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QUESTION 1 3x— A4 — X 5 dx can be determined using the partial fractions 1 + 2 1—x I+x 1—x The indefinite integral f The value of 4 1s (A) -3 (B) -1 <€) 1 (D) 3
QUESTION 2 When using proof by mathematical induction to show that n(2n—1)(2n +1) is divisible by 3V ne Z™ the inductive step requires proving () (k+1) (B) (k+1) (k+1) (k+1) k +1)(2k)(2k +2) is divisible by 3. ( (2k)(2k +3) is divisible by 3. k +1)(2k +1)(2k +2) s divisible by 3. ( (D) (k+1)(2k +1)(2k+3) is divisible by 3.
QUESTION 3 According to a recent census the mean hours worked per week by all Australian workers is 35.6 hours. A mean of 36.1 hours worked per week is calculated from a random selection of 500 Australian workers. Based on this data which of the following is correct? (A) x=35.6 u=36.1 (B) X =35.6 X=36.1 (C) x=36.1 u=35.6 (D) x=36.1 X =356
QUESTION 4 Consider points A and B as shown. NP e e e e o o - 10 . X The position vector representing the midpoint of AB is 5 (A) |85 10 5 10 8.5 | : o[ ; (B) 5 10 5 8.5 (D)
QUESTION 5§ Determine f 4x(3x2 +5)3 dx (A) %(3x2+5)4+c (B) %(3x2—|—5)4—|—c ©) 2(3x2—|—5)2—|—c (D) 72x> (3x2 + 5)2 te
QUESTION 6 Givenz=2—2iand w=-3 + i calculate z> — (A) 3-9i (B) 3-7i (C) 11-9i (D) 11-7i
QUESTION 7 The diagram shows a slope field. | / / / \ \ \ \ The differential equation represented by the slope field is d 5 A Y2 dx % dy 5y° B) L _2V dx ¥ dy Sy C) === ©) B 2 dy 5)° (D) Y _2V dx x2
QUESTION 8 An equation of the line passing through the points A(2 4 5) and B(3 -2 1) is (A) (B) ©) (D) 2i+4j+5k+t(3i—2j+k) tER —31+2j—k+t(i—6j—4k) t€R x—1 y+6 z+4 2 4 5 -3 y+2 z-1 6 4
QUESTION 9 The scores on a test are assumed to be normally distributed. Researchers use the results from a random sample of scores to calculate a confidence interval for the population mean. However a shorter confidence interval width is required so the researchers decide to use a second sample for their calculations. Assuming that the standard deviations for both samples are the same the researchers can ensure that a shorter confidence interval width is produced by (A) decreasing the sample size and decreasing the confidence level. (B) decreasing the sample size and increasing the confidence level. (C) 1increasing the sample size and decreasing the confidence level. (D) increasing the sample size and increasing the confidence level.
QUESTION 10 The subset of the complex plane that represents arg|z +i — 1| +% =0forzeCis (A) Im(z) (B) Im(z) 5 Re(z) 5 Re(2) ©) Im(z) (D) Im(z) Re(z) Re(z)
QUESTION 11 (7 marks) The vertices of a regular hexagon are positioned on the circumference of a unit circle as shown on the Argand plane. Im(2) Re(2) Consider the complex number w as shown on the plane. a) Determine w expressing your answer in the form 7 cis(6). b) Convert w into Cartesian form. Each vertex of the hexagon is a solution of an equation of the form z” = @ where z € C. c) State the value of n. d) State the value of a. e) Verify that w satisfies the equation z ” = a using the results from 11¢) and 11d). [1 mark] [2 marks] [1 mark] [1 mark] [2 marks]
QUESTION 12 (8 marks) Consider the vertices A B and C of the rectangular prism as shown. a) State the coordinates of A B and C. [1 mark] b) Determine a unit vector n that is normal to the plane containing A B and C. [3 marks] ¢) Verify that 7 is perpendicular to 4B . [2 marks] d) Determine the Cartesian equation of the plane that contains A B and C. [2 marks]
QUESTION 13 (4 marks) The expected value of an exponential random variable X with parameter A > 0 can be determined using the rule E(X)= j;)oox)\e_)‘xdx Use integration by parts to determine £(X). Express your answer in simplest form.
QUESTION 14 (4 marks) The motion of an object that moves in a straight line is given by v(x) —cos ! (2x) where v 1s the velocity (m s™!) and x is the displacement (m) from the origin. a) Determine a(x) where a is the acceleration (m s2) of the object. [2 marks] b) Use the result from 14a) to determine a(0) given —2rn < a(0) < 0. Express your answer in simplest form. [2 marks]
QUESTION 15 (6 marks) The points O(0 0 0) A(-6 2 -2) and C(3 1 2) are represented in three-dimensional space in the diagram. Not drawn to scale z OABC forms a parallelogram in three-dimensional space. a) Determine the coordinates of B. [1 mark] M is the midpoint of BC. b) Determine the vector that represents OM. [1 mark] N divides AM in the ratio 2:1. c) Determine the vector that represents ON. [2 marks] d) Use a vector method to show that O B and N lie on a straight line. [2 marks]
QUESTION 16 (6 marks) Given cos(6) =0V n€ Z use mathematical induction to prove 1— (— l)n cos(2n9> cos(@) — cos(30) + cos(59) — ..+ (_ 1)n+1 cos<<2n — 1)0) N 2005(«9)
QUESTION 17 (7 marks) Determine the smallest positive value of a given fa [sec(2x) + tan (2x) cosec(2x)+1 2 dx =1 —da
QUESTION 18 (6 marks) Consider the function P(z) —22% + az® +62% +az + b where abeZ One of the roots of P(z) 1Sz =—i Determine the possible value/s for a and b such that all remaining roots of P(z) have an imaginary component.
