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# Questions

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QUESTION 1 The graphs of f(x) = ¢* and g(x) =x~— 1 are shown. The area of the shaded section bounded by these graphs between the linesx=0and x =1 is A) 1-e (B) e-2 5 ©) B 1 (D) e=3

QUESTION 2 X i e +1 Determine I e (A) (B) ©) (D) x—e “+c x+te¥+c l+xe*+c x+xe*+c

QUESTION 3 Determine 2 j (4x+ 6)3 dx (A) 16(4x+6)* +¢ (B) 8(4x+6)*+c (4x +6)* 2 (4x + 6)* 8 ©) +C T~ (D)

QUESTION 4 Pulse rates of adult men are approximately normally distributed with a mean of 70 and a standard deviation of 8. Which of the following choices correctly describes how to determine the proportion of men that have a pulse rate greater than 787 (A) Determine the area to the left of z= 1 under the standard normal curve. (B) Determine the area to the right of z = 1 under the standard normal curve. (C) Determine the area to the right of z = —1 under the standard normal curve. (D) Determine the area between z =—1 and z = 1 under the standard normal curve.

QUESTION 5 The equation of the tangent to the curve f(t) =te` att=11is (A) y=et (B) y=2et—e ©) y=et—e® +1 (D) y=2et—2¢*+1

QUESTION 6 If the probability of success in a Bernoulli trial is 0.30 the variance is A) 0.70 (B) 0.46 (C) 0.30 (D) 0.21

QUESTION 7 The life expectancy (in years) of an electronic component can be represented by the probability density function 1 p(x) =142 0 otherwise > 1 The probability that the component lasts between 1 and 10 years is (A) 0.010 (B) 0.100 (C) 0.900 (D) 0.990

QUESTION 8 A test includes six multiple choice questions. Each question has four options for the answer. If the answers are guessed the probability of getting at most two questions correct is represented by 6 0 6 6 1 5 (A) 5 0.25 x 0.75° + 1 0.25 x 0.75 6 0 6 [0 1 s (6 2 4 (B) N 0.25 x 0.75° + 1 0.25 x 0.75° + 5 0.25“x 0.75 ©) 1 6 0.25% % 0.75° + (Jo.zsl x 0.755] @ oo 6 6 0.25% % 0.75° + (Jo.zsl x 0.75° + [2]0.252 x 0.754J

QUESTION 9 Determine I X—H dx x2 +2x (A) ln[2x+2j+c (B) In(2x+2)+c (C) %ln(xz +2x)+¢ (D) 2In(x? +2x)+c

QUESTION 10 Two types of material (A and B) are being tested for their ability to withstand different temperatures. A random selection of both materials was subjected to extreme temperature changes and then classified according to their condition after they were removed from the testing facility. The results are shown in the table. Material Broke completely Showed defects ot | 5 | 3| o T s s An approximate 95% confidence interval for the probability that material A will break completely or show defects is given by [c—l.% /w cﬂ.% /Mj n n The values of ¢ and » are @A) 9 andos 95 ®) Y andos 200 ©) 140 and9s 200 @) 9% and 200 200

QUESTION 11 (3 marks) Determine the derivative of each of the following with respect to x. 1 D 7 b) y=x`xe* Express your answer in factorised form. [1 mark] [2 marks]

QUESTION 12 (5 marks) An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration a (in m s ™) of the object is given by a(f) =m cos(mt) t=>0 where ¢ is time in seconds. The velocity at =115 0.5 m s ! a) Determine the initial acceleration. [1 mark] b) Determine the initial velocity. [2 marks] c) Determine the displacement after one second. [2 marks]

QUESTION 13 (7 marks) A function is defined as f(x) = x(In(x))? x > 0. The graph of the function is shown and has a local maximum at point 4 and a global minimum at point B. The derivative of the function is given by f`(x) = 2 In(x) + (In(x))? x > 0. a) Verify that there 1s a stationary point at x = 1. [2 marks] b) Determine the coordinates of A. [3 marks] The graph of the function has a point of inflection at x = ¢` c) Determine p. [2 marks]

QUESTION 14 (3 marks) Determine the area of the triangle shown. C Not drawn to scale

QUESTION 15 (4 marks) Solve the following equations. a) 4e* =100 [1 mark] b) 2log x—log (x-1)=1 [3 marks]

QUESTION 16 (4 marks) Consider the following graph of f(x). Diagram 1 Diagram 2 Diagram 3 Justify your decisions using mathematical reasoning.

