DRAFT. Mathematical Methods 2017 Sample paper. Question Booklet 1

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1 1 South Australian Certificate of Education Mathematical Methods 017 Sample paper Question Booklet 1 Part 1 Questions 1 to 10 Answer all questions in Part 1 Write your answers in this question booklet You may write on page 18 if you need more space Allow approximately 90 minutes The list of mathematical formulae is on page 19 GENERAL INFORMATION Examination material Total marks 148 Question Booklet 1 (19 pages) Question Booklet (0 pages) Part 1 one SACE registration number label Questions 1 to marks Reading time 10 minutes Part You may make notes on scribbling paper Questions 11 to marks Writing time 3 hours Show all working in the question booklets Appropriate steps of logic and correct answers are required for full marks State all answers correct to three significant figures, unless otherwise instructed Use black or blue pen You may use a sharp dark pencil for graphs and diagrams DRAFT SACE Board of South Australia 016 Graphics calculator For office use only Attach SACE registration number label to this box 1. Brand Model Supervisor check Re-marked. Brand Model

2 PART 1 (Questions 1 to 10) (70 marks) QUESTION 1 (8 marks) Find d y for the following functions. There is no need to simplify your answers. dx (a) y 3sin x6cos x. 5. (b) y ( 3x 3 x 4) (3 marks) (c) y 1 4 e x x. (3 marks) page of 19

QUESTION (5 marks) (a) (i) Each time a fair dice is rolled, X 1 if the uppermost face is 3 and X 0 otherwise. On the basis of the information above, is X a Bernoulli random variable?

3 QUESTION (5 marks) (a) (i) Each time a fair dice is rolled, X 1 if the uppermost face is 3 and X 0 otherwise. On the basis of the information above, is X a Bernoulli random variable? Tick the appropriate box. Yes No (ii) A fair coin is tossed five times. X is the number of times that heads is uppermost. On the basis of the information above, is X a Bernoulli random variable? Tick the appropriate box. Yes No (b) X represents a Bernoulli random variable with probability of success p 0.. (i) State the mean of X. (ii) Determine the standard deviation of X. (c) X represents a Bernoulli random variable with probability of success p. State the value of p for which the standard deviation of X is maximised. page 3 of 19 PLEASE TURN OVER

4 QUESTION 3 (6 marks) In the 1860s Dr C.R.A. Wunderlich studied the body temperature of many thousands of healthy people. Body temperature was measured under standard conditions. Dr Wunderlich concluded that, in healthy people, body temperature was normally distributed with a mean of 37 C and a standard deviation of 0.5 C. (a) Using Dr Wunderlich s conclusions, find the proportion of the population of healthy people who had a body temperature: (i) above 37 C. (ii) above 38 C. (b) In 199 a further study was undertaken in which the body temperature of 148 healthy people was measured under standard conditions. The sample mean x was found to be 36.8 C. (i) Let be the mean body temperature of the 199 population of healthy people. Find a 95% confidence interval for, assuming that the standard deviation (s) is still 0.5 C. (ii) On the basis of the confidence interval you found in part (b)(i), explain whether or not the 199 data are consistent with the value of = 37 C found in the 1860s. page 4 of 19

5 QUESTION 4 (5 marks) Find, from first principles, ( ) f x if f x x 3. (5 marks) page 5 of 19 PLEASE TURN OVER

6 QUESTION 5 (8 marks) Fitts s law models the time taken by computer users to move the cursor a fixed distance in order to point to an on-screen target. Applying Fitts s law for a particular computer user, the average time taken (t), in seconds, to move the cursor 10 cm to a target that is w cm wide is given by the model t 10. 5ln w. cursor 10 cm w Diagram not drawn to scale. (a) In the space provided below, sketch the graph of t vs w for 0. 5 w 4. (3 marks) (b) (i) The width of an existing target is changed from cm to 4 cm. State by how many seconds the average time taken by this computer user will change. page 6 of 19

7 (ii) Show that according to Fitts s law, if the width of the target is changed from k cm to k cm, the change in t is constant. (3 marks) page 7 of 19 PLEASE TURN OVER

8 QUESTION 6 (9 marks) Consider the function f xsin ( x ) 1 for 0 x. 3 (a) Sketch the graph of the function y f x on the axes below. 4 y x 4 (b) Find the exact values of the zeros of f x. (3 marks) page 8 of 19

9 (c) For f( x) 0, find the exact value of the area contained between f x and the x-axis. (4 marks) page 9 of 19 PLEASE TURN OVER

10 QUESTION 7 (6 marks) (a) The discrete random variable X has the following probability distribution: x P( X x) a (i) Find the value of a. (ii) Find the mean, X. page 10 of 19

