view_agenda
Questions
Find the perfect question by filtering through every previous QCAA Mathematical Methods external exam question
notifications
Question Classifications
We’re conducting a final review of the classifications and formatting for our recently onboarded questions. If you have any feedback, please share it with us here.
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 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 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 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 15 (4 marks) Solve the following equations. a) 4e* =100 [1 mark] b) 2log x—log (x-1)=1 [3 marks]
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 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 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 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 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 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 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 3 Determine JlOe‘“dx` ] 0643{‘4`] 4x +1 (A) +c (B) 40e* +c (C) %e“-“ +c (D) 2™ +¢
QUESTION 7 Determine J (2x+3)dx | (A) 2 (B) 4 C) 14 (D) 16
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
QUESTION 10 The solution of 2 = 42 is (A) (B) ©) (D) 1.48 231 3.37 4.54
QUESTION 13 (4 marks) A sandy beach has a fence on one side and ocean on the other. The width of the beach is the distance (in metres) from the fence to the water`s edge. The width w(t) at a certain point is given by w(t) =a +bsin(£t —zj 0<r<24 6 3 where ¢ is time (in hours) since 6 am. The width of the beach is 8 metres at 8 am and 3 metres at 5 pm. a) Determine a and b. [2 marks] b) Determine the rate of change of the width of the beach at 8 am and the first time after this when this rate of change is repeated. [2 marks]
QUESTION 17 (4 marks) A snail is travelling along a straight path from point 4. The snail`s velocity (cm min ) is modelled by v(t)= 1.4ln(1 e ) where ¢ is time (in minutes) for 0 <z <15. An ant passes point 4 12 minutes after the snail and follows the snail`s path. The ant moves with a constant acceleration of 2 cm min~2 and passes the snail at # =15 minutes. Determine the ant`s velocity at point A4.
QUESTION 19 (4 marks) Flying foxes enter and leave a fruit-growing region every evening. The rate at which the flying foxes enter the region is modelled by the function A(t)= 42sin[0.03t —%) +71 0<1<240 The rate at which the flying foxes leave the region is modelled by the function L(t)= 42sin(0.04t —gj +42 0<1<240 Both A(¢) and L(¢) are measured in animals per minute and ¢ is measured in minutes after 7 pm. There are 100 flying foxes in the region at 7 pm. Determine the maximum number of flying foxes in the region and the time that this occurs.
QUESTION 2 The approximate area under the curve y = 2x? between x = 1 and x = 4 is found using rectangles (of width equal to one unit) as shown in the diagram. -1 0 1 2 3 4 5 What is the approximate area found using these rectangles? (A) 58 units` (B) 45 units` (C) 42 units® (D) 28 units?
QUESTION 3 If y = cos?(3x) then Z—i` = (A) —65sin(3x) cos(3x) (B) —2sin(3x) cos(3x) (C) 2 sin(3x) cos(3x) (D) 6 sin(3x) cos(3x)
QUESTION 4 0 3 ; . i If [ f(x)dx = 4 and [ f(x)dx = —10 which of the following is true? (A [3 fC)dx = —14 B) [° fe)dx =6 © [P fx)dx=6 D) [ f(x)dx =14
QUESTION 5 The slope of the tangent to the graph of y = In(x?) atx = e? is 1 (A) 2 2 (B) 2 4 ©) i 1 (D) —
QUESTION 6 7 dx Calculate f_ 1 TG0 (A 1 4 B) 3 € 2 (D) 4
QUESTION 8 Given y = 5e2* rearrange the function to make x the subject. (A) x=In (%) ®) x=j@ (€) x=In(V5y) D) x=In ((%)Z>
QUESTION 11 (5 marks) Determine: 4 3In(5x) 2) <= 3In(5Y) d b) — (x3cos(x* + 1)) dx c) [3e*dx [1 mark] (You do not need to simplify.) /3 marks] [1 mark]
QUESTION 12 (3 marks) Simplify the following: a) logg9 + 2logg 2 logz 16 b) log; 64 [1 mark] [2 marks]
QUESTION 13 (4 marks) Solve the following: a) logs(2x+4) =2 [2 marks] b) In(7 —x) —In10 = In(x) [2 marks]
QUESTION 14 (5 marks) 1 Letv(t) = = + 3sin(t) t = 0 represent the velocity of an object moving in a straight line. T Att = 3` the position of the object is 4. a) Determine the acceleration function. [2 marks] b) Determine the displacement function. [3 marks]
QUESTION 19 (6 marks) Determine the equation of the tangent to the curve y = x In(x) that passes through the point (0 —e).