QUESTION 17 (6 marks) The volume of water in a tank is represented by a function of the form Vt)= Aekt where V` is in litres and ¢ 1s in minutes. Initially the volume is 100 litres and it is decreasing by 50 litres per minute. : : : : : 50 .. : Determine the time at which the volume is decreasing at the rate of 3 litres per minute. Express your answer in the form In(a).

QUESTION 18 (6 marks) The function f(x) has the form given by f(x) = 3 log2 (x+a)+b The function g(x) has the form given by g(x) =—log (x + ¢) + 5 A section of the graphs of the two functions is shown. X / a oo r Determine the values of a b and c.

QUESTION 19 (6 marks) A horizontal point of inflection is a point of inflection that is also a stationary point. In(x) kx Determine the value/s of £ for which the graph of f(x) = r xrl has only one horizontal point X+ of inflection.

QUESTION 20 (6 marks) At the end of the first stage of its growth cycle a species of tree has a height of 5 metres and a trunk radius of 15 cm. In the second stage of its growth cycle the tree stays at this height for the next 10 years. However the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function 1 r(t)=1.5+ sm(?j Assume the density (mass per unit volume) of the tree trunk is approximately 1 g/(:m3 and the tree trunk is in the shape of a cylinder. Determine the ratio of the trunk`s mass at the end of the second stage to its mass at the end of the first stage.

QUESTION 1 The limit of 12 -1 as h approaches 0 is closest to A) 0.0 ! (B) 1.0 C) 2.5 (D) 3.0

QUESTION 2 The pH of a substance is a measure of its acidity and is given by the formula pH = —log; [H ] where [H ]is the concentration of hydrogen ions in moles per litre. If a solution has a pH equal to 0.2 the concentration of hydrogen ions in moles per litre is closest to (A) 0.32 (B) 0.63 (C) 0.70 (D) 1.8

QUESTION 3 Let R be the region enclosed by the graph of y = xe” the x-axis and the lines x =—1 and x = 1. The area of R is closest to A) 0.74 (B) 1.26 C) 2.35 (D) 3.09

QUESTION 4 Consider the function f(x) =log (x+¢g) wherep>1and 0 <g <. Which of the following could be the graph of f(x)? (A) (B) © (D)

QUESTION 5 An object moves in a straight line with a velocity v given by v(t)=40(e ` - g )ms ` where £ >0 The object is at the origin initially. The displacement—time graph in the first 6 seconds is (A) ------ ©)

QUESTION 6 Oil is leaking from a tanker at the rate of »(¢) = 9000¢ % litres per hour where ¢ is in hours. Determine how much oil leaks from the tanker (to the nearest litre) from time # = 0 to time ¢ = 10. (A) 38910 litres (B) 8756 litres (C) 7782 litres (D) 1556 litres

QUESTION 7 The records of a shoe manufacturer show that 10% of shoes made are defective. Assuming independence the probability of getting 2 defective shoes in a batch of 20 is A) 0.1937 (B) 0.2852 (C) 0.3917 (D) 0.6083

QUESTION 8 Determine the size of angle 4 in the triangle. C 9.9 8.8 A 11.3 B Not drawn to scale (A) 48.5° (B) 614° (C) 118.6° (D) 131.5°

QUESTION 9 The displacement of a particle (in metres) at time ¢ (in seconds) is represented by the function s@)=tln@t)—t 0<t<4 Determine the approximate acceleration of the particle at time ¢ = 3. (A) 0.66ms > (B) 0.33ms> (C) -033ms™ (D) —0.66m s

