ࡱ> XZWFc 9bjbj D؎U\؎U\1 $ $ 444HHH8LHV !6!6!6!###b4#a#^###6!6!+Q)Q)Q)#l6!46!Q)#Q)Q)`,͛6!`6$$ \&0Veh%͛4͛Q)###m'###V#############$ > b:  Student Name: SPECIALIST MATHEMATICS Calculus: CAS Active Writing time: 75 minutes Total Marks: 50 SECTION 1 Question 1 The antiderivative of  EMBED Equation.3  is: A  EMBED Equation.3  B  EMBED Equation.3  C  EMBED Equation.3  D  EMBED Equation.3  E  EMBED Equation.3  Question 2 By using the appropriate substitution,  EMBED Equation.3  e2x EMBED Equation.3  dx is equal to: A  EMBED Equation.3  (u EMBED Equation.3  + u EMBED Equation.3 ) du B  EMBED Equation.3  (2u EMBED Equation.3  + u EMBED Equation.3 ) du C  EMBED Equation.3  (u EMBED Equation.3  + 2u EMBED Equation.3  + u EMBED Equation.3 ) du D  EMBED Equation.3  (u EMBED Equation.3  + 2u EMBED Equation.3 ) du E  EMBED Equation.3  (u EMBED Equation.3  + u EMBED Equation.3 ) du Question 3 The values of the constants, a and b, if y = a sin2x + b cos2x satisfies the differential equation  EMBED Equation.3  EMBED FXE300.Equation  EMBED Equation.3  2 EMBED Equation.3  + 3y = 9 sin2x + 2 cos2x, are: A a = 3, b = 1 B a = 1, b = 2 C a = 2, b = 1 D a = 1, b = 2 E a = 3, b = 2 Questions 4 and 5 refer to the following information. A tank containing 500L of water has 40 kg of sugar dissolved in it. Fresh water is pumped in at a rate of 5L/min and the resulting solution is pumped out at the same rate. The mixture is kept uniform. Question 4 If x g is the amount of sugar in the container at any time, t minutes, then the differential equation describing the rate of change of sugar in the container, in g/min, is: A  EMBED Equation.3  B  EMBED Equation.3  C  EMBED Equation.3  D  EMBED Equation.3  E  EMBED Equation.3  Question 5 The concentration of sugar in the container after 10 minutes is nearest to: A 0.07 g/L B 36.20 g/L C 362 g/L D 30 g/L E 0.35 g/L Question 6  EMBED Equation.3  is equal to: A  EMBED Equation.3  B  EMBED Equation.3  C  EMBED Equation.3  D  EMBED Equation.3  E  EMBED Equation.3  where c is a constant. Question 9 Using a suitable substitution,  EMBED Equation.3  can be expressed as  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Question 10 Point  EMBED Equation.3  is any point that lies on a particular curve. The gradient of a line joining point  EMBED Equation.3  and P is two more than the gradient of the tangent to the particular curve at P. The coordinates of point P satisfy the differential equation given by  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Question 11 Using an appropriate substitution,  EMBED Equation.3  can be expressed as  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Question 12  The graph of the function with the rule  EMBED Equation.3  is shown above. The antiderivative function has the rule  EMBED Equation.3 . For the graph of  EMBED Equation.3 , it is true to say that it has no points of inflection has no stationary points of inflection has no stationary points has two stationary points has one local maximum and one local minimum Question 13 The equation of the normal to the curve  EMBED Equation.DSMT4  at the point where  EMBED Equation.DSMT4  is given by A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  SECTION 2 Question 1 Let  EMBED Equation.3  where f has a maximal domain. Sketch the graph of  EMBED Equation.3  on the set of axes below. Indicate clearly on your graph the location of any x-intercepts the coordinates of any stationary points expressed as an exact value the equation of any asymptotes  4 marks If there are two solutions to the equation  EMBED Equation.3  where  EMBED Equation.3 , find a. Express your answer as an exact value. 1 mark That part of the graph of f for which  EMBED Equation.3 , is rotated around the x-axis to form a volume of revolution. i. Write down a definite integral that could be evaluated to find this volume of revolution. Find the exact value of the definite integral in part i.  EMBED Equation.3  Find the rule and the domain of the function  EMBED Equation.3  2 marks On the set of axes below, sketch the graph of  EMBED Equation.3 .  Indicate clearly on your graph the coordinates of any axes intercepts equation of any asymptotes 2 marks Find  EMBED Equation.3  and hence explain why the graph of  EMBED Equation.3  does not have a point of inflection. 1 mark Total 13 marks Question 2: A gelatine mixture of concentration 1.4 kg/L is added to a 120-litre tank of water at a constant rate of 3 L/min. At the same time, the solution is pumped out at 3 L/min, while the mixture is kept uniform by continuous stirring. Let G = amount of gelatine (kg) at time t minutes (i) Calculate the input rate of the gelatine in kg/min _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 1 mark (ii) Calculate the output rate of the gelatine in kg/min _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 1 marks (iii) Construct a differential equation to show the rate of change of concentration of gelatine _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 1 mark (iv) Assume that the water can set to jelly if the gelatine concentration reaches 1.2 kg/L. When does the water set to jelly? _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 3 marks (v) Find an expression for the amount of gelatine, G (kg), in terms of the time (t) in minutes _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 3 marks [3 marks] (vi) Sketch a graph of the mass of gelatin G (kg) against time t (minutes).  EMBED FG3.Document  3 marks Total Marks: 12 marks QUESTION 2 The graph of f: [0, 1]  EMBED Equation.3  R, where: f(x) =  EMBED Equation.3  is drawn below.  EMBED FG3.Document  (i) Write down the range of f(x) _____________________________________________________________________ 1 mark [1 mark] (ii) Use calculus to show that  EMBED Equation.3  is the point of greatest slope of f(x) _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 3 marks [4 marks] Joshua wants to build an ornamental pond for his garden. He uses as a model, the function: f: [0, 1]  EMBED Equation.3  R, where: f(x) =  EMBED Equation.3  and rotates it about the y-axis. All lengths are in metres. (iii) Using calculus find the volume of the pond when filled to capacity. Give your answer in cubic metres (m3) correct to 3 decimal places. _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 4 marks [4 marks] (iv) Once constructed, Joshua fills the pond with water at the rate of 0.012 m3/min. Show that the rate at which the water level is rising at any time t is given by:  EMBED Equation.3  m/min _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ [4 marks] 4 marks Total Marks: 12 marks      PAGE 16    +ABCWXYgqsúndQGjGhkmEHU$j*cW= hkmCJUVmHnHujhkmEHU$j&cW= hkmCJUVmHnHujhkmUhkmh0hel5\ hel5\helhkm5CJ\h{')5CJ\ hkmCJhel5CJ\h?hel5CJaJh{')5CJaJh?helCJaJhelCJ$OJQJhrhelCJ$OJQJ helCJ%&'()*+BCXYrs @1$gdkmgdel$a$gdel$a$gdel$a$gdel  9r gdelgdelgdkm " > Z \ ] ^ i j * G  & F dh1$gdkm & F dhP1$gdkm & F 1$gdkm @dh1$gdkm @1$gdkm      ! & ' : ; < = B C V W X Y \ ] ^ i j ùypga helCJh0helCJh0hu3CJ hT|CJjH hkmEHU$jxxn; hkmCJUVmHnHuj hkmEHU$jDxn; hkmCJUVmHnHujhkmEHU$jgxn; hkmCJUVmHnHujhkmEHU$jnn; hkmCJUVmHnHuhkmjhkmU!     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