11 (b) People pay a fee to enter a local showground. Upon entry, each person receives 10 tokens that are redeemable for activities such as rides and games. In one game, 30 balls numbered from 1 to 30 are placed into a bag. Players give two tokens to the game s operator each time they randomly draw one numbered ball from the bag. The number on the ball is checked and the ball is put back into the bag. The number on the ball determines the outcome of the game, as shown in the table below: Number on the ball Numbers of tokens won from the operator by the player a multiple of 9 9 a multiple of Probability of occurring If the number on the ball drawn is neither a multiple of 9 nor a multiple of 11, the player wins no tokens from the operator. (i) Let X represent the number of tokens won through playing this game. What is the expected value of X? (ii) In one day the game is played 1000 times. Predict whether or not the operator will have more tokens at the end of the day than at the start of the day. Explain your answer. page 11 of 19 PLEASE TURN OVER

12 QUESTION 8 (5 marks) Shown below is a graph of y f( x) for 0 x 5: y x (a) On the graph above, represent f x d x. 1 0 (b) Indicate which one of the graphs on the page opposite could represent y F( x), where Fx f xd x for 0 x 5. Graph Explain your answer. (3 marks) page 1 of 19

13 Graph A y x Graph B y x Graph C y x page 13 of 19 PLEASE TURN OVER

14 QUESTION 9 (7 marks) (a) Without using a calculator, show that x dx ln. The graph of y f( x), where f x 1, x 0, is shown below: x y A B C x Three rectangles, each of 1-unit width, have been included and their areas can be used to calculate an estimate for the area bounded by f( x), the x-axis, and the vertical lines at x 3 and x 6. page 14 of 19

15 (b) Determine the sum (S) of the areas of rectangles A, B, and C. (c) The sum (S) of the areas of rectangles A, B, and C could be used to approximate the value of ln. (i) Explain why ln S. (ii) Describe a method involving the area of rectangles that would result in a more accurate approximation of ln than using S as an approximation. Do not carry out the method. page 15 of 19 PLEASE TURN OVER

16 QUESTION 10 (11 marks) The diagram shows a large circular lake with centre O and radius 4 km. P AB is a diameter of the lake. A person is at point A and must travel to point B. A row boat is available at point A. Travel routes include: running around the lake to point B A O B rowing across the lake to point B rowing across the lake to another point, such as point P, and then running around to point B. The person rows at 6 km/h and runs at 1 km/h. The relationship between time, distance, and constant speed is time Let PAB radians. distance. speed Let t be the time taken, in hours, by the person to travel from point A to point B by any of the routes described above. (a) (i) Show that AP 8cos. (ii) Hence show that t 3 ( cos ). page 16 of 19

17 (b) Find the value of for which d t d 0 for 0. (3 marks) (c) Draw a sign diagram for d t d for 0. (d) What route should the person take in order to travel from point A to point B in the least amount of time? Justify your answer. (3 marks) page 17 of 19 PLEASE TURN OVER

18 You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. Question 3(b)(ii) continued ). page 18 of 19

19 You may remove this page from the booklet by tearing along the perforations so that you can refer to it while you write your answers. LIST OF MATHEMATICAL FORMULAE FOR USE IN STAGE MATHEMATICAL METHODS Properties of Derivatives d dx f x g x f x g x f x g x d f x f x g x f x gx dx gx g x d dx f g x f g x g x Quadratic Equations If b b 4ac ax bx c 0 then x. a Discrete Random Variables The mean or expected value of a discrete random variable is: X xp. x, where px is the probability function for achieving result x. The standard deviation of a discrete random variable is: X xx px, where X is the expected value and px is the probability function for achieving result x. Bernoulli Distribution The mean of the Bernoulli distribution is p, and the standard deviation is:. p 1 p Binomial Distribution The mean of the binomial distribution is np, and the standard deviation is: np 1 p, where p is the probability of success in a single Bernoulli trial and n is the number of trials. The probability of k successes from n trials is: n k n k Pr X kc p 1 p, where p is the probability of success in the single Bernoulli trial. k Population Proportions The sample proportion is ˆp X n, where sample of size n is chosen, and X is the number of elements with a given characteristic. The distribution of a sample proportion has a mean of p and a standard deviation of p1 p. n The upper and lower limits of a con dence interval for the population proportion are: ( 1 ) ( 1 ) pˆ pˆ pˆ pˆ pˆ z p pˆ + z, n n where the value of z is determined by the con dence level required. Continuous Random Variables The mean or expected value of a continuous random variable is: X xfxd x, DRAFT is the probability density function. where f x The standard deviation of a continuous random variable is: where f X xx f xd x, x is the probability density function. Normal Distributions The probability density function for the normal distribution with the mean and the standard deviation is: 1 x 1 f x e. All normal distributions can be transformed to the standard normal distribution with 0 and 1 by: X Z. Sampling and Condence Intervals If x is the sample mean and s the standard deviation of a suitably large sample, then the upper and lower limits of the con dence interval for the population mean are: s s x z x z, n n where the value of z is determined by the con dence level required. page 19 of 19 end of question booklet

Specialist Mathematics 2017 Sample paper

Specialist Mathematics 2017 Sample paper South Australian Certificate of Education Specialist Mathematics 07 Sample paper Question Booklet Part (Questions to 9) 75 marks Answer all questions in Part Write your answers in this question booklet