2023 CGS Sec 4 AM Prelim P2 Solutions

Name: Name: Solutions Register No.: Class: CRESCENT GIRLS’ SCHOOL SECONDARY FOUR 2023 PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/02 Paper 2 25 August 2023 Candidates answer on the Question Paper. 2 hours 15 mins No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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  . At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. Question 1 2 3 4 5 6 7 8 9 10 11 Marks Table of Penalties Qn. No. Presentation –1 Significant Figures/ Units –1 Parent’s/Guardian’s Signature For Examiner’s Use This document consists of 20 printed pages. 90

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 =++ cbxax , a acbbx 2 42 −−= Binomial expansion 1 2 2( ) ... ... 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ! ( )...( 1) !( )! ! n n n n r n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 1cossin 22 =+ AA AA 22 tan1sec += 22cosec 1 cotAA=+ BABABA sincoscossin)sin( = cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= Formulae for ABC C c B b A a sinsinsin == Abccba cos2222 −+=  = Cabsin2 1

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 3 1 (i) The equation of a curve is 2 1() 34 xfx x += + , .xp Find '( )fx , simplifying your answer as a single fraction. Hence determine the gradient of the tangent at the point on the curve where x = 0. [5] 11 22 22 2 1(3 4). ( 1) (2 ) ( 1) (3)2'( ) (3 4) x x x x fx x − + + − + = + = 2 2 2 (3 4) 31 1 (3 4) xx x x x + −+ + + = 2 22 (3 4) 3( 1) 1(3 4) x x x xx + − + ++ = 22 43 1(3 4) x xx − ++ When x = 0, gradient of tangent is 22 4(0) 3 0 1(3(0) 4) − ++ 3 16=− (ii) State the value of p. [1] 113p=−

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 4 2 By using a suitable substitution, show that 434 xxee −= has only one solution and find its value correct to 2 significant figures. [5] ( ) 2 34 xxee −= Let xye= 23 4 0yy− − = (3 4)( 1) 0yy− + = 4 3y= or 1y=− 4 3 xe = or 1xe =− (NA since 0)xe  4ln 3x =  x = 0.083 (to 2 s.f.)

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 5 3 The diagram below shows a circle 22 9xy+= and a straight line 5y kx=+ . Find the range of values of k. [5] Sub 5y kx=+ into 22 9xy+= 22 ( 5) 9x kx+ + = 2 2 2 10 25 9 0x k x kx+ + + − = 22(1 ) 10 16 0x k kx+ + + = Since line cuts the curve at 2 distinct points, 2 40b ac− . 22(10 ) 4(1 )(16) 0kk − +  22100 64 64 0kk− −  24(9 16) 0k − (3 4)(3 4) 0kk+ −  4 3k− or 4 3k  Since k < 0, 4 3k− . x y y = kx + 5 0 x2 + y2 = 9

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 6 4 (a) Show that d ln(sin cos ) tand4 x x xx + = −  . [4] d 1 dln(sin cos ) (sin cos )d sin cos dx x x xx x x x + = + + = cos sin sin cos xx xx − + = cos sin cos cos sin cos cos cos xx xx xx xx − + = 1 tan 1 tan x x − + = tan tan4 tan tan4 x x   − + = tan 4 x − (shown) (b) Solve tan 34 x − =− for 0 x  . [2] tan 34 x − =− tan 3 4x  − − =−  tan 3 4x  −= Basic angle = 1.2490 1.24904x −= 2.03x= (to 3 s.f.)

Crescent Girls’ School 2023 Prelim Sec 4 A Math P2 7 5 (i) Differentiate 13 cos 2xx with respect to x. [3] d 1 d 1 1 d3 cos 3 cos cos (3 )d 2 d 2 2 dx x x x x xx x x    =+        = 1 1 13 sin cos 32 2 2x x x   −+       = 3 1 1sin 3cos2 2 2x x x−+ (ii) Hence, find the exact value of 3 0 1sin d .2x x x   [4] d 1 3 1 13 cos sin 3cosd 2 2 2 2x x x x xx =− + 3 1 1 d 1sin 3cos 3 cos2 2 2 d 2x x x x x x=− 1 1 d 1sin 2cos 2 cos2 2 d 2x x x x x x=− 3 33 00 0 1 1 1sin d 2cos d 2 cos2 2 2x x x x x x x  =−  = 3 0 114sin 2 cos22x x x   −