QUESTION 10 The approximate value of x where the graph of the function y = X +6x%+Tx— 2 cos(x) changes concavity is (A) -3.26 (B) -2.85 (C) -2.20 (D) —1.89

QUESTION 11 (4 marks) A sugar company samples the packets of sugar it produces and finds that 5% of packets are underweight. Consider a batch of 20 packets. a) Determine how many packets of sugar the company can expect to be underweight in a batch of 20 packets. [1 mark] b) Determine the variance of the batch. [1 mark] c) Determine the probability that at most one of the packets of sugar in the batch is underweight. [2 marks]

QUESTION 12 (7 marks) The rates of change in population for two cities are given by 45 City A: A`() = — y ® t+1 City B: B`(¢) = 105¢ %% where ¢ is the number of years since 2018 and both 4`(f) and B`(f) are measured in people per year. At the beginning of 2018 City A had a population of 5000 and City B had a population of 3500. a) Determine the population models for both cities. [3 marks] b) Use the information in 12a) to predict the population of City B at the beginning of 2028. /I mark] c) Use the information in 12a) to predict the year in which the population of both cities will be the same. [3 marks]

QUESTION 13 (6 marks) An online retailer claims that 90% of all orders are shipped within 12 hours of being received. On a particular day 121 orders were received and 102 orders were shipped within 12 hours. a) State the sample proportion of orders shipped within 12 hours. [1 mark] The distribution of the sample proportion of all orders that are shipped within 12 hours of being received on any day is approximately normal. b) Assuming the online retailer`s claim is true find the probability that in a random sample of 121 less than 85% of all orders are shipped within 12 hours. [3 marks] c) Use the result from 13b) to evaluate the reasonableness of the online retailer`s claim. [2 marks]

QUESTION 14 (6 marks) Let X denote the time in minutes between the arrival of trains at a station. The cumulative distribution function of X is defined by —— 5<x<10 F(x)= X 0 otherwise a) Determine the probability density function of X. [3 marks] b) Determine the probability that 5 < X <7. [1 mark] c¢) Determine the mean time between the arrival of trains at the station. [2 marks]

QUESTION 15 (3 marks) A field is divided into five sections as shown. The width of each section is 1 metre. The perpendicular height in metres of each section is given in the diagram. The area of the field was approximated using the trapezoidal rule and found to be 11.12 m` Not drawn to scale a) Determine the height marked x on the diagram. [2 marks] b) Determine the area of the field given the shape of the field is modelled by the function X L 0<x<5 x? +1 [1 mark] J(x)=

QUESTION 16 (4 marks) Bottles of soft drink should contain a volume with a mean of 591 mL but some variation is expected. Any bottle at or below the 20th percentile of the volume distribution is rejected. A percentile is a measure in statistics that shows the values below which a given percentage of observations occur. Thirty-five per cent of the bottles contain 593 mL or more of soft drink. Assuming the volumes are normally distributed determine the smallest volume (in mL) that will be accepted.

QUESTION 17 (4 marks) In a survey of 326 lecturers 303 said that on at least one occasion a mobile phone had rung in a lecture they were giving. Determine the sample size required to conduct a follow-up survey that provides 95% confidence that this one-point estimate is correct to within + 0.02 of the population proportion.

QUESTION 18 (4 marks) The diagram shows the quadrilateral ABCD. B 4.1 cm 7.6 cm Not drawn to scale D Determine the perimeter of the quadrilateral.

QUESTION 19 (7 marks) Consider the following information when completing this question. 2 The length of a curve y = f(x) over the interval [a b] = Ib 1+ (%j dx a \f 67 You are driving along a road with a vertical distance above sea level D (in metres) given by the function D) =300 + In(x> -3x + ¢) where x 1s the horizontal distance from an initial point of measurement (in kilometres) at sea level. Assume that if x 1s positive you are east of the initial point of measurement and if x is negative you are west of the 1nitial point of measurement. You start your drive along the road at a horizontal distance of 10 kilometres west of the initial point of measurement and drive until you are at a horizontal distance of 10 kilometres east of the initial point. Determine the time you spend driving downhill if you drive downhill at an average speed of 40 km/h.

