CCHM 4048 02 AS

2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 [Turn over Name: Class: Class Register Number: PRELIMINARY EXAMINATION 2022 SECONDARY 4 MATHEMATICS 4048/02 Paper 2 Wednesday 31 August 2022 Candidates answer on the Question Paper 2 hours 30 minutes This document consists of 23 printed pages and 1 blank page. Solutions

2 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 Mathematical Formulae Compound interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length = , where is in radians Sector area = , where is in radians Trigonometry Statistics Mean = f fx   Standard deviation = 22        −  f fx f fx nrP      + 1001 lr  2 4 r hr 3 1 2 3 3 4 r Cba sin 2 1 r  2 2 1 r  C c B b A a sin sin sin == Acbcba cos 2 222 −+=

3 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 [Turn over [Turn over for Question 1]

4 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 1 (a) Express as a single fraction in its simplest form ( )31 68 xx −− . Answer …………………………………….. [2] (b) Simplify 1 6 3 9 27b w −   . Answer …………………………………….. [2] (c) Simplify 2 2 5 12 4 25 20 4 pp pp −+ −+ . Answer …………………………………….. [3] ( ) ( ) 31 68 914 24 24 4 9 9 24 95 24 xx xx xx x −− −=− −+= −= ( )( ) ( )( ) 2 2 5 12 4 25 20 4 2 5 2 5 2 5 2 2 52 pp pp pp pp p p −+ −+ −−= −− −= − 1 3 1 6 3 9 1 9 3 6 3 2 3 2 27 27 27 3 b w w b w b w b −   = = = 95 24 x− 3 23 w b 2 52 p p − −

5 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 [Turn over (d) Solve the equation 31 122xx +=−− . Answer …………………………………….. [3] 31 122 31 122 2 12 22 4 xx xx x x x +=−− −=−− =− −= = ( ) ( )( ) ( )( ) 2 2 31 122 3 2 2 122 6 3 2 4 4 6 8 0 4 2 0 4 or 2 (rej. 2) xx xx xx x x x x xx xx x x x +=−− − + − =−− − + − =− + − − + = − − = = =  Alternative 4x=

6 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 2 A is the point (−5, −2) and B is the point (−1, 3). (a) Find the distance between point A and point B. Answer ……………………………….. units [2] (b) Line p passes through point A and point B. Show that the equation of line p is 4 5 17yx=+ . Answer [3] ( )( ) ( ) 2 2 distance 5 1 2 3 16 25 41 6.40 units (3 s.f.) AB= − − − + − − =+ = = ( ) ( ) 32gradient 15 5 4 −−= − − − = ( )( ) ( ) equation: 531 4 531 4 4 12 5 5 4 5 17 (shown) yx yx yx yx − = − − − = + − = + =+ Alternative equation: 5 4 sub. ( 1,3) 53 ( 1)4 17 4 5 17 44 4 5 17 (shown) y x c c c yx yx =+ − = − + = =+ =+ 6.40

7 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 [Turn over (c) The equation of line q is 8 7 43yx−= . Find the coordinates of the point of intersection of line p and line q. Answer (…………….…… , ….…….….…) [3] (d) The equation of another line r is 2 2.5y x k−= , where k is a constant and 8.5k  . Without solving algebraically, explain why line p and line r will never meet. Answer ……………………………………………………………………………………………... ……………………………………………………………………………………………... …………………………………………………………………………………………. [2] 4 5 17 --- (1) 8 7 43 --- (2) yx yx =+ −= (1) x 2: 8 10 34 --- (3)yx=+ (3) - (2): 7 10 9xx=− 39 3 x x = = sub. x = 3 into (1): 4 15 17 8 the point of intersection is (3,8). y y =+ =  Line r : 2 2.5 -- (1) (1) 2: 4 5 2 5gradient of line 4 y x k y x k r −=  = + = 2 17k  Since line r and line p have the same gradient, they are parallel lines or lines that overlap. As 2 17k  , they do not have the same y-intercept and are parallel lines. Therefore, line p and line r will never meet.t. 3 8

8 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 3 An empty pool has a capacity of 20 000 litres. Hose A can fill the pool at a rate of x litres per minute. Hose B can fill the pool at a rate of (x – 8) litres per minute. (a) Write down an expression, in terms of x, for the number of minutes it would take to fill the pool using hose A. Answer …………………………… minutes [1] (b) Write down an expression, in terms of x, for the number of minutes it would take to fill the pool using hose B. Answer …………………………… minutes [1] (c) It takes 2 hours longer to fill the pool using hose B than it does using hose A. Write down an equation to represent this information and show that it reduces to 23 24 4000 0xx− − = . 20000number of minutes to fill pool using hose A minx= 20000number of minutes to fill pool using hose A min 8x= − ( ) ( ) ( ) ( ) 2 2 20000 20000 1208 20000 20000 8 =1208 160000 1208 120 8 160000 120 960 160000 0 3 24 4000 0 (shown) xx xx xx xx xx xx xx −=− −− − =− −= − − = − − = 20000 x 20000 8x−

9 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 [Turn over (d) Solve the equation 23 24 4000 0xx− − = , giving your solutions correct to two decimal places. Answer x = …………..….… or .….……………. [3] (e) Calculate how long it would take to fill the empty pool using hose A and hose B together. Give your answer in hours and minutes, correct to the nearest minute. Answer ....…….……hours …….………minutes [2] ( ) ( ) 2 24 24 4(3)( 4000) 2(3) 24 48576= 6 40.73327... or 32.73327... 40.73 (2 d.p.) or 32.73 (2 d.p.) x − −  − − −=  =− =− 20000time 8 20000= 28 20000 2(40.73327...) 8 272.23277...mins 4 h 32 mins (nearest minute) xx x = +− − = − = = 40.73 32.73− 4 32

10 2022 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4048/02 4 ABC is a triangle where D is a point on AC. AB= a , AC = b and AD : DC = 3 : 2 X is a point on BD such that BX : XD = 1 : 2 . (a) Express, as simply as possible, in terms of a and/or b, (i) CB Answer ………………………………………….. [1] (ii) DB Answer ………………………………………….. [2] (iii) CX Answer ………………………………………….. [2] A B C D X = CB AB AC=− −ab 3= 5 DB AB AD=− −ab 22= 53 2 2 3 5 3 5 2 2 2 5 3 5 24 35 CX CD DX DB =+ − =− −  =− − − b+ b + a b b + a b = a b b a −ab 3 5−ab 24 35 −ab