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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 6 7 dx Calculate f_ 1 TG0 (A 1 4 B) 3 € 2 (D) 4
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 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 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 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 9 Determine fj zsin Exax (A) 3 (B) 6 (Cc) -3 (D) -6
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 6 Sa | ; | dx a#0is a x+a (A) 1.7918 (B) 1.6094 (C) 1.3863 (D) 1.0986
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)