QUESTION 19 (7 marks) A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram. y 4 The bowl has a shape that can be generated by rotating the curve y = S x —l about the y-axis — X for 4 <x<7.6 cm. The bowl is being filled with a liquid at the rate of 77 cm’s L. Determine the rate at which the depth of liquid is increasing when the depth of liquid reaches one-third of the height of the bowl.
QUESTION 1 The position x (m) at time 7 (s) of a 7 kg particle moving in a straight line is given by x=3 —5 +2t—4 for 0 <1 <10 Determine the time when the particle has a momentum of 620 kg m s . A) 173s (B) 2.60s (C) 3.66s (D) 371s
QUESTION 2 The Leslie matrix for a certain endangered species is given. 0.8 24 03 L=04 0 0 0 055 O A group of the species was moved into a secure property at the start of 2018. The initial female population is given. 150 N() = 80 40 The best estimate of the total female population at the start of 2025 is (A) 3000 (B) 4000 (C) 5000 (D) 6000
QUESTION 3 The masses of packages of cheese produced by a company are assumed to be normally distributed with a known mean of ¢ grams and a standard deviation of 7.37 grams. The packages of cheese are labelled to contain 500 grams. Given there 1s a 25% probability that the mean mass of 20 randomly selected packages will be less than the labelled amount the value of u 1s (A) 498.89 (B) 500.25 (C) 50111 (D) 504.98
QUESTION 4 D . o . . . . Tt A particle is moving with simple harmonic motion described by the equation x =1.32 cos[;} where x (m) is the displacement of the particle from a central position over time ¢ (s) t>0 The maximum speed of the particle is (A) 2.07ms! (B) 4.15m 5! (C) 430ms™! (D) 5.28 m 5!
QUESTION 5 The gradient of the tangent at point A on the curve y2 =4x1s 1.36 The x-coordinate of A is A) 0.12 (B) 0.46 (C) 0.54 (D) 147
QUESTION 6 Solve the matrix equation for X. 3% 27l 1 o |00 @ 1 4 o [3 ® 1 5 13 —14 2 ln 12J 54 56 (D) l }
QUESTION 7 The heights of all students at a school were measured. A mean height of 157.0 cm was calculated from this data. A random sample of 35 students from this school was selected. The mean height of this sample was 159.7 cm with a standard deviation of 8.7 cm. The smallest confidence level that could be used to produce a confidence interval that contains u« based on this sample 1s A) 85% (B) 90% C) 95% (D) 99%
QUESTION 8 Letu=1+iandv=-12+5i Re(u5 — |v|) 1S A) 17 (B) -4 <€) 8 D) 9
QUESTION 9 Two objects P and Q move in three-dimensional space such that their positions r over time ¢ are described by the following vectors until they collide. re =2 —40)i+ 22—t +3)j— (6 - 5t)k ro = (—t2+20)i + Bt + t)j + t?k The objects will collide at (A) t=0 (B) t=1 C) t=2 (D) t=3
QUESTION 10 The time taken by the Year 7 students at a particular school to complete a standardised test is known to be normally distributed. A researcher claims that the population mean is 8.2 minutes. The mean time taken to complete this test by a sample of 10 of these students is 8.1 minutes with a standard deviation of 1.2 minutes. The 95% confidence interval for u based on this sample is (A) (7.36 8.84) minutes (B) (7.46 8.94) minutes (C) (7.86 8.33) minutes (D) (7.96 8.44) minutes
QUESTION 11 (5 marks) Teams A B C D and E participated in a competition with the following results: » A defeated D. * B defeated A C and E. e C defeated A and E. e D defeated B C and E. » E defeated A. To rank the teams at the end of the competition the organisers constructed a dominance matrix N that is partially completed. a) By allocating 1 to represent ‘defeated’ and 0 to represent either ‘was defeated by’ or ‘no result’ complete matrix N. [1 mark] Losing teams ooooooooooooooooooooooooooooooooooo ----- A » B g | heeiihociihociiociiond B8l ieaes Siwisss EEEesie) GHESEES wseitese a2 N= cuC .......... | Feeeet et Bevene ool Tuaeed g ooooooooooooooooooooooooooooooooooo = D [: i i B ................................... E The organisers need to rank the teams into individual places from first to fifth place. They decide to use the ranking model N + N? to achieve this. b) Use the model N + N? to rank the teams. [2 marks] c) Use the result from 11b) to identify a limitation of the organisers’ ranking model. [1 mark] d) State a mathematical refinement the organisers could consider to overcome the limitation of the ranking model identified in 11c). [1 mark]
