Mathematics. Pre-Leaving Certificate Examination, Paper 1 Higher Level Time: 2 hours, 30 minutes. 300 marks L.17 NAME SCHOOL TEACHER

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1 L.7 NAME SCHOOL TEACHER Pre-Leaving Certificate Examination, 205 Name/v Printed Checke To: Update Name/v Comple Paper Higher Level Time: 2 hours, 30 minutes 300 marks For examiner Question 2 School stamp Mark Running total 9 Grade Total 205. L.7 /20 Page of 9

2 Instructions There are two sections in this examination paper: Section A Concepts and Skills 50 marks 6 questions Section B Contexts and Applications 50 marks 3 questions Answer all nine questions. Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part. The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination. You will lose marks if all necessary work is not clearly shown. You may lose marks if the appropriate units of measurement are not included, where relevant. You may lose marks if your answers are not given in simplest form, where relevant. Write the make and model of your calculator(s) here: 205. L.7 2/20 Page 2 of 9

3 Section A Concepts and Skills 50 marks Answer all six questions from this section. Question (a) (25 marks) The number of users that log on to a new social networking website doubles every week. In week, there are 24 users and in week 2, there are 48 users. (i) How many users access the website in week 2? (ii) In which week does the number of users first exceed 0 million? (iii) Users access the website on average 0 times per week. The website earns 0 4 cent per user visit. How much revenue does the website generate in the first 2 weeks? (b) The sum to infinity of the series 5 + 0x + 20x x 3 + is 00. Find the value of x. page running 205. L.7 3/20 Page 3 of 9

4 Question 2 (25 marks) The function f is defined as f : R R : x x 2, where x R \ {4}. x 4 (a) Show that the curve y = f (x) has no local maximum or local minimum point. (b) Find the equations of the asymptotes and draw a sketch of the curve on the axes below. y x (c) Show that no two tangents to the curve are perpendicular to each other L.7 4/20 Page 4 of 9

5 Question 3 (a) Four complex numbers, z, z 2, z 3 and z 4, are shown on the Argand diagram. The same scale is used on both axes. The number z has modulus less than. (i) 2 Which of the numbers, z 2, z 3 or z 4, best represents z? Justify your answer. Im( z) z 2 z z 4 (25 marks) z 3 Re( z) (ii) Plot the relative positions of iz and z on the Argand diagram above, where z is the complex conjugate of z, and label each point. π π cos + i sin 8 8 (b) Let w =, where i 2 =. 4 π π cos + i sin (i) Express w in the form a + bi, where a, b R. (ii) Find the two complex numbers x + yi for which (x + yi) 2 = w, where x, y R. page running 205. L.7 5/20 Page 5 of 9

6 Question 4 The diagrams show the graph of the function g(x) = 4x 3. y (25 marks) gx () x (a) State whether or not g(x) is injective. Give a reason for your answer. (b) Under what criteria is g(x) bijective? For what domain and codomain does g(x) meet this criteria? (c) Find the inverse function g (x) and sketch the graph of g (x) on the diagram above L.7 6/20 Page 6 of 9

7 Question 5 (a) (25 marks) The graphs of the functions f : x 2x 3 and g : x 3 + 3x x 2 are shown in the diagram. (i) Use your graph to solve the inequality 3 + 3x x 2 < 2x y f() x 2 gx () x (ii) Use algebra to solve the inequality 3 + 3x x 2 < 2x 3. (iii) Find the set of values of m for which 2x 3 = mx + has only one solution. page running 205. L.7 7/20 Page 7 of 9

8 Question 6 (a) The graphs of four polynomial functions are shown below. Graph A Graph B (25 marks) y y 3 x 3 x Graph C y Graph D y 3 x 3 x State which of the graphs above is that of the function f : x x( x) 3 (3 x) 2, where x R, and justify your answer L.7 8/20 Page 8 of 9

9 (b) Let g(x) = ax 3 + bx 2 + cx + d where a, b, c, d are constants and x R. (i) Given that g(x) has only one local turning point, show that b 2 = 3ac. (ii) Hence, find the co-ordinates of the local turning point. (iii) Hence, show that the point of inflection occurs at x = b c. page running 205. L.7 9/20 Page 9 of 9

10 Section B Contexts and Applications 50 marks Answer all three questions from this section. Question 7 (50 marks) A quantity is subject to exponential growth or decay if it increases or decreases at a rate proportional to its current value. Many physical phenomena such as the growth of human population, the spread of viruses, the rates of radioactive decay, etc. are modelled on exponential functions. (a) dn The equation that describes exponential growth is = kn, dt which states that the growth rate of the quantity N at time t is proportional to the value N(t). (i) Given that y = Ae kt dy, find in terms of y. dt (ii) Hence, using your result from part (i), show that ln N = kt + c, where c is a constant. (iii) Hence, show that the solution to the equation can be written as N(t) = N 0 e kt, where N 0 is the quantity at t = 0. (b) Exponential decay therefore occurs in the same way when the growth rate is negative. The rates of certain types of chemical reactions, whose rates depend only on the concentration of one or another reacting substance, consequently follow exponential decay. The concentration of the reacting substance is represented by the equation ln A = kt + ln A 0, where A 0 is the initial concentration and A is the concentration after time t. In an experiment to observe the rate of reaction in a chemical process, the concentration of the reacting substance was measured at various times t from when the experiment began. The data is shown in the table below. Time (hours) Concentration (moles litre ) L.7 0/20 Page 0 of 9