QUESTION 1 Determine the asymptote of the graph of the function f(x) = log;(x +5) (A) x=-5 (B) x=5 ©) y=-5 D) y=5
QUESTION 2 A population of bacteria after t hours is given by P(t) = 5000e°%18¢ The rate of increase of the population (to the nearest unit) at 15 minutes is (A) 74 399 bacteria/hour. (B) 13 392 bacteria/hour. (C) 5230 bacteria/hour. (D) 941 bacteria/hour.
QUESTION 3 Using the trapezoidal rule with four sections the approximate area under the curve y = x? + 5 between x = 0 and x = 2 is (A) 12.67 units® (B) 12.75 units® (C) 13.00 units? (D) 13.75 units?
QUESTION 4 The area of the region enclosed by the graphs of y = xvVx + 1 and y = 2x 1s (A) 1.267 units® (B) 0.467 units® (C) 0.200 units` (D) 0.029 units®
QUESTION 5 Two particles move along the x-axis. For 0 < t < 5 their respective position functions are given by: x ; = sin(2t) and x = e~ For how many values of t do the particles have the same position? A) 4 B) 3 €) 1 (D) 0
QUESTION 8 A particle travels along a straight line with a velocity of v(t) = 3e(_7t) X sin(2t) ms™. Determine the acceleration of the particle at t = 2 seconds. (A) —1.03ms™? (B) —0.01 ms™? (C) 2.49ms? (D) 3.51ms~?
QUESTION 11 (4 marks) The Palermo Technical Impact Hazard Scale (P) is a logarithmic scale used by astronomers to rate the potential hazard of a Near-Earth Object (NEO). Potential impacts with positive Palermo Scale values will generally indicate situations that merit some level of concern. For the NEO asteroid Apophis P is given by: P = logso (8.71 Zi10—6) where p; represents the impact probability. a) Determine P for a p; value of 1 in 100 000. [1 mark] John believes that increasing the P value in 11a) by 2 will make the chance of impact 1000 times more likely. b) Evaluate the reasonableness of this statement. [3 marks]
QUESTION 13 (6 marks) 1 11643t [2 marks] —3 Let N(t) = [ 1800t(16 + 3t2)=2 dt a) Determine Z—A: it M(t) = b) Using the result from 13a) determine N (t) [1 mark] ¢) Determine N`(4) [1 mark] The function N (t) models the total number of customers N served by staff after ¢t hours during an 8-hour workday (0 <t < 8). At time t = 0 no customers had been served. d) Determine N(4) [2 marks]
QUESTION 17 (3 marks) During one 30-day period the rate at which pollution passes into a nearby lake is measured every six days and results are given in the table below. Rate of pollution in units per day p(t) This information can be modelled as a quadratic function. Determine the total amount of pollution entering the lake during this 30-day period.
QUESTION 18 (5 marks) The amount of a certain drug in the bloodstream M (mg) at any time ¢t (hours) is modelled closely by M(t) = Ate™t where A and b are parameters. a) Determine the exact values of A and b if the maximum amount of the drug in the [3 marks] bloodstream was 120 mg at t = 2 hours. [2 marks] b) Evaluate the reasonableness of your solution.