QUESTION 12 (9 marks) For a certain experiment the number of yeast cells VN after # hours in a test tube can be modelled by the differential equation d—N:LN(IOOO—N) fort >0 dr 1000 1000 1 1 a) Given ————=—+———— show that the general solution of the differential N(1000—N) N 1000—N equation can be expressed as N —— |=t+4c 1000N| ln| [2 marks] A scientist commenced this experiment at 9:00 am on a certain day and observed that 100 yeast cells were present at this time. b) Show that the solution of the differential equation can be expressed as ~ 1000 1+9¢” [3 marks] c) Determine the time of day when 900 yeast cells were present. [2 marks] The scientist predicted that the number of yeast cells would eventually exceed 1200. d) Evaluate the reasonableness of the scientist’s prediction. [2 marks]
QUESTION 13 (6 marks) Data records show that the speeds of cars at a particular location on a highway are normally distributed with a mean of 98.7 km h™! and a standard deviation of 4.1 km h™. The speed limit at this location is 100 km h ™. A police officer plans to record the speeds of 20 randomly selected cars at this location. a) Determine the expected number of cars in the sample that will be travelling within + 1 km h™! of the population mean. [2 marks] b) Determine the probability that the mean speed of this sample will exceed the speed limit. [2 marks] There 1s a 5% probability that the mean speed of this sample will exceed £. c) Determine the value of 4. [2 marks]
QUESTION 14 (5 marks) The time ¢ (months) that it takes before a phone owner cracks the screen on their phone can be modelled by an exponentially distributed random variable ) = 0.16e~ 1 t>0 0 otherwise a) Show that f(¢) is a probability density function. [1 mark] b) Determine the probability that a phone owner cracks the screen on their phone within 1 year. [2 marks] Three-quarters of phone owners take between 1 and m months before they crack the screen on their phone. c) Determine the value of m. [2 marks]
QUESTION 15 (4 marks) The position vectors of points P and Q are 2{ — 3j + k and 2i + 2j — 4k respectively. Let O be the origin. a) Determine the angle POQ. [2 marks] Points O P and Q are joined to form a triangle. b) Determine the area of triangle POQ. [2 marks]
QUESTION 16 (6 marks) Consider the identity cos(40) = Acos? (0)+ Bsin? (0)+ C where A Band C € Z a) Determine the values of 4 B and C using De Moivre’s theorem. [5 marks] b) State an appropriate method of verifying your results from 16a). [1 mark]
QUESTION 17 (7 marks) An object is released from rest at a height of 100 m above the ground. The motion of the vertical descent of the object is modelled by v 9 810 004 (v>0) dx where v is the velocity (m s) and x is the displacement from the ground (m). Determine the velocity of the object when it strikes the ground.
QUESTION 18 (6 marks) The mass of a certain species of kangaroo is known to be normally distributed with a mean mass of u kg and standard deviation of ¢ kg. When one of the kangaroos is randomly selected the probability that its mass is greater than 83.2 kg is 0.145. When a sample of 12 kangaroos is randomly selected the probability that the sample mean mass is less than 74.1 kg is 0.079. A 90% approximate confidence interval for u is calculated using a random sample of n of the kangaroos that has a sample mean mass of 79.1 kg and a sample standard deviation equal to o. Determine the possible range of values that n could have been given that the confidence interval did not contain u.
QUESTION 19 (7 marks) An object is swinging at the end of a 0.5 m length of string in a vertical circular path with a constant angular speed completing each revolution in 0.24 seconds. The object is projected from a height of 0.3 m above the ground in a vertical plane and just passes over a narrow pole as shown in the diagram. The pole is 2.05 m high and its base is 14 m horizontally from where the object was projected. _______ iy . ~. Not drawn to scale ~ 14 m 0.6 m A flat-topped vehicle of length 0.6 m and height 0.4 m is initially at rest against the pole as shown in the diagram. At the instant that the object is projected the vehicle moves in a horizontal direction away from the pole in the same vertical plane with an acceleration of magnitude of @ m s 2. The object strikes the middle of the top of the vehicle. Assuming that air resistance is negligible use vector calculus to model the motion of the projectile in order to determine the value of a.
