KSS 2023 Prelims 4NA AMaths P1-1

Candidate Name: __________________________ ( ) Class: _________ KRANJI SECONDARY SCHOOL Preliminary Examination Secondary 4 Normal (Academic) ADDITIONAL MATHEMATICS 4051/01 Paper 1 Tuesday 1 Aug 2023 1 hr 45 min KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY K RANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRAN JI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY K RANJI SECONDARY KRANJI SECONDARY KRANJI KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI SECONDARY KRANJI READ THESE INSTRUCTIONS FIRST: Do not open this question paper until you are told to do so. Write your name and register number in the spaces at the top of this page. Write in dark blue or black pen. You may use 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. 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 70. ___________________________________________________________________________________ Setter: Mdm J Yap _________________________________________________________________ This question paper consists of 14 printed pages including the cover page [Turn over 4N(A) Session 2 Question Marks 1 2 3 4 5 6 7 8 9 10 11 12 13 Total

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ,02  cbxax a acbbx 2 42  2. TRIGONOMETRY Identities AA AA 22 22 tan1sec 1cossin   cosec2 A = 1 + cot2 A BA BABA BABABA BABABA tantan1 tantan)tan( sinsincoscos)cos( sincoscossin)sin(      A AA AAAAA AAA 2 2222 tan1 tan22tan sin211cos2sincos2cos cossin22sin    Formulae for ABC Cab Abccba C c B b A a sin2 1 cos2 sinsinsin 222   

3 1 Express 18 7 (1 ) ( 41 ) x xx   in partial fractions. [4] 2 A line has equation 12y x and a curve has equation 235y xx  . Determine, with reasons, whether the line intersects, is a tangent to, or does not intersect the curve. [3]

4 3 (i) Mrs Lee has two rectangular flowerbeds. Flowerbed A has length (5 3)x m and breadth ( 2)x m. Flowerbed B has an area of 6 m2. Find the range of values of x such that the area of flowerbed A is greater than the area of flowerbed B. [4] (ii) If x is an integer, state the smallest possible value of x. [ 1 ]

5 4 The polynomial f( )x is given by 32f( ) 2 5 4 3 .xxxx  (i) Divide f( )x by 3.x [ 2 ] (ii) What can you deduce about 3?x [ 1 ] (iii) Solve the equation f( ) 0 .x  [ 2 ]

6 5 Factorise 3 8x  and explain why 2x is the only real root of the equation 3 80x  . [4] 6 (i) Express 224 3x x in the form 2(),ax h k where a, h and k are constants. [ 3 ] (ii) Hence state the maximum value of the curve 224 3y xx   . [1]

7 7 The volume, V cm3, of water in a container is given by 34 23 ,3 xVx  where x cm is the depth of water in the container. Given that water is being poured into the container at a constant rate of 52 cm3/s, find the rate at which the depth of water is increasing when x = 5 cm. [4]

8 8 A curve is such that 2 2 d 12 2d y xx  . The point (2, 15) lies on the curve and the gradient of the curve at this point is 27. Find the equation of the curve. [5]

9 9 (a) Find the principal values, in degrees, of (i) 1 3sin ( ),2  [1] (ii) 1cos ( sin 45 ).  [ 1 ] (b) (i) Write down the period and the amplitude of 3cos 2 1 .yx [ 2 ] (ii) Sketch the graph of 3cos 2 1yx for 0 180x   . [ 3 ]

10 10 In the triangle ABC, angle B = 90, (7 2 )AB cm and (7 2 )BC  cm. (i) Find the exact length of AC. [ 3 ] (ii) Express tan A in the form 7 3 ab , where a and b are integers. [3]