QUESTION 20 (5 marks) A farmer has a paddock with straight fences on two sides (fence A and fence B) perpendicular to each other and bounded by a creek on the other side. Fence B is 4 kilometres long. The creek boundary can be modelled using d=In(5x+¢e)—1 where d is the perpendicular distance in kilometres from fence A to the creek and x is the distance in kilometres along fence A from the point O. The farmer wants to divide his paddock area in half with a straight fence parallel to fence B. Not drawn to scale creek Fence B O Fence A Determine where the farmer should locate the fence.
QUESTION 1 el (*) is equivalent to (A) 0 (B) 1 (C) x (D) —
QUESTION 2 If f(x)=e°** determine the value of / (2). (A) e? (B) 2e (C) -e? (D) —2e?
QUESTION 4 20 ; If the gradient of the function f(x) is given by — then f(x) is equal to x? 60 (A) —7+e xX (B) 2 +¢ Xx 10 (C) -y te X (D) he x
QUESTION 5 Determine } aI 1 2x (A) lin 2 (B) tins 2 1 1 D) =In3 (D) In
QUESTION 6 qa —1 Substitutions for / are used to estimate the limit of as h — 0. Which sequence is the most appropriate? (A) —4 -2 -1 -0.5 -0.25 -0.125 ... (B) -0.05 -0.1 -0.2 -0.4 -0.8 ... (C) 2 1 0 -1 -2 -3... (D) 1 2 3 4 5 6...
QUESTION 7 Determine the mean of the continuous random variable X with the probability density function 1 fO)=48 0 otherwise O<x<4 (A) (B) (C) (D) WloO wplRe WlWw CwleH
QUESTION 9 Determine fj zsin Exax (A) 3 (B) 6 (Cc) -3 (D) -6
QUESTION 10 The continuous random variable Y has the probability density function l+y 0<y<¥ 3-1 f(y) = 0 otherwise ] Determine P(O< y< 5 (ay 1 5 (iB) 3 8 (Cc) > 8 3 D) = (D) 1
QUESTION 12 (4 marks) The region bounded by the x-axis and the curve of f(x) =x(4—x) represents the plan of a garden bed. All measurements are in metres. f(x) a) Estimate the area of the garden bed using sums of the form > f (x; )Ox; where x = 1 Xy = 2 x3=3 and x 4=4. [1 mark] b) Use a definite integral to determine the area of the garden bed. [3 marks]
QUESTION 14 (6 marks) The rate of change in the number of bacteria in a science experiment is represented by < =e 420 where ¢ represents the time (hours) since starting the experiment and P represents the number of bacteria present (thousands). Initially there are 60 000 bacteria present i.e. P(O) = 60. a) Determine the equation for P(t). [2 marks] b) Determine the change in the number of bacteria during the third hour. Express your answer in terms of e. [2 marks] c) Determine how long it will take for the number of bacteria present to double after starting the experiment. [2 marks]
QUESTION 16 (5 marks) Solve for x in the equation 4+ 7e?* = 3e”.
QUESTION 18 (4 marks) A person enters the lowest carriage of a miniature Ferris wheel with a six-metre diameter. The bottom carriage is one metre off the ground. When top speed is reached it takes three seconds for a carriage to travel from the lowest to the highest point of the ride. It is claimed that: The vertical motion of the Ferris wheel produces a maximum vertical acceleration on each rider that is more than half the acceleration of free fall. Free fall occurs when gravity is the only force acting resulting in an acceleration of 9.8 ms”. Evaluate the reasonableness of the claim.
QUESTION 19 (7 marks) Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days. The number of daily customers for each shop can be modelled by the equation y = A In( Bx) where x is the day and y is the number of customers. The constants A and B are different for each shop. Determine algebraically whether the total number of customers for Jaxon and Shari’s shops will be the same every day in the future.