QUESTION 1 Which of the following is a population parameter? (A) s B) u C) x (D) z
QUESTION 2 When the polynomial P(z) = z° —iz* — z—i is divided by z — i the remainder is A) -2 (B) 0 ©) 2i (D) 4
QUESTION 3 An object has a velocity v(7) = e i+ G]fi where ¢ represents time (7 > 0). The displacement r(7) of the object could be (A) —2ei+In(r)k -2t % (B) —2¢ i —LZIG
QUESTION 4 The number of sunflower seeds in each packet produced by a company is known to be normally distributed with a standard deviation of 100. A worker counts the number of seeds in a random sample of four packets and calculates the sample mean. Based on this sampling the standard deviation of the distribution of the sample mean is (A) 25 (B) 50 C) 75 (D) 100
QUESTION 5 The augmented matrix shown is produced when a Gaussian elimination technique is used to solve a certain system of equations with three variables. 1 3|4 0 -1 5 |[-6 0 0 110 The geometric interpretation of the solution to this system of equations is best represented by (A) % | é ©) J J (D) E i
QUESTION 6 The subset of the complex plane that represents |z| = |z — 2| for z € C is (A) Im(2) (B) Im(z) ©) Im(2) (D) Im(2) A
QUESTION 7 The mass of a particular variety of cake is claimed to be normally distributed with a mean of 660 grames. A random sample of five of these cakes is found to have a mean mass of 600 grams. Which option represents an approximate confidence interval for z based on this sample? A B 600 660 600 660 grams grams ©) A (D) A 600 660 600 660
QUESTION 8 Let P(n) be the proposition that Y (r+1)3 =nx3 Vnez* Which option represents a correct formulation of the assumption that P(k) is true Vk € Z* in a proof using mathematical induction? @A) Y (k+1)3* T = kx3k (B) Z (k+1)35 = nx3 © Y (r+1)3 T =kx3 @) YF (r+1)3 =3
QUESTION 9 — e— e— e— e— e— e— e— — — e— e— o— e— e— e— a—ll a—ll a—ll a—ll a—ll a—l e—l e— - /| /S e e = — = = = — — — ) ~N\\\ /) /s rrem——— = = = W ~ ~ ~ ~ N \ N a_ \_A ~ ~ \ _‘_p ~ N N\ 8 \ /| /|- — = = = — S o o o o | e o e e e e — o s e e e am— | — om— c— c— c— c— — c— The slope field for the differential equation — The solution curve to the differential equation that passes through the point (3 —5) also passes through point (A) P B) Q C) R (D) S
QUESTION 10 The 2016 Australian census recorded the number of bedrooms per household. The results are summarised in the histogram as shown. Based on this data the mean number of bedrooms per household was calculated to be 3.5. Frequency 0 1 2 3 4 5 6 Number of bedrooms Fifty samples of size 40 were randomly selected from the census data and the sample means recorded. The histogram that most likely represents the distribution of the sample means is A 20 (B) 204 Frequency S Frequency S ] 0 0 1.5 2.0 25 3.0 3540 45 5.0 5.5 3.0 3.1 3.2 33 34 35 36 3.7 3.8 Number of bedrooms Number of bedrooms <€) 20 (D) 204 > 2 = = Q O =10 S 10 & o ~ L 0 0 3.1 3233343536237 3839 3.1 32 33 34 35 3.6 3.7 3.8 39 Number of bedrooms Number of bedrooms
QUESTION 11 (5 marks) Let f(x)=tan (%j for suitable values of x where f(x)e [—% %j : a) Determine f(2). [1 mark] b) Determine f'(2). [2 marks] c) Use the results from Questions 11a) and 11b) to determine the equation of the tangent to the graph of y = f(x) at x = 2. [2 marks]
QUESTION 12 (8 marks) Consider the plane x—y—2z=15. a) Determine a vector n that is perpendicular to the plane. [1 mark] b) Determine the vector equation of the line / that is perpendicular to the plane and contains the point A(-2 1 3). [1 mark] c) Use the result from Question 12b) to express the equation of the line / in parametric form. /I mark] The line / and the plane intersect at point S. d) Show that the coordinates of S are (2 -3 - 5). [3 marks] ¢) Determine AS. [1 mark] f) Use a property of parallel vectors to verify that AS and n are parallel. [1 mark]
QUESTION 13 (6 marks) Use z =a+bi and w = ¢+ di where a b ¢ d € R to prove ’z — w‘z = ‘Z‘Z + |w|2 — 2Re(zv_v)
QUESTION 14 (6 marks) An object is projected vertically upwards from ground level. After the object has been in motion for t seconds its position vector through the air in metres is modelled by A . r(t)=5t(8-1)j a) Determine the velocity of the object through the air v (t) in metres per second. [2 marks] b) Determine the number of seconds until the object reaches its maximum height. [2 marks] c) Determine the maximum height that the object reaches in metres. [2 marks]
QUESTION 15 (4 marks) 4x—17 2 —x—6 Use partial fractions to determine J dx where xe R x # -2 x # 3. Express your answer in the form ln| f (x)‘ +c.
QUESTION 16 (6 marks) Use mathematical induction to prove that 22 +3n—11is divisible by3VneZ .
QUESTION 17 (7 marks) The area between the graphs of the functions y =4x and y = 2x? is rotated about the y-axis to form a solid of revolution with a volume of V units. Determine the exact value of V.
QUESTION 18 (6 marks) This differential equation can be used to determine the current / (amperes) at time ¢ (seconds) with voltage V (volts) in an electric circuit containing a resistance R (ohms): dl k—+RI=V dt where k& R and V are positive constants and ¢ > 0. Assuming that there is no current in the electric circuit initially show that the size of the current can never be greater than %
QUESTION 19 (7 marks) The velocity vectors of two objects A and B (in m s_l) at time ¢ (in s) are given respectively by v = 6sin(3t)i +6cos(3t) j v = cos(¢)i —sin(¢)j Objects A and B are initially at (-2 0 2) and (0 1 — 1) respectively. Determine the position of Object A when it is 4 metres away from Object B for the first time.