QUESTION 20 (5 marks) Assuming the approximate normality of sample proportions (p and p ) and based on two independent samples the approximate confidence interval for the difference of two proportions is given by pd-p) p 1-p ) pd-p) p 0-p )) ————————— P =t i B i iy If the approximate confidence interval for the difference between two proportions does not contain 0 this provides evidence that the two proportions are not equal. The data in the table shows the observed frequencies of two drink preferences for independent samples of people who live in Town A and Town B. Using the approximate 99% confidence interval for the difference of two proportions determine if there is evidence to conclude that drink preference is associated with the town where the person lives.

QUESTION 1 2log;o (x)—log;o(3x) is equal to (A) logm[?] (B) log (x2 - 3x) 2logyg (x) log;(3x) (D) —log;o(x) ©)

QUESTION 2 The table shows the time a technician has spent servicing photocopiers. What is the probability that a given service required at least 10 minutes but less than 20 minutes? (A) 0.15 (B) 0.35 (C) 0.70 (D) 0.85

QUESTION 3 Determine JlOe‘“dx` ] 0643{‘4`] 4x +1 (A) +c (B) 40e* +c (C) %e“-“ +c (D) 2™ +¢

QUESTION 4 The second derivative of the function f(x) is given by f” (x)=— The interval on which the graph of f (x) 1S concave up is A x<0 (B) x<0 ) x>0 (D) x>0

QUESTION 5 The graph of f (x) is shown. -5 Which of the following could be the graph of / (x)? (A) (B) ©) (D)

QUESTION 6 A random variable X 1s the number of successes in a Bernoulli experiment with #z trials each with a probability of success p and a probability of failure g. The probability distribution table of X is shown.

QUESTION 7 Determine J (2x+3)dx | (A) 2 (B) 4 C) 14 (D) 16

QUESTION 8 The continuous random variable X has the probability density function - 1<x< -~ —_— = S Il = 2 b | W 0 otherwise The mean of X 1s (A) ln(%) it ©) h{%) (D) 1

QUESTION 9 A basket contains 10 green apples and 30 red apples. Three apples are drawn at random from the basket with replacement. Determine the probability that exactly two of the three apples are green. 3 (A) o 9 (B) o 10 ©) o (D) 27 64

QUESTION 10 Handspans of teenagers are approximately normally distributed with a mean of 15 cm and a standard deviation of 2 cm. Which of the following groups is expected to be the largest? (A) teenagers with handspans that are between 7 cm and 11 cm (B) teenagers with handspans that are between 11 cm and 15 cm (C) teenagers with handspans that are between 13 cm and 17 cm (D) teenagers with handspans that are between 17 cm and 21 cm

QUESTION 11 (5 marks) Determine the derivative with respect to x of the following functions. a) y= (e“` + l)3 b y= &g\) (Give your answer in simplest form.) X [2 marks] [3 marks]

QUESTION 12 (5 marks) Solve for x in the following. a) log (5x+7)f 5 [2 marks] b) loglu(x‘i‘3)+]0g]0(x_3): logm(gx—29) [3 marks]

QUESTION 13 (5 marks) Consider the functions f(x)=x”and g(x)= 4x. a) Determine the x-coordinates of the points of intersection of the graphs of the two functions. [2 marks] b) Use the results from Question 13a) to calculate the area enclosed by the graphs of f(‘c‘) and g(‘c) [3 marks]

QUESTION 14 (4 marks) Consider the function f(x)=In(3x+4) for x > _?4 a) Determine f`(x). [1 mark] b) Determine the x-intercept of the graph of f/(x). [2 marks] c) Determine the gradient of the tangent to the graph of f(x) at the x-intercept. [1 mark]

QUESTION 15 (4 marks) In the isosceles triangle ABC angle C is 120° and side a is 4 cm. a) Draw the triangle indicating all given information. [1 mark] b) Calculate the area of the triangle in cm?. (Give your answer in simplest form.) [3 marks]

QUESTION 16 (4 marks) A tangent is drawn at the point (1 e) on the graph of the function y = ¢` as shown. Not to scale Determine the area of the shaded triangle.

