the whole of Mr Lo’s journey. 18