AJC H2 MATH P2

Page 2 of 9 Anderson Junior College Preliminary Examination 2011 H2 Mathematics Paper 2 Section A: Pure Mathematics [40 marks] Answer ALL questions 1. At a water treatment plant, sewage water is being pumped into a processing tank, with an unknown capacity, at a constant rate of 100 litres per hour. The tank processes the water and discharges it at a rate proportional to the volume of sewage water currently in it. Due to a crack in the tank, sewage water is leaking out of the tank at a constant rate of 3 litres per hour. At time t hours, the volume of sewage water in the tank is u litres. (i) By setting up and solving a differential equation, show that the general solution is 1 97 e ktuA k  where A and k are constants, and k > 0. [3] (ii) The processing container is initially empty. After an hour, the volume of sewage water in the container is 70 litres. Given that the sewage water that leaked out of the tank is not processed, find the volume of water processed by the container when t = 2. [4] (iii)Sketch, on a single diagram, the graph of the solution found in part (ii) and find the minimum tank capacity for the model to be valid. Give your answer to the nearest litre. [2] 2. (a) The function f is defined by 1f( ) ( 1 )s i n 1x xx   , 10 2x  (i) By using differentiation or otherwise, show that the function f -1 exist. Hence find the value of f -1(1). [3] (ii) Find the equation of the tangent to the curve y = f(x) at the point where x = 0. [2] (iii) Write down the equation of the tangent to the curve y = f -1(x) at the point where x = 1. [1]

Page 3 of 9 (b) The functions g and h are defined by 2 3 11 , for , 0 4g( ) 5 ( 6) , for , 4 7 xx xx xx x      

1h( ) 1 for ,x xx axa   (i) Find the range of g. [1] (ii) Given that the composite function hg exists and a is an integer, find the greatest value of a such that the range of hg is a subset of (1, ) . [2] 3. With reference to the origin O, the position vectors of the points A and B are 23ij k and 2 ik whereis a negative integer. The cosine of the angle between the position vectors of A and B is 3 70  . The point M lies on the line segment AB such that 3AMM B , and the point C lies on OM produced such that 43OM OC . (i) Show that 4  . [2] (ii) Find the position vector of C. [2] (iii)Geometrically, what is the significance of p , where ACA B p AB       ? By finding the value of p , deduce the distance between O and the line passing through A and B. [4] 4. The complex number z is such that 943 34iz i zi    . (i) Sketch, on an Argand diagram, the locus of the points representing z. [2] (ii) Find the range of values of  arg zi . [2] Another complex number w is such that *4ww  . (iii) Sketch the locus of the points representing w on the same Argand diagram and hence, find the least value of zi w . [3]

Page 4 of 9 5. l is the line of intersection between two perpendicular planes 1 and 2. The equations of l and 2 are given as follows: :l 1 , 1 a   r 2 : 5 1 b c    r where a, b and c are constants. (i) State the value of c. [1] (ii) Show that 51ab  . [1] (iii)Given that the vector i + 2j is parallel to 1, find the value of a and b. [3] (iv) The plane 3 is parallel to 2 and is at a distance of 4 units from 2. Find a vector equation of 3 in scalar product form. [2]

Page 5 of 9 Section B: Statistics [60 marks] Answer ALL questions 6. Using the letters of the word “CORRELATION”, find the number of (i) arrangements such that the first letter is a vowel and the last letter is a consonant. [2] (ii) 6-letter code words that can be formed. [3] 7. (i) The Yummy Confectionery manufactures candies from different fruits and randomly packs them in packets of 20. It is known that 56% of the candies are made from mangoes, 20% from strawberries and the rest from lemons. A supermarket buys 60 packets of candies from Yummy Confectionery. Find the probability that the average number of lemon candies in each packet is not less than 5. [3] (ii) The confectionery wishes to test the popularity of a new flavour target among people of all ages by surveying 300 people. Comment on the suitability of the sample if the survey is administered to 300 randomly selected students from various secondary schools. [1] Give a reason why it would be difficult to use a stratified sample and suggest a suitable sampling method. [2] 8. A researcher wants to find out how the ground temperature affects the rate of chirps made by a ground cricket. The data gathered is as follows: Ground temperature, t, (degree Celsius, oC) 20.9 22.0 24.0 27.0 29.1 31.6 34.1 Rate of chirps, x, (number of chirps / min) 16.2 16.5 16.9 17.7 20.2 25.3 31.2 (i) Draw a scatter diagram to illustrate the data. [1] The researcher wants to fit a model of the form Btx Ae where A and B are constants. (ii) Calculate the product moment correlation coefficient, r, between ln x and t. [1] (iii) Use your answers in part (i) & (ii) to explain why the model is suitable and find the values of the least square estimates of A and B. [3]

Page 6 of 9 (iv) Estimate the percentage increase in the rate of chirps when the ground temperature is increased by 5 oC. On a particular day, there is a change in the ground temperature from 10 oC to 15 oC. Can the researcher use the answer found above to estimate the percentage change in the rate of chirps? Justify your answer. [2] (v) The researcher wants to convert the units for the temperature, t, from Celsius to Fahrenheit, by using the formula 5 329cf , where c is the temperature in Celsius and f is the temperature in Fahrenheit. Explain how this affects the value of r. [1] 9. The lifespan of a Liquid Crystal Display (LCD) television, A, is normally distributed with a mean of 45000 hours and a standard deviation of 2000 hours. The lifespan of a Plasma television, B, is also normally distributed with a standard deviation of 1850 hours. It is known that there is a 50% chance that the lifespan of a Plasma television is less than 30000 hours. (i) A LCD television and two Plasma televisions are randomly selected. Show that the probability that twice the lifespan of the LCD television exceeds the total lifespan of the two Plasma televisions by at least 25000 hours is 0.852. State an assumption needed for your calculations [3] (ii) A batch of 50 Plasma televisions is produced. Find the probability that more than 14 but fewer than 22 of the Plasma televisions have a lifespan of more than 30000 hours. [2] (iii) Find the least value of n such that the probability that the