MGS 2024 Sec 4 Prelim AM P2 Solution

This question paper consists of 19 printed pages and 1 blank page. Class Index Number Name : __________________________________________ METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2024 Secondary 4 Tuesday ADDITIONAL MATHEMATICS 4049/02 13 August 2024 PAPER 2 2 hours 15 mins Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. Write in dark blue or black pen You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. Give non-exact numerical answers correct to 3 significant figure, 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. 90

Page 2 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities 22cos ec 1 cotAA= + AAA A A 2222 sin2 1 1cos2sin cos2cos − = − = − = Formulae for ∆ABC sin sin sin abc ABC= = a2 = b2 + c2 − 2bc cos A ∆ = 2 1 bc sin A 02 = + +c bx ax a ac b bx 2 42 − ± −= ( ) nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− 2 21 2 1 ( ) ! ( 1)...( 1) !! ! n n nn n r r rnr r  − −+= = − 1cos sin22 = +A A AA 22 tan1sec + = B A B A B Asin cos cos sin)sin( ±= ± B A B A B Asin sin cos cos)cos( = ± B A B AB A tan tan1 tan tan)tan(  ±= ± A A Acos sin2 2sin = A AA 2tan1 tan22tan −=

Page 3 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 1 (a) Solve the equation 21 1xx+− = . [3] 21 1xx+− = 2 11xx+=+ 2 112x xx+=+ + M1 2xx= 2 4xx= M1 ( 4) 0xx −= 0x= or 4x= A1 (b) Express ( ) 2 235 32 − − in the form 3pq + where p and q are integers. [4] ( ) 2 235 32 − − 12 20 3 25 3 2 32 32 −+ += × −+ M1 ( ) ( )37 20 3 3 2 34 −+ = − M1 ( )37 3 74 60 40 3= − +−− M1 3 3 14= − A1

Page 4 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 2 The diagram shows a circle passing through the vertices of a triangle ABC. The tangent to the circle at A meets BC extended at point D. The tangent at C meets AD at E. (i) Prove that angle CED is twice of angle ABC. [3] Let angle EAC = x and angle ECD = y angle ECA = x (base angles, isosceles triangle) M1 angle ABC = x (alternate segment theorem) M1 angle CED = 2x (exterior angle = sum of interior opposite angles) A1 Therefore, angle CED is twice of angle ABC. (ii) By proving a pair of similar triangles, show that 2BD CD AD×= . [3] angle ABD = angle CAD (alternate segment theorem) M1 angle ADB = angle CDA (common) Therefore, triangle ABD is similar to triangle CAD. M1 AD BD CD AD= 2BD CD AD×= A1

Page 5 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 3 The mass, m grams, of a radioactive substance detected in a piece of stone is given by the formula ktme α −= , where 0α ≠ , k is a constant and t is the time interval in months. (i) The mass of the substance is reduced to half its original value four months after it was first being detected, find the value of k. [2] ktme α −= (4)1 2 keαα −= 1ln 42 k =− M1 (4)1 2 keαα −= 0.1732867...k = 0.173k = A1 (ii) Find the initial mass of the substance given its mass after 1 month is 0.25 g. [2] 0.1732867(1)0.25 eα −= 0.1732867 0.25 e α− = M1 0.29730177...α = 0.297α = A1 (iii) Calculate the time taken for the mass to reduce to 0.01 g. [2] 0.17328670.01 0.29730177 te−= 0.17328670.01 0.29730177 te−= M1 0.01ln 0.17328670.29730177 t =− 19.575...t = 19.6t = A1

Page 6 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 4 (a) Solve the equation 3cos 2 sin 2xx= + , for 0 360x°≤ ≤ ° . [4] 23(1 2sin ) sin 2xx−= + M1 26sin sin 1 0xx+ −= (3sin 1)(2sin 1) 0xx− += M1 1sin 3 19.5 ,160.5 x x = = °° 1sin 2 210 ,330 x x =− =°° A2

Page 7 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 (b) Find all the angles between 0 and 5 which satisfy the equation 2sin 2 3cos 0xx+= . [4] 2sin 2 3cos 0xx+= 22sin cos 3cos 0xx x += M1 cos (2sin 3cos ) 0xx x += M1 cos 0 3, 22 x x ππ = = or 2sin 3cos tan 1.5 2.16 xx x x =− =− = A2

Page 8 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 5 (a) (i) Given that m is a constant, expand 4(3 ) mx+ , in ascending powers of x, simplifying each term in your expansion. [2] 4 4 3 22 3 4 (3 ) 44 43 (3 ) (3 )( ) (3)( ) ( )12 3 mx mx mx mx mx +   = ++ + +     M1 22 33 4481 108 54 12m x mx mx mx= ++ + + A1 (ii) Given also that the coefficient of x is equal to the coefficient of x 2, find the value of m. [1] 2108 54 2 mm m = = A1 (b) (i) By considering the general term in the binomial expansion of 9 13 2x x  − , explain why there are no even powers of x in this expansion. [3] 99 1(3 ) 2 r rxr x − −  M1 Powers of x 9 92rr r=−−=− M1 2 is always even for all integer values of The difference between an odd and even number is always odd. Thus, there are no even powers of in the expansion rr x A1

Page 9 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 (ii) Using the value of m found in part (a)(ii), find the term independent of x in the expansion of 9 4 1(3 ) 3 2mx x x +−  . [4] 9 22 33 44 234 1 3 1(81 108 54 12 ) 3 2 (81 216 216 96 16 )(..... ..... .....) m x mx mx mx x x x xxx x x −− ++ + + −  = ++ ++ M1 92 1 92 1 2 10 5 rxx r r r −− = −= − = = 92 3 92 3 2 12 6 rxx r r r −− = −= − = = M1 45 3699 11cofficient = 216 (3) ( ) 96 (3) ( )56 22  × − +× −   M1 130977= 2− A1

Page 10 of 20 Methodist Girls’ School Additional Mathematics Paper 2 Sec 4 Preliminary Exam 2024 6 A circle C has equation 22 6 4 12xy xy++−= . (i) Find the centre and the radius of C. [3] 22 22 6 4 12 ( 3) ( 2) 25 xy xy xy ++−= + +− = M1 centre is ( 3,2) radius = 5 units − A2 The points P (–8, 2) and Q (1, –1) lie on the circumference of C. (ii) Determine whether PQ is a diameter of C. [2] 22( 8 1) (2 1) 9.4868 10 is not a diameter of PQ C −− + + = ≠ M1 A1