QUESTION 17 (3 marks) In any five-day working week Leonardo either catches a bus to work or uses another form of transportation. On average he catches the bus to work on three of the five days. His decision on any given day is independent of his decision on any other day. Determine the probability that Leonardo catches a bus to work on exactly one day in a given five-day working week.

QUESTION 18 (4 marks) The graph of y = f(x) where f(x) is the quadratic function f(x)= ax” + bx +4 is shown. Two regions of the area between the graph of y = /(x) and the x-axis are shaded. Not to scale : .2 43 . 5 Region P has an area of %umts“ and Region Q has an area of ?) units”. Determine the values of a and b.

QUESTION 19 (4 marks) A firm aims to have 95% confidence in estimating the proportion of office workers who respond to an email in less than an hour to within + 0.05. A survey has never been undertaken before so no past data is available. The firm believes that if the proportion is 0.5 then this will result in the largest variability in the sample proportion. Based on this determine the sample size needed using the approximate value of z = 2 for the 95% confidence interval. Justify the choice of 0.5 for the proportion.

QUESTION 20 (7 marks) The population of rabbits (P) on an island in hundreds is given by P(7) = 1In (3t)+6 1> 0 where # is time in years. Determine the intervals of time when the population is increasing and the intervals when it is decreasing.

QUESTION 1 The scores obtained on a test can be assumed to be normally distributed with a mean of 102 and a standard deviation of 19. What proportion of scores are over 113? (A) 0.2813 (B) 0.5789 (C) 0.7187 (D) 0.8216

QUESTION 2 A substance is being heated such that its temperature 7 in °C after # minutes is given by the function 7` = 2 The first integer value of 7 for which the instantaneous rate of change of temperature is greater than 100 °C per minute is (A) t=10 (B) =9 C) t=8 (D) t=7

QUESTION 3 A random sample of people were surveyed about the most important factor when deciding where to shop. The results appear in the table. Quality of merchandise Shopping environment If the sample size was 1200 the approximate 95% confidence interval for the proportion of people who identified price as the most important factor is (A) (0.395 0.405) (B) (0.386 0.414) (C) (0.377 0.423) (D) (0.372 0.428)

QUESTION 4 3 Using the trapezoidal rule with an interval size of 1 the approximate value of the integral J 0.5%dx is 0 A) 1.25 (B) 1.26 C) 131 (D) 1.88

QUESTION 5 Solve for x given that logy(x—1)=2. (A) 7 (B) 8 © 9 (D) 10

QUESTION 6 When seeds of a certain variety of flower are planted the probability of each seed germinating is 0.8. If eight seeds are planted what is the probability that at least six seeds will germinate? A) 0.797 (B) 0.503 (C) 0.294 (D) 0.001

QUESTION 7 : 11 : Determine f(x) given f`(x)= 6x% +—+— and f (1)=5. ~ 27 (A) f(\) =2x + %-4— ln(_r) -1 X B) f(x)=2x - L (x)+4 X ©) f(x) =2 —l+i}+2 X x- “ 3 2 D) f(x)=2x+=+5-2 X X~

QUESTION 8 The displacement (in metres) of a particle is given by s(‘t) =-3cos(7)+2sin (t) where 7 1s in seconds. : : : T : The instantaneous velocity of the particle at time 7 = . seconds is et (A) 3ms! (B) 2ms”! (C) 2ms”! (D) 3ms`

QUESTION 9 The graphs of the functions f(x)=2¢ +5 and g(x)= point A. 3 . : : : — intersect at point A. Determine the coordinates of B A) (1.609 15) (B) (1.099 1) (C) (0.4065 2) (D) (-0.693 6)

