Regent 4E5N AM Prelim Examination Paper 2 QP

1 NAME: ________________________________ INDEX NUMBER: ________ CLASS: ________ SETTER: MS SU RY _________________________________________________________________________ ADDITIONAL MATHEMATICS 4047/02 Paper 2 15 September 2020 2 hours 30 minutes Candidates answer on the Question Paper. No Additional Materials are required. _________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Give your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figure s, 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 of the marks for this paper is 100. _________________________________________________________________________ This document consists of 20 printed pages. TARGET PARENT’S SIGNATURE REGENT SECONDARY SCHOOL PRELIMINARY EXAMINATION 2020 SECONDARY FOUR EXPRESS 100

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation , Binomial expansion where n is a positive integer and 2. TRIGONOMETRY Identities AA 22 tan1sec += 22cosec 1 cotAA=+ ( ) BABABA sinsincoscoscos = Formulae for 02 =++ cbxax a acbbx 2 42 −−= ( ) nrrnnnnn bbar nbanbanaba ++     ++     +     +=+ −−− ......21 221 ( ) ! )1)...(1( ! ! ! r rnnn rrn n r n +−−=−=      1cossin 22 =+ AA ( ) BABABA sincoscossinsin = ( ) BA BABA tantan1 tantantan  = AAA cossin22sin = AAAAA 2222 sin211cos2sincos2cos −=−=−= A AA 2tan1 tan22tan −= ABC C c B b A a sinsinsin == Abccba cos2222 −+= Abcsin2 1=

3 1 The roots of the quadratic equation 22 4 3 0xx− + = are  and  . (i) Show that 22 1+= . [4] (ii) Find the quadratic equation whose roots are 2 3 + and 2 3 + . [3]

4 2 (i) In the binomial expansion of 7 kx x  + , where k is a positive constant, the coefficients of x and x3 are the same. Find the value of k. [6]

5 (ii) Using the value of k found in part (i), find the coefficient of x7 in the expansion of ( ) 7 215 kxx x −+  . [4]

6 3 Recorded values of the mass, m grams, of a radioactive substance, t hours after observations began, are shown in the table below. t (hours) 2 4 6 8 10 m (mg) 48.2 41.5 35.7 30.8 26.5 It is known that m and t are related by the equation 0 ktm m e −= , where 0m and k are constants. (i) Plot mln against t and draw a straight line graph. [3] Use your graph to estimate (ii) mass of the substance when the observations began, [2] (iii) the value of k, [2] (iv) the time taken for the substance to lose half of its original mass. [2]

7 3 (i)

8 4 The equation of a curve is 226y x kx k= + + − , where k is a constant. (i) Find the range of values of k for which the curve lies completely above the x-axis. [3] (ii) In the case where k = 2, find the values of m for which the line 4y mx=− is a tangent to the curve. [4]

9 5 The diagram shows a pentagon ABCEF formed by a rectangle ABDF and two isosceles triangles DEF and BCD, where CDE is a straight line. The angle BDC is θ, BC = CD = 4 cm, DE = EF = 3 cm. (i) Show that P cm, the perimeter of the pentagon, can be expressed in the form . [3] (ii) Express P in the form 14 sin( )R ++ where 0R and .900   [3]  

10 (iii) Find the value of  for which P = 23 cm. [3]