ANDSS 4E5N2025AM WA2 (Qn+Ans)

This document consists of 7 printed pages. Setter: Mr Aaron Wong O ANDERSON SECONDARY SCHOOL Weighted Assessment 2 2025 Secondary Four Express/ Five Normal Academic CANDIDATE NAME: CLASS: / INDEX NUMBER: ADDITIONAL MATHEMATICS 4049 23 April 2025 45 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid/tape. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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 30.

WA2 4E/5N Add Maths 2025 2 1 Differentiate each of the following with respect to x. (a) 621 33 xx −−+ [1] (b) 31ln 52 x x + − [2] (c) 2 3sin 2 53 x x − [3]

WA2 4E/5N Add Maths 2025 3 2 The equation of a curve is 2(2 3) 4 1yx x= −+ . (a) Show that 2 2 d 16 12 2 d 41 yx x x x −+= + . [2] (b) Find the range of values of x for which 2(2 3) 4 1yx x= −+ is an increasing function. [4]

WA2 4E/5N Add Maths 2025 4 3 It is given that 43eexxyP Q −= + , and that 2 43 2 dd 2 12e 3edd xxyy xx −+= + . Find the value of each of the constants P and Q. [4]

WA2 4E/5N Add Maths 2025 5 4 The equation of a curve is 31 2 xyx x −= + − , where 0x> . (a) Find d d y x . [2] (b) Find the equation of the tangent to the curve at 1x= . [1] (c) Find the equation of the normal to the curve at 1x= . [2]

WA2 4E/5N Add Maths 2025 6 5 The table shows experimental values of two variables, x and y, which are connected by the equation e2 x kyA= + , where A and k are constants. x 2 4 6 8 y 2.135 2.368 3 4.718 (a) On the grid provided on the next page, draw a graph of ( )ln 2y− plotted against x. [3] (b) Use your graph to estimate the value of A and of k. [4] (c) On the same diagram, draw the straight line representing the equation 52e x y − −= . Hence find the value of x for which ln 5 xx Ak += − . [2]

WA2 4E/5N Add Maths 2025 7 END OF PAPER x 1 2 2 4 6 8 0 −1 −2 −3

WA2 4E/5N Add Maths 2025 1 2025 Secondary 4E/5N Additional Mathematics WA2 Mark Scheme 1 Differentiate each of the following with respect to x. (a) 621 33 xx −−+ [1] 62 5 3 53 d1 13 (6) ( 2)d3 3 22 xx x xx xx −− −  − + = −− = + (b) 31ln 52 x x + − [2] ( ) ( )( )d 31 dln ln 3 1 ln 5 2d 52 d 32 3 152 x xxx xx xx +  = +− − − = + +− OR ( ) ( ) ( )( ) 2 3(5 2 ) ( 2) 3 1 52d 31ln 31d 52 52 17 52 3 1 xx xx xxx x xx − −− + −+  = +− − = −+ (c) 2 3sin 2 53 x x − [3] ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 2 2 22 2 2 3 2 2 3 2 105 3 6cos 2 3sin 2 d 3sin 2 25 3 d 53 53 2 5 3 6cos 2 10 3sin 2 25 3 6 5 3 cos 2 15 sin 2 53 xxx x x x x x x x xx x x x xx x x −−  −= − − −− = − −− = − [B1] [M1] – applying logarithmic law [M2] 1m for ( )( ) 25 3 6cos 2xx− 1m for ( ) 2 10 3sin 2 25 3 x x x   − [A1] [A1] [M1] – differentiate directly [A1]

WA2 4E/5N Add Maths 2025 2 2 The equation of a curve is 2(2 3) 4 1yx x= −+ . (a) Show that 2 2 d 16 12 2 d 41 yx x x x −+= + . [2] ( ) ( ) ( ) ( ) 2 2 2 2 2 2 d 1124 1 2 3 8d2 41 24 1 4 2 3 41 16 12 2 41 y xx xx x x xx x xx x = ++ −  + ++ − = + −+= + (b) Find the range of values of x for which 2(2 3) 4 1yx x= −+ is an increasing function. [4] For y to be an increasing function, d 0d y x > , hence, ( ) ( )( ) 2 2 2 2 2 16 12 2 0 41 28 6 1 0 41 2 41 21 0 41 xx x xx x xx x −+ > + −+ > + −− > + Since 24 10x +> , then ( )( ) ( )( ) 216 12 2 0 2 41 210 41 210 xx xx xx − +> − −> − −> 11 or 42xx<> [M1] [A1] – Making a single fraction, leading to RHS [M1] [M1] [M1] [A1] [Shown]

WA2 4E/5N Add Maths 2025 3 3 It is given that 43eexxyP Q −= + , and that 2 43 2 dd 2 12e 3edd xxyy xx −+= + . Find the value of each of the constants P and Q. [4] 43 43 2 43 2 ee d 4e 3ed d 16 e 9 ed xx xx xx yP Q y PQx y PQx − − − = + = − = + Substituting d d y x and 2 2 d d y x into 2 43 2 dd 2 12e 3edd xxyy xx −+= + ( ) 4 3 4 3 43 4 3 43 1 6e 9e 2 4e 3e 1 2 e 3 e 24 e 3 e 12e 3e x x x x xx x x xx PQ PQ PQ − −− −− ++ − = + += + By comparing coefficients, 24 12P= 1 2P= and 1Q= [M1] [M1] – correct substitution or comparison of coefficients [A1] – Both P and Q [M1]