QUESTION 10 An object travels in a straight line so that its velocity at time # seconds is given by v(#)=2¢+sin(2¢). Determine the expression for acceleration as a function of time. (A) a(t)=2+2cos(2r) (B) a(t):2—%cos(2f) (©) a(t)=1 +2cos(2¢) (D) a(t)ztz— cos(27) b | —

QUESTION 11 (5 marks) Consider the function f(x)=¢ sin(x) 0<x<27 a) State the exact values of the x-intercepts of the graph of f`(x). [2 marks] b) Write an expression for the area enclosed between the graph of f (r) and the x-axis. [2 marks] ¢) Determine the area enclosed between the graph of f(x) and the x-axis to the nearest square unit. [1 mark]

QUESTION 12 (4 marks) The velocity function of an object in m s is given by v(t) = cos[6r + %] +2 0<1<S. Initially the object is at the origin. a) Determine the displacement function. [2 marks] b) What is the displacement of the object from the origin in metres (m) after three seconds? [2 marks]

QUESTION 13 (7 marks) The amount of gravel (in tonnes) sold by a construction company in a given week is a continuous random variable X and has a probability density function defined by: f`(x)={c(l_x2)* 0<x<l 0 otherwise 3 a) Show that ¢ = 5 b) Determine P(X <0.25). c¢) Determine the variance of X. [1 mark] [2 marks] [4 marks]

QUESTION 14 (7 marks) The heights of students at School A are normally distributed with a mean of 165 cm and a standard deviation of 15 cm. a) Determine the probability that a student chosen at random from School A is shorter than 180 cm. [1 mark] b) Determine the minimum integer value of the height of a student who is in the top 2% of this distribution. [3 marks] The heights of students at School B are also normally distributed. A student at School B has the same height as the height determined in Question 14b) but their corresponding z-score is 3. c) Determine which student`s height ranks higher in terms of percentile for their school. [3 marks]

QUESTION 15 (4 marks) A new internet search engine gives a ranking R to each website based on the function R = log; (5011r2 ) where / is the number of hits (visits) the website has received. If a website currently has 100 hits determine how many more hits they need to increase their ranking by 1.

QUESTION 16 (4 marks) In the diagram DC represents a 60 metre vertical tower. A and B are two points in the same horizontal plane as the foot C of the tower. The angle above the horizontal from A to D is 28° and the angle above the horizontal from B to D is 35°. 35° Not to scale ‘B The bearing of C from A is 050°T and the bearing of C from B is 300°T. Determine the distance between A and B to the nearest metre.

QUESTION 17 (4 marks) Rabbits and foxes are among two species of mammals that live on an isolated island. Rabbits represent a significant food source for the foxes. The populations of rabbits and foxes were monitored each month for two years. The graph shows the population of foxes (in thousands) and the population of rabbits (in thousands) at any time ¢ (in months) over the two years. The two populations can be modelled using trigonometric functions. (0 14.5) (12 14.5) - o NN (15 11) Population (thousands) 0 3 10 15 Time (¢) in months since 1 January 20 Foxes ------- Rabbits Jane believes that there were periods of time over the two years when the total population of foxes and rabbits on the 1sland exceeded 25 000. Evaluate the reasonableness of Jane`s claim.

QUESTION 18 (3 marks) The number of animals in a population (in thousands) is modelled by the function P such that . 100 P(1)= 1+4e Determine the number of animals in the population when the population is growing the fastest. — where 7 18 in years.

QUESTION 19 (4 marks) A random variable X defined over the interval a < x < b is uniformly distributed if its probability density function is defined by: I —— a<x<bh f(x)=1(b=a)` 0 otherwise The expected value and variance of a uniform random variable X are 2 E(X):@ Var(X)= (b;;) A manufacturer has observed that the time that elapses between placing an order with a supplier and the delivery of the order is uniformly distributed between 100 and 180 minutes. Determine the probability that the time between placing an order and delivery of the order will be within one standard deviation of the expected time.

QUESTION 20 (3 marks) The random variable B is normally distributed with a mean of 0 and a standard deviation of 1. Determine the probability that the quadratic equation x? +3x+2B =0 has real roots.