QUESTION 1 The time taken to complete orders at a pizza store is normally distributed with a mean time () of 10 minutes. The owner of the pizza store records the time taken to complete orders for a random sample of 20 pizzas each day over a 30-day period. From this data an approximate 90% confidence interval for 4 is calculated at the end of each day. How many of these confidence intervals would be expected to contain z? (A) 3 (B) 18 C) 27 (D) 30
QUESTION 2 Determine the area of the shaded region between the graphs of the functions y = lsec[£] and y = ZCOS(EJ as shown. 3 3 2 -3 Not to scale (A) 5.29 units? (B) 5.5 units® (C) 5.65 units” (D) 5.71 units?
QUESTION 3 Given n € Z™ for which proposition can the initial statement for mathematical induction be proven? (A) x* - is divisible by (x+ )V (x+y)#0 2 H2 22 > | 2 B) 12422432+ 4n :gn(Zn +3n—1) ©) (n+1)* +(n+2) is divisible by 3 ! _ B ® 2 (2r-1)(2r+1) n+l
QUESTION 4 The mean time that visitors spend at an art exhibition is 39 minutes and the standard deviation is 6 minutes. Determine the approximate probability that the mean time spent at the exhibition by a random sample of 35 visitors is between 38 and 40 minutes. (A) 0.13 (B) 0.16 (C) 0.68 (D) 0.84
QUESTION 5 1 1 A vector normal to the plane that contains the vectors | 3 | and | 0 | 1s 0 2 (A) 6i+2j+3k (B) 6i+2j-3k C) 6i-2j+3k (D) 6i-2j-3k
QUESTION 6 The Cartesian equation of a sphere is given by X+ y2 +z2+2x-2 y=1T. The centre and radius of the sphere are (A) (-1 1 0) and 3 respectively. B) (-1 (C) (1 -1 0) and 3 respectively. ( — (D) —1 1 0) and 9 respectively. 1 —1 0) and 9 respectively.
QUESTION 7 The altitude angle of OA is represented as . z A Not to scale X Given the coordinates of A are (3 4 6) the altitude angle of OA in radians is A) 093 (B) 0.88 ) 0.69 D) 0.66
QUESTION 8 =2 The imaginary part of [Cis (%D 1s (A) (B) ©) (D) —-6.83 -0.71 0.71 1.17
QUESTION 9 Two vertical forces act on a skydiver with a mass of 85 kg as shown. Alir resistance A = y Weight When the magnitude of the air resistance is 62 N the magnitude of the acceleration of the skydiver is (A) 073ms 2 (B) 2.65ms (C) 9.07ms™> (D) 12.44ms™>
QUESTION 10 A random variable is normally distributed with a mean #. An approximate 95% confidence interval for from a sample from this distribution is (209.7 221.9). An approximate confidence interval for 1 based on the same sample using a confidence level greater than 95% could be (A) (206.5 223.3) ®) ( ) (©) (210.6 221.0) D ( ) 208.5 223.1 215.8 228.0
QUESTION 11 (4 marks) OABC is a triangular-based pyramid as shown. A(l 2 5) C(0 5 2) Not to scale Use a vector method to determine the area of the shaded face of the pyramid.
QUESTION 12 (6 marks) The life span of batteries manufactured by a company is assumed to be normally distributed with an unknown mean and standard deviation. A supervisor at the company randomly selects 7 batteries and uses the life spans from this sample to calculate an approximate 95% confidence interval for the population mean of (2321.4 2423.6) hours. a) Determine the mean life span for this sample of batteries. [1 mark] The standard deviation of the life spans of batteries in this sample is 125 hours. b) Determine n. [3 marks] c) Use the result from Question 12b) to explain whether the assumption that the life span of batteries is normally distributed is required to support the supervisor’s calculations. [2 marks]
QUESTION 13 (6 marks) The area under the graph of the function f(x)= 0.2¢7%** for 1< x <9 is shaded. f(x) 1 9 & Not to scale a) Use Simpson’s rule with four intervals to determine an approximation for this area. [4 marks] b) Use a calculus approach to evaluate the reasonableness of your area approximation from Question 13a). [2 marks]
QUESTION 14 (5 marks) The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year. A Leslie matrix L modelling the population distribution of the Tasmanian thornbill has been partially completed. = O s S O B a) State the values of x and y. [1 mark] At the start of 2021 a study began into the population of Tasmanian thornbills. The study: * estimated that the initial female population was 510 in their first year 480 in their second year and 420 in their third year found that the ratio of male to female was approximately 1:2. b) Estimate the total population of Tasmanian thornbills at the start of 2025. [4 marks]
QUESTION 15 (8 marks) Water 1s poured into a cone-shaped cup at a rate of 2 em® s71. The cup has a height of 12 cm and a radius of 6 cm as shown. ] 12 cm h Not to scale J As the cup fills the ratio of the height of the water / to the surface radius of the water » remains constant. a) Given that 7 =2r determine a function for the volume of water in the cup ¥ in terms of 4. Express your answer in simplified form. [1 mark] b) Use the results from Question 15a) to show that the rate at which the height of water in the cup is increasing with respect to time is given by — [3 marks] h c) Determine the rate at which the height of water in the cup is increasing with respect to time when the volume of water in the cup reaches half of the total capacity of the cup. [4 marks]
QUESTION 16 (6 marks) 1 2 2 LetA=|2 1 1 2 2 1 Given A* = pA® +gA? +rA + 31 use matrix algebra to determine the value of the scalars p ¢ and r. p q g P-4
QUESTION 17 (7 marks) An object with a mass of 2 kg is released from rest at the top of a 1 metre long frictionless plane inclined at 30° to the horizontal. A force of P newtons acting parallel to the plane opposes the motion of the object as it travels down the plane. When the object is x metres from the top of the plane its velocity is v m s 4 Va4 -—x? ’ Given ‘P| = determine x when v = 2.
