AHS A Math Paper 1

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   NAME: _______________________ _____ ( ) CLASS: 4 ( ) 9o 11 September 2018 Tuesday 2 hours Additional Materials: 6 Writing Papers   READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen on both sides of the writing paper provided. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters and glue or correction fluid. Answer all the questions. Write your answers on the separate Writing Paper provided. 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 and attach the question paper on top of the scripts. 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 80. --------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Question Marks Question Marks Question Marks 1 7 13 2 8 Table of Penalties 3 9 4 10 Units 5 11 Presentation 6 12 Accuracy Parent’s Name & Signature: Date: Total: 80 This paper consists of 6 printed pages. S4 ANGLICAN HIGH SCHOOL SECONDARY FOUR PRELIMINARY EXAMINATIONS 2018 ADDITIONAL MATHEMATICS PAPER 1 [4047/01] [Turn over

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42  Binomial expansion nrrnnnnn bbar nbanbanaba                  ......21)( 221 , where n is a positive integer and ! )1)...(1( )!(! ! r rnnn rnr n r n       2. TRIGONOMETRY Identities 1cossin 22  AA AA 22 tan1sec  AAec 22 cot1cos  BABABA sincoscossin)sin(  BABABA sinsincoscos)cos(  BA BABA tantan1 tantan)tan(   AAA cossin22sin  AAAAA 2222 sin211cos2sincos2cos  A AA 2tan1 tan22tan   Formulae for ABC C c B b A a sinsinsin  Abccba cos2222  Area of Cab sin 2 1

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   Answer ALL questions 1. The product of the two positive numbers, x and y, where x > y, is 24. The difference between their squares is 14. Form two equations, and hence, find the exact values of the two numbers. [5] 2. Show that  2 1827 7 37 cd    where c and d are integers. [4] 3. (a) (i) Sketch the two curves 30.5y x and 8y x on the same axes for x > 0. [3] (ii) Find the coordinates of the intersection point. [2] (b) Solve the equation 23 xe . [3] 4 (i) Given that the line 2y  intersects the graph of 1 5 logyx at the point P, find the coordinates of P. [2] ( i i ) Sketch the graph of 1 5 logyx . [2] (iii) State the range of values of x for which y < 0. [1] 5 (i) Sketch the graph of 2 169yx . [2] ( i i ) Express 24 181 9xx  in the form  2 169pxq x , where p and q are constants. [2] (iii) A suitable straight line can be drawn on the graph in (i) to solve the equation 24 181 9xx  . (a) State the equation of this straight line. [1] (b) On the same axes, sketch the straight line and state the number of solutions of the equation 24 181 9xx  . [2] [Turn over

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   6. The diagram shows a circle passing through points A, B, C and D. The tangents from E meets the circle at B and D. Given that AD BF and triangle ABD is isosceles, where AB BD . Prove that (i) ABFD i s a p a r a l l e l o g r a m . [ 3 ] (ii) triangle BCD is similar to triangle DFE. [ 3 ] (iii) BD  EF = CD  DE.             [1] 7. (a) Sketch the graph of , for the interval . [2] (b) In the diagram, triangle ABC is a right-angle triangle, where  90ABC . D is a point of AB such that AD is 7 cm and BD is 2 cm. Given that 3 1cos ADC , (i) Find the exact length of BC. [ 1 ] (ii) Find the value of ACDtan in the form ba , where a and b are integers. [ 2 ] 2c os 12 xy   02 x   A B C D 7 cm 2 cm A B C D E F

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   8. The minute hand of a clock is 50 cm, measured from the centre of the clock, O, to the tip of the minute hand, M. The displacement, d cm, of M from the vertical line through O is given by btad sin , where t is the time in minutes past the hour. (i) Find the exact value of a and of b. [ 3 ] (ii) Find the duration, in each hour, where 25d . [ 3 ]   9. (i) Prove that 44 22 cos sin 4cot2 c osec2 sin cos      . [ 3 ] (ii) Hence, or otherwise, solve   2cosec4cossin sincos 22 44  for 0 180   . [3] 10. A curve is such that 2 2 dy dx  6x – 2 and P(2, – 8) is a point on the curve. The gradient of the normal at P is 1 2 . F i n d t h e e q u a t i o n o f t h e c u r v e . [ 7 ] 11. Find and simplify dy dx for the following: (i) y = ln cos x (ii) 2xxye e [ 4 ] O DE F I L M 50 cm d cm [Turn over

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   12. In the diagram, the curve x = (y  1)2 + 4 and the line y + x = 7 intersect at A and B. (i) Find the coordinates of A and of B. [ 3 ] (ii) Calculate the area of the shaded region. [ 4 ]                                       13. A particle moves in a straight line so that its acceleration, a ms-2, is given by 21 3at , where t is the time in seconds after passing a fixed point O. The particle first comes to instantaneous rest at 5st  . Find, (i) the velocity when the particle passes through O. [ 2 ] (ii) the total distance travelle d by the particle when it next comes to rest. [5] (iii) the minimum velocity of the particle. [ 2 ] *** End of Paper ***

AM-2018-AHS-EOY-P1 Anglican High School 2018 Sec 4 Prel AM Paper 1   Answer key 1  42 x  = 3 2 y   2  16 5 7    3  (a)(i)                                       (ii) (8, 1)    (b)  1.61 or 0 x    4  (i) (0.04, 2) (iii) x >1 (ii)    5  (i)  (ii)