MGS 2024 Sec 4 Prelim AM P1

Class Index Number Name : __________________________________________ This question paper consists of 21 printed pages and 3 blank pages. METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2024 Secondary 4 Monday ADDITIONAL MATHEMATICS 4049/01 12 August 2024 Paper 1 2 h 15 min 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. 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 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation , Binomial Expansion , where n is a positive integer and . 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA= + 22cosec 1 cotAA= + 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  − −+= = − 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 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 1 (i) Express 24 24 49yx x=++ in the form 2()ax b c++ . [2] (ii) Hence, state the maximum value of 1 y and the corresponding value of x at which 1 y is maximum. [2]

Page 4 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 2 By using a suitable substitution, solve the equation 22 4(1 )xxee −= − . [3]

Page 5 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 3 Show that the equation 2 2 28x px p x++=+ has real roots for all real values of p. [3]

Page 6 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 4 A and B are acute angles such that 1sin( ) 4AB−= and 3sin cos 5AB = . Without using a calculator, find the value of (i) cos sinAB , [2] (ii) sin( )AB+ , [1] (iii) 2 tan cotAB . [2]

Page 7 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 5 A curve has the equation 21 3 xey x + = + , where 3− ≠x . (i) Find an expression for d d y x . [2] (ii) Hence, stating your reasons clearly, determine if the graph of 21 3 xey x + = + is decreasing or increasing for which x > 0. [2]

Page 8 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 6 Find the values of k for which the curve 22 (3 1) 4yx kx= +++ lies entirely above the line 223 5y xk= −+ . [5]

Page 9 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 7 A curve is such that f " ( ) 6 10xx= − . The tangent to the curve f( )yx= at the point (1, –2) passes through the point ( )0, 13 . Find the equation of the curve. [6]

Page 10 of 24 Methodist Girls’ School Additional Mathematics Paper 1 Sec 4 Preliminary Examination 2024 8 (a) Given that log 9x p= and log 6x q= , find the value of 2log p q x . . [3] (b) Solve the equation lg(3 1) lg(2 1) 2 lg 2xx++ −=− and explain why there is only one solution. [5]