QUESTION 1 Consider the graph of f”(x) for a <x<b. cooee e - Which statement describes all the local maxima and minima of the graph of f(x) over a <x<b? (A) one local minimum and one local maximum (B) one local minimum and two local maxima (C) one local minimum only (D) one local maximum only

QUESTION 2 A binomial random variable arises from the number of successes in n independent Bernoulli trials. A context not suitable for modelling using a binomial random variable is recording the number of (A) heads when a coin is tossed 12 times. (B) left-handed people in a sample of 100 people. (C) times a player hits a target from 20 shots where each shot is independent of all other shots. (D) red marbles selected when three marbles are drawn without replacement from a bag containing four blue and five red marbles.

QUESTION 3 The area between the curve y =9 — x” and the x-axis is (A) 12 units® (B) 18 units® (C) 36 units? (D) 54 units?

QUESTION 4 The weekly amount of money a company spends on repairs is normally distributed with a mean of $1200 and a standard deviation of $100. Given that P(Z <-2.5) =0.0062 and P(Z >1) = 0.1587 where Z is a standard normal random variable determine the probability that the weekly repair costs will be between $950 and $1300. (A) 0.6525 (B) 0.6587 (C) 0.8351 (D) 0.8413

QUESTION 5 Which normal distribution curve best represents a normal distribution with a mean of 1 and a standard deviation of 0.5?

QUESTION 6 Which graph represents the function f(x)=-3—In(x+3)? (A) (B) -—+—+—+—+0 —43211012345 (D) Y 5 4 3 2 1 O 0 -— X -4 -3 -2 -1 1 23456 728 910 -4-3-2-1.0 1 2 4 56 78 910 3 -1 B %)) -3 -3 -4 -4 -5 -5 -6 -6 -7 7 -8 -8 —_ N W KA W -1.01 234567238910 © —_— N W KA WD

QUESTION 7 A circle with radius 7 and internal angle 6 has a shaded segment as shown. = If 4 1s in radians the area of the shaded segment is B) —(0-sin(0)) 0 2[5 (D) %(9—1)

QUESTION 8 In a survey 80 respondents exercised daily while 120 did not. When calculating the approximate 95% confidence interval for the proportion of people who exercise daily the margin of error 1s 0.4(1-0.4) (A) 1.96 ——— 200 0.4(1-0.4) (B) 0.95——— 200 0.67(1-0.67) C) 1.96 |——— © 120 0.67(1—0.67) (D) 0.95 |————= 120

QUESTION 9 The approximate area under the curve f (x) =+/2x +1 between x = 0 and x = 4 using the trapezoidal rule with four strips is (A) 2+-B+5+47 B) 2+2(V3+5+7) () 4+2(\B+£+\ﬁ) (D) 4+3+5+47

QUESTION 10 A survey plans to draw conclusions based on a random sample of 1% of Queensland`s adult population. To be regarded as a random sample every (A) adult in the population will be placed in an alphabetical list and every 100th person will be selected for the sample. (B) adult in the population can choose to participate until the sample size has been reached. (C) subgroup within the population will be represented in a similar proportion in the sample. (D) adult in the population will have an equal chance of being selected for the sample.

QUESTION 11 (5 marks) Solve for x in the following. a) ln(2x) =3 [2 marks] b) log (4x+16)log (x* —2)=1 [3 marks]

QUESTION 12 (3 marks) The probability that a debating team wins a debate can be modelled as a Bernoulli distribution. Given that the probability of winning a debate is 3 a) Determine the mean of this distribution. [1 mark] b) Determine the variance of this distribution. [1 mark] c¢) Determine the standard deviation of this distribution. [1 mark]

QUESTION 13 (9 marks) a) Determine the derivative of f`(x)= 32 [1 mark] : ln(x) : : b) Given that g(x)=——= determine the simplest value of g`(e). [3 marks] X ¢) Determine the second derivative of 4(x) = xsin(x). (Give your answer in simplest form.) /5 marks]