QUESTION 1 If f(x) = sin(3x) determine the value of f {Z). (A) 2.772 (B) 1.148 (C) 0.929 (D) 0.383
QUESTION 4 The displacement (m) of a moving particle is given by d = e°°! _1 where ¢ is time (s). The acceleration (ms~”) of the particle when t= 4 is (A) 7.3891 (B) 6.3891 (C) 3.6945 (D) 1.8473
QUESTION 5 Solve In(x) + In(3.70) = In(9.25) for x. (A) 0.92 (B) 1.71 (C) 2.50 (D) 5.55
QUESTION 6 Sa | ; | dx a#0is a x+a (A) 1.7918 (B) 1.6094 (C) 1.3863 (D) 1.0986
QUESTION 8 The number of koalas in a conservation park is modelled by N= 15 In(7t+ 1) t = 1 where ¢ represents the time (years) since the park opened. There were 20 koalas in the park when it opened. Determine the approximate rate of change in the number of koalas when f= 3. (A) 46 (B) 26 (C) 25 (D) 5
QUESTION 9 If f(x) = e?*(x+1) and f(x) = ae**(x + 1) determine the expression for a. (A) 3x45 (B) 3x+3 (C) S5x+5 (D) 5x+3
QUESTION 12 (4 marks) The graph shows the water level under a bridge over a 12-hour period. a) b) Water level (metres) 12:00 am 4:00 am 8:00 am 12:00 pm Time Determine the equation of the cosine function that models the water level as a function of time after 12:00 am. [1 mark] How long in the 12-hour period shown is the rate of change of water level more than 0.55 metres per hour? [3 marks]
QUESTION 13 (4 marks) The curved lines represent graphs of the equations y = x7 —4x+8 and y= 10cos(x +10). y a) Determine the coordinates of the points of intersection A and B. [1 mark] b) State an integral expression representing the area enclosed by the two graphs. [2 marks] c) Determine the area enclosed by the two graphs. [1 mark]
QUESTION 15 (4 marks) Determine the derivative of f (x) =Inx? + In(x —35 y. Express the derivative as a single fraction in its simplest and factorised form.
QUESTION 16 (6 marks) A particle is moving in a straight line. The velocity (ms!) of the particle is given by 20sin (2t v(t) =e) 6 —5cos(2r) The initial position of the particle is +6.0 m from the origin. t > 0 where ¢ is time (s) after moving from its initial position. a) Use calculus methods to determine an equation for the position of the particle from the origin at any time ¢. b) Determine the position of the particle relative to the origin when it first reaches maximum velocity. [3 marks] [3 marks]
QUESTION 18 (5 marks) A company makes windows using glass that has a mass of 5.6 kg per square metre. A customer orders an unusual window in a partial parabolic shape as shown. Not to scale 8m Determine the mass of the window.
QUESTION 19 (6 marks) Over a suitable domain a hill has a cross-sectional area given by fr (x) dx = a +c where: ¢ a b and c are constants b+0 ¢ h(x) represents vertical distance (m) x represents horizontal distance (m). It is known that 4(0) = 1.22 and h(40) = 25. Where the gradient of the hill is 0.86 there is a tree stump. A second tree stump is located further up the hill. The difference in hill gradient between the two tree stumps is 0.44. A surveyor predicts that the vertical distance separating the two tree stumps is between 7.5 m and 8.5 m. Evaluate the reasonableness of this prediction.
No results found
open_with
![Formula-Book-Image](https://examinsightsimages.syd1.cdn.digitaloceanspaces.com/Mathematical_Methods/Mathematical_Methods_Formula_Sheet_1.webp)
![Formula-Book-Image](https://examinsightsimages.syd1.cdn.digitaloceanspaces.com/Mathematical_Methods/Mathematical_Methods_Formula_Sheet_2.webp)
![Formula-Book-Image](https://examinsightsimages.syd1.cdn.digitaloceanspaces.com/Mathematical_Methods/Mathematical_Methods_Formula_Sheet_3.webp)