QUESTION 18 (6 marks) Consider the polynomial P(z) =2 +az” +bz+c where a b ce Rand z e C. Two of the roots of P(z) are also roots of z* + 2% + 22 + 2 +1. The remaining root of P(z)is z=2. 2 Given z° —1=(z~- 1)(2 +2+z20 4z + 1) determine a possible expression for P(z). Leave your answer in expanded form.
QUESTION 19 (7 marks) Consider the following information. Continuous random variable X variance The waiting time (minutes) until workers at a certain call centre receive their nth phone call where ne Z¥ is a random variable 7 with probability density function kn tn—l _% >0 f(t)= (n—l)!e t 0 otherwise where k 1s a positive constant. The waiting time until workers receive their 5th call is collected from a random sample of 80 workers. Determine the probability that the mean waiting time from this sample is more than 16 minutes.
QUESTION 1 Let z=a+3i and w= -3+ bi where a b € R. If z = w then (A) a=-3 b=-3 B) a=-3 b=3 (C) a=3 b=-3 (D) a=3 b=3
QUESTION 2 Which statement regarding sample means is true? (A) The distribution of X is always normally distributed. (B) The distribution of X is always normally distributed. (C) The value of x changes when different samples are selected. (D) The value of u changes when different samples are selected.
QUESTION 3 A particle travels in a straight line over time ¢ with a constant acceleration a(t). Which function could represent the particle’s displacement x(t)? (A) x(t)=7 (B) x(t)=t* 1 (C) X(t) = ; D) x(¢)=+r
QUESTION 4 When using proof by mathematical induction to prove De Moivre’s theorem expressed as (rcis(@))n = r cis(n0)Vn € Z* which statement would be correct in the proof of the inductive step? (A) (reis(8)) = rrcis(k6) i 01s(k+ 6) k+1 ( (reis(0)) = ©) (r01s(9))k+1= eis(k6+1) (reis(9)) c1s((k+1)0)
QUESTION 5 Four random samples of different sizes were taken to estimate a certain population mean given a known population standard deviation. A 95% confidence interval was calculated for each sample. Sample 1 — Sample 2 | Sample 3 P Sample 4 P -~ T 1T 1T 71 > 10 15 20 25 30 35 40 Which sample used the largest sample size? (A) (B) ©) (D) Sample 1 Sample 2 Sample 3 Sample 4
QUESTION 6 The Cartesian equation for a sphere with centre (-2 3 —4) and radius 9 is (A) (x=2) +(y+3) +(z-4) =9
QUESTION 7 Two forces act concurrently on a 2 kg object placed at the origin. y Not to scale The magnitude of the acceleration of the object is (A) 2m 572 (B) 23 ms™ (C) 4ms™> (D) 43 ms™
QUESTION 8 Use the substitution u = tan(x) to determine Itan(x)secz (x)dx. (A) %tan(x) +c (B) %tan2 (x)+c ©) tan(x) +c (D) tan’ (x)+c
QUESTION 9 A random variable X 1s normally distributed with a mean of 36 and a standard deviation of 4. The respective mean and standard deviation of the distribution of X from repeated random samples of size 9 are (A) 4 and 4 9 (B) 4and 4 3 (C) 36 and g (D) 36 and %
QUESTION 10 A plane is represented by the equation x —2z = 5. A vector normal to this plane is 1 (A) | -2 5 (B) o O = -2 ~ O N 7~ N 7 N -/ O M AN AN _/ -5
QUESTION 11 (6 marks) The position vector of a particle r|(cm) over time ¢(s) is given by r ()= (2t +1)i +(¢+3) j— (2t -3)k a) Determine the velocity vector of the particle. [1 mark] b) Determine the time when the position vector of the particle is perpendicular to its velocity vector. [2 marks] The position vector of a second particle rz(cm) over time t(s) is given by ry(t)=(16—4t)i —(3t—13)j+ 2k c) Determine whether the two particles collide. [3 marks]
QUESTION 12 (6 marks) Given zy=a +bi zy=c+di Va b c d € R and z # 0 prove the identity _|Zl‘ _|Zz| A D
QUESTION 13 (6 marks) : : : 22 a) Use partial fractions to determine j (2x-3)(x+4) ax 0 b) Use the result from Question 13a) to determine J. 22 Express your answer in simplest form. [4 marks] [2 marks]
QUESTION 14 (4 marks) —0.5(y—4 The slope field for the differential equation Y = w x#0using—6<x<6and-6<y<6 is shown. ax * a) Determine the value of the slope at point A. [2 marks] : dy —0.5(y-4) b) Use the slope field to sketch the solution curve for . given that X when x =-6 y=3.5 & [2 marks]
QUESTION 15 (4 marks) Consider the equation z>=1where z € C. a) Sketch the solutions to z>=1on the Argand diagram. Im(z) Re(2) The solutions to z°=1 can be expressed in the form z = a + bi where a b € R. b) Determine the largest possible positive value of ab. [2 marks] [2 marks]
QUESTION 16 (7 marks) Consider this system of equations that corresponds to three planes. x+5y=14+2z x+z=3y+3 8y—A=3z a) Use a Gaussian technique to determine the value of 4 for which this system of equations has infinitely many solutions. [4 marks] b) Use the result from Question 16a) to determine the infinitely many solutions. Express your answer in the form of a vector equation of a line. [3 marks]
QUESTION 17 (5 marks) The region between the x-axis and the curve of the function y =1+ sin(Zx) for 0 < x < Z is rotated about the x-axis to form a solid of revolution. 2 Determine the volume of this solid. Express your answer in simplest form.