QUESTION 14 (6 marks) The rate that water fills an empty vessel is given by L 0.25¢ * (in litres per hour) 0 <t <8In(6) where ¢ is time (in hours). dt a) Determine the function that represents the volume of water in the vessel (in litres). [2 marks] The vessel is full when 7 = 81n(6). b) Determine the volume of water to the nearest litre the vessel can hold when full. [2 marks] The table shows the approximate rate the water flows into the vessel at certain times. c) Use information from the table and the trapezoidal rule to determine the approximate volume of water in the vessel after three hours. [2 marks]

QUESTION 15 (4 marks) The derivative of a function 1s given by f `(x) =e* (x — 4). Determine the interval on which the graph of f (x) is both decreasing and concave up.

QUESTION 16 (3 marks) A section of the graphs of the first and second derivatives of a function are shown. Sketch a possible graph of the function on the same axes over the domain 0 < x < 27. Explain all reasoning used to produce the sketch. y

QUESTION 17 (4 marks) . . b b— Determine the value of b given I 3x%dx =117 and I l3x2a`x =56 forb>1. a a

QUESTION 18 (4 marks) A percentile 1s a measure in statistics showing the value below which a given percentage of observations occur. The continuous random variable X has the probability density function 2x—2 1<x<2 f(x)={ 0 otherwise Determine the 36th percentile of X.

QUESTION 19 (7 marks) Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion e.g. if AUV W is similar to AXYZ then ur _yw _uw LU=4X 2V =LY and LW = £Z and — =— XY YZ XZ Two parallel walls AB and CD where the northern ends are 4 and C respectively are joined by a fence from B to C. The wall AB is 20 metres long the angle ABC = 30° and the fence BC is 10 metres long. A new fence is being built from A4 to a point P somewhere along CD. The new fence 4P will cross the original fence BC at O. Let OB = x metres where 0 < x <10. Determine the value of x that minimises the total area enclosed by AOBA and AOCP. Verify that this total area is a minimum.

QUESTION 1 The position (in cm) of a particle is given by x = cos(4t) where ¢ 1s time (in seconds). The velocity of the particle when ¢ =5 is (A) 1.6323 cms ! (B) 0.4081 cm s (C) —0.9129 cms ™ (D) —3.6518cms !

QUESTION 2 Identify the correct features of the function f(x)= x (A) f`(-1)=0 f (-1)<0 B f(=1)=0/(-1)> ©) f`(-1)<0 f (-1)<0 (-1)<0 ./ (-1) (D) f 1)>0 1 1)> 1)<0

QUESTION 3 The derivative of the function f`(x) is given by f”(x) = sin(x3 ) for the domain —1.8 < x <1.8. The number of points of inflection that the graph of f (x) has on this interval is (A) 1 (B) 3 €) 4 (D) 5

QUESTION 4 The distribution for a sample proportion p has a mean of 0.15 and a standard deviation of 0.0345. The sample size is (A) 10 (B) 14 (C) 107 (D) 116

QUESTION 5 The continuous random variable X has the probability density function cos(x) —rx T 48 < & f(x)={ 2 > 2772 0o otherwise The standard deviation of X is (A) 0.467 (B) 0.684 (C) 1.211 (D) 1.467

QUESTION 6 A stall at the school fete sells cups of lemonade. Assuming the amount of lemonade in a cup 1s normally distributed with a mean of 60 mL and a standard deviation of 3 mL 80% of the cups contain more than (A) 52.4mL (B) 57.5mL (C) 61.6mL (D) 62.5mL

QUESTION 7 A marble moves in one direction in a straight line with velocity v = 21n(t + 1) (in metres per second) where 7 is time (in seconds) since the marble passed through the origin. Determine the distance from the origin the marble has rolled after 4 seconds. (A) 040 m (B) 1.60m (C) 322m (D) 8.09 m

QUESTION 8 Determine the equation of the asymptote of the function f (x) =logg (x — 3) —4. (A) x=-4 (B) x=-3 (C) x=3 (D) x=4