Greendale Sec 2024 Prelim P2 with Solutions

Name:_______________________________________ ( ) Write in BLOCK Letters Class: S____. TG:______ This document consists of 17 printed pages including this cover page. Greendale Secondary School 2024 GREENDALE SECONDARY SCHOOL Preliminary Examination 2024 Additional Mathematics 4049/02 Paper 2 26 August 2024 Secondary 4 Express / 5 Normal Academic 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your index number and name 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 staples, paper clips, glue or correction fluid. 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. 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 number of marks for this paper is 90. My Target BEFORE My Reflection: My Target AFTER Qn 1 2 3 4 5 6 7 8 9a 9b 10a,b 10c Strand G2 G3 A3 G1 C1 G1 A4 C1 A2 A4 C2 C1 Marks 6 6 8 8 9 9 10 10 5 6 8 5 90

Greendale Secondary School 2 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial expansion nr r nnnn n b b ar nb anb ana b a + +     + +     +     + = + −−− ......21) ( 2 21 , where n is a positive integer and .! ) 1 ( ... ) 1 ( ! )! ( ! r r n n n r r n n r n + − −=−=      2. TRIGONOMETRY Identities AAec AA A A 22 22 22 cot1cos tan1sec 1cossin + = + = = + A AA AA A A A A A A B A B AB A B A B A B A B A B A B A 2 2222 tan1 tan22tan sin2 1 1cos2sin cos2cos cos sin2 2sin tan tan1 tan tan)tan( sin sin cos cos)cos( sin cos cos sin)sin( −= − = − = − = = ±= ± = ± ±= ±   Formulae for ∆ABC C c B b A a sin sin sin= = A bc c b acos22 2 2− + = Area of ∆ = A bcsin2 1

Greendale Secondary School 3 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 Answer ALL questions. 1 The two variables x and y are related by the equation 3210 xy −+= . The diagram shows part of a straight line graph obtained when plotting lg y against x . The straight line cuts the lg -axisy and -axisx at points A and B respectively. (a) Find the coordinates of A and of B. [3] (b ) Point C lies on the straight line and the x-coordinate of point C is 1 2 . Find the value of y at point C. [1] (c) Find the value of tanθ. [2]

Greendale Secondary School 4 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 2 In the diagram below, the line ADF is a tangent to the circle at D. DE is a diameter and O is the centre. ABC is a straight line intersecting the circle at points B and C. (a) By showing that triangle ABD is similar to triangle ADC, prove that 2AD AB AC= × . [3] (b ) Given that G is the midpoint of DC, show that angle 90DGO= ° . [3] A B C D O F E G

Greendale Secondary School 5 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 3 (a) It is given that 1 5, 4 25 mn= + − where m and n are rational numbers. Find the value of m and of n. [2] (b) The sides AB and BC of a triangle, right-angled at B, are of length ( )32 5+ units and ( )2 25+ units respectively. Find the exact value of tan . ACB [3] (c) It is given that 11 15 and 165 11.a b ab+= + = − Without using a calculator, express a + b in the form 2,pq+ where p and q are integers. [3]

Greendale Secondary School 6 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 4 (a) Show that sin 2 cos 2 1 2cossin cos xx xxx ++ =+ . [3] (b) Hence, solve 2sin 4 cos 4 1 3cos 2 1sin 2 cos 2 θθ θθθ ++ = −+ for 180 180θ− °< < ° . [5]

Greendale Secondary School 7 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 5 It is given that ( ) 2ln 1 , 0xt t=+> . (a) Find d d x t . [2] (b) By writing t in terms of x, find d d t x , in terms of x. [4] (c) Show that dd dd xt ktx⋅= , where k is an integer to be found. [3]

Greendale Secondary School 8 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 6 The diagram shows a quadrilateral ABCD in which angle ADC θ= radians and angle angle 90DAB BCD= = °. 8 cm and 12 cmBC DC= = . (a) P is the perimeter of quadrilateral ABCD . Show that 20sin 4cos 20P θθ= ++ . [4] A B C D 8 12

Greendale Secondary School 9 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 (b ) Express P in the form sin( )Rk θα++ , where 0R> and 0 2 πα<< . [2] (c ) State the maximum value of P and the corresponding value of θ. [2] (d) Find the largest θ such that the perimeter of ABCD is at most 30 cm. [1]

Greendale Secondary School 10 Sec 4 Express/5 Normal Academic Preliminary Examination 2024 Additional Mathematics Paper 2 7 (a) Given that 323 3 7 ( 1)( )( 1)x x x x Ax B x C− − += + + −+ for all values of x, find the value of A, of B and of C. [3] (b) The polynomial 322 10x ax x b+−+ is exactly divisible by 2x− but leaves a remainder of 3 when divided by x + 1. Find the value of a and of b. [3]