QUESTION 18 (5 marks) It is proposed that the following expression is divisible by (1 + cis(@)) forneZ (1 + cis(@)) = 0. 2n+l > cis(r6) =0 Evaluate the reasonableness of the proposition.
QUESTION 19 (7 marks) The function f (x) passes through the origin. The gradient function of f (x) is defined as g (x) = sin_l(ex ). Determine f (x)
QUESTION 1 A solution of the equation 22 = ai whereae R isz=-2-2i. The other solution is (A) -8i (B) —2+2i C) 2+2i (D) 8i
QUESTION 2 The win/draw/loss results after a netball competition involving five teams is represented in matrix M. = I Winning teams ~ O w2 T 0 Losing teams Q 1 1 R 2 2 S 0 0 0 Key: Team P drew with Team Q defeated Team R and Team T and lost to Team S The model M+M?+M? is used to rank the teams. The final positions from first to fifth are (A) S Q PR T B) S Q PT R © S P QT R D) S P Q R T
QUESTION 3 2 Determine the solution of the differential equation — & M : T given y =0 when x = —. dx cos(2x) 5 (A) y= —2ln‘cos(2x)|—2_35 B) y= —2ln‘cos(2x)|+ 2.35 C) y= —%1n‘cos(2x)|— 0.59 (D) y= —%ln‘cos(2x)|+ 0.59
QUESTION 4 The time taken for students to answer questions in a class is assumed to be a random variable X with an exponential distribution that has the probability density function f(x):{/leix x>0 0 otherwise The mean of X 1s l The mean length of time taken for students to answer questions in this class is 15 seconds. The probability that the next question in this class is answered between 8 seconds and 17 seconds is (A) 0.05 (B) 0.12 (C) 0.22 (D) 0.26
QUESTION 5 A random sample of the petrol price per litre at 50 petrol stations produced a sample mean of $1.52 and a standard deviation of $0.14. Based on this sample and using a z-value of 1.5 an approximate confidence interval for u is (A) ($1.47 8%1.57) (B) ($1.48 $1.56) (C) ($1.49 $1.55) (D) ($1.50 $1.54)
QUESTION 6 A 4 kg object moves in a straight line over time z‘(s) where 0 < ¢ < 5 with velocity v = 9 + 8¢ — 1> (m g7 ) Determine the momentum of the object when ¢ = 3. (A) 24kgms! (B) 27kgm s (C) 96kgm s (D) 100 kg m s
QUESTION 7 Givena = (3n + 2); + 2} b= (n — 2)} and axb = (1 — 2n)l€ the possible values of n are (A) —-5and 2 3 5 B) —land — (B) 3 (C) land —% (D) 5and —%
QUESTION 8 Determine the gradient of the tangent to the curve y2 —3x =5 at the point (1 22 ) (A) 041 (B) 0.53 (C) 1.06 (D) 8.49
QUESTION 9 Consider the matrix equation. Matrix X 1s T 1 ] — AN 402 — N wn wvw wn n S o~ N N <t n N A | ~~ ~ ~ ~ < /m O a ~ —~ ~ N’
QUESTION 10 In a town the mean number of residents per household is 3.79 people with a standard deviation of 1.47 people. Using a random sample of 45 households from the town determine the probability that the mean number of residents per household will be more than 4. (A) 0.17 (B) 0.33 (C) 0.83 (D) 0.96
QUESTION 11 (6 marks) An aerial view of the surface of a dam 6 km in length i1s symmetrically positioned on a Cartesian plane as shown. A dam wall 1s located along the y-axis. 2 2 -2 The surrounding edge of the dam can be modelled by the ellipse (x 7 ) + % =1 for0<x<6. y Dam wall X Not to scale a) Use Simpson’s rule with four strips to determine an approximate area of the surface of the dam. [4 marks] b) Evaluate the reasonableness of this approximation. [2 marks]
QUESTION 12 (5 marks) A scientist collects data for a species of tree frog in a protected area. Details for the female tree frog population are shown in the table. Swvame | 00 | 0 | 02 | 0 The scientist uses a Leslie matrix model to make predictions about the female tree frog population. a) State the initial population matrix. [1 mark] b) Determine the Leslie matrix. [1 mark] A species 1s considered to be endangered if the female population in a restricted area is predicted to fall to less than 125 in the next 20 years. c) Determine whether this species of tree frog is considered to be endangered. [3 marks]
QUESTION 13 (5 marks) An article claims that the mean starting salary of graduates in Australia is currently $64 800 with a standard deviation of $4500. To check the validity of this claim an employment agent intends to collect data on the starting salaries of a random sample of 360 graduates. a) Determine the probability that the sample mean starting salary will be between $64 000 and $65 000. [2 marks] From the data the agent calculates a confidence interval for the population mean starting salary of (564 589 $65 811). b) Determine the sample mean. [1 mark] c) Comment on the reasonableness of the article’s claim based on this confidence interval. [2 marks]
QUESTION 14 (5 marks) An object is moving in a straight line with an acceleration represented by the differential equation dv dt a) Determine the general solution of the differential equation. [3 marks] = —(4 +? ) where v is the object’s velocity (m 57! ) over time #(s) where ¢ > 0 until it comes to rest. The initial velocity of the objectis 1.5 m s b) Determine the time when the particle comes to rest. [2 marks]
QUESTION 15 (5 marks) Consider points A(3 —1 3)and B(1 1 6). a) Determine AB. [1 mark] b) Determine the Cartesian equation of the line that passes through points A and B. [2 marks] Point A lies on the plane ¢ and AB is perpendicular to this plane. c) Determine the Cartesian equation of the plane. [2 marks]
QUESTION 16 (6 marks) An object with a mass of 12 kg lies on a frictionless inclined plane. A rope is attached to the object at an angle of 25° above the plane as shown. Not to scale The force of the rope T N prevents the object from moving. When the rope is detached the object moves down the plane with an acceleration of 5.6 m §72 Determine the magnitude of T.
QUESTION 17 (6 marks) The mass of a population of elephants is known to be normally distributed. A biologist randomly selects a number of elephants from this population and measures their masses. The mean mass of the sample is 5206 kg with a standard deviation of 356 kg. The biologist uses the data to calculate a 90% confidence interval for the population mean mass of (5159.1 5252.9) kg. Determine a 99% confidence interval for the population mean mass based on the same data. pop
QUESTION 18 (5 marks) Consider the polynomials P(z) = 2+ (i- a)22 —2biz +3i and O(z) = z—2i where a be R. P(z Given @ has a remainder of a — bi evaluate the reasonableness that (z — (a —bi )) 1s a factor of P(z).
QUESTION 19 (7 marks) A research organisation plans to use a drone to drop a scientific instrument vertically from a stationary position above the ocean surface. The acceleration (m s_z) of the falling instrument can be modelled by 9.8—0.1v where v is its velocity (m s~ ) In order for the instrument sensors to activate its speed as it hits the ocean surface must reach at least 20 m s~ L. However if it hits with a speed above 50 m s7! the sensors will be damaged. Determine the range of the drone’s flying height above the ocean surface to ensure that the sensors are activated but not damaged.
QUESTION 1 Determine the value of y = arccos (sin (4?”)) in the domain y € [0 7r]. (A) (B) ©) NFENFRIERE (D)
QUESTION 2 The time taken for customers to assemble a particular piece of furniture is normally distributed with a mean of 4 hours and a standard deviation of 36 minutes. A random sample of 9 customers is selected. Each customer assembles the piece of furniture. The mean and standard deviation of the distribution of the mean time taken to assemble the piece of furniture for this sample is (A) g hours 4 minutes. (B) ghours 12 minutes. (C) 4 hours 4 minutes. (D) 4 hours 12 minutes.
QUESTION 3 Which of the following equations describes circular motion? (A) r(t) =sin(t) T+ cos(2t)j (B) 1(t) = 2cos(t)i+ sin(t)jf (C) r(t)=2cos(t)i+sin(2t)j (D) r(t) =2cos(2t) i+ 2sin(2t) ]
QUESTION 4 The following diagram shows a vector 04 in three dimensions and its projection OF in the xy plane. Point C lies on the z-axis. 3 X The angle that represents the altitude of the vector 04 is (A) <«AOB (B) <«AOC (C) «£BAO (D) «BOC
QUESTION 5 A matchbox company advertises that the mean number of matches in their boxes is 50. Peter takes samples of the number of matches in 350 boxes. Based on his sample mean and corresponding sample standard deviation results he calculates a 95% confidence interval for the mean of (45.2 49.6) matches. Which of the following statements best describes a statistical inference from this confidence interval? (A) The company is using false advertising. (B) 95% of the company’s matchboxes do not contain 50 matches. (C) There is a statistical basis for being suspicious of the company’s advertising. (D