11 (i) Find, correct to two decimal places, the values of ln A for each of the concentration measurements in the data set shown and plot ln A against t on the axes below t lna (ii) Using your graph, or otherwise, find the value of k, correct to two significant figures. (iii) The concentration of the reacting substance when the chemical process is complete was determined to be 0 05 moles litre. Using your value for k, find the time it took for the experiment to be fully completed, correct to the nearest minute. page running 205. L.7 /20 Page of 9

12 (iv) Show that, according to the exponential model used, the rate of change of A(t) is always decreasing over time. What limitation does this model have to describe this chemical process? (c) For any quantity which follows exponential decay, the term half-life can be used to describe the time required for the decaying quantity to fall to one half of its initial value. Under different experimental conditions, it took 36 minutes for 3 g of the reacting substance to decay to g. Using the formula ln A = kt + ln A 0, or otherwise, find, t /2, the half-life of this reaction L.7 2/20 Page 2 of 9

13 Question 8 (a) The diagram shows a right circular cone with a fixed slant height of l cm. The slant height makes an angle with the vertical, π where 0 < <. 2 (50 marks) l (i) Express the radius and height of the cone in terms of l and 3 π l and, hence, show that the volume of the cone is (sin sin 2 ). 6 (ii) Find the value of for which the volume of the cone is a maximum. Give your answer in the form tan a, where a N. (iii) Find the maximum volume of the cone. Give your answer in surd form. page running 205. L.7 3/20 Page 3 of 9

14 (b) A water tower is an elevated structure which supports a water tank constructed at sufficient height to pressurise a water supply system for the distribution of drinking water. It also serves as a reservoir to help with water needs during peak usage times. The water level in a tower typically falls during daytime usage, and then a pump fills it back up during the night. The reservoir tank in a water tower is in the shape of an inverted right cone, as shown in the diagram. The slant height of the cone is l and it makes an angle of 45 with the vertical. 45 During daytime usage, it is observed that the rate at which the water level drops, in metres per hour, is given by l 45 O x dx = 2 dt where x is the fall in the slant height, in metres, from the maximum capacity (volume) of the tank and t is the time, in hours, from the instant that the water level begins to fall from this point. (i) Find the maximum capacity of the reservoir. (ii) Hence, show that the volume of water in the reservoir tank at time t is π (l x) L.7 4/20 Page 4 of 9

15 (iii) Find, in terms of l and t, the rate at which the volume of water in the reservoir tank is decreasing with respect to time. (iv) Find the rate at which the volume of water in the reservoir tank is decreasing with respect to time when of its capacity is remaining. 8 (v) Given that a particular example of this type of water tower has a slant height of 30 m, how long would it take, correct to the nearest minute, to empty the reservoir tank if no water is replenished? page running 205. L.7 5/20 Page 5 of 9

16 Question 9 Mary and Paul took out a mortgage of 400,000 to buy a new house. They agreed to repay this loan, plus interest, by a series of equal monthly payments, starting one month after they received the loan and continuing for 20 years. The effective annual rate of interest charged is 6%. (50 marks) (a) (i) Show that the rate of interest, compounded monthly, that corresponds to an effective annual rate of 6% is 0 487%, correct to three decimal places. (ii) Show how to use the sum of a geometric series to calculate the monthly repayments that Mary and Paul have to make on their mortgage. (iii) Using amortisation, or otherwise, verify your answer to part (ii) above L.7 6/20 Page 6 of 9

17 (b) After two years of paying their monthly repayments on their mortgage in full, Mary and Paul s financial situation worsened and they were unable to make any further repayments. (i) Using amortisation, or otherwise, find out the amount outstanding at the point that Mary and Paul were unable to make any further repayments. (ii) Mary and Paul s lender charges an additional rate of interest of 5% per annum on the outstanding amounts of loans in arrears, compounded monthly. How long will it take for the outstanding amount that Mary and Paul owe to exceed the original sum of money borrowed? (iii) Give one justification why Mary and Paul s lender would charge a higher rate of interest on loans in arrears. page running 205. L.7 7/20 Page 7 of 9

18 You may use this page for extra work L.7 8/20 Page 8 of 9

19 You may use this page for extra work. page running 205. L.7 9/20 Page 9 of 9

20 Pre-Leaving Certificate 205 Higher Level Paper Time: 2 hours, 30 minutes 205. L.7 20/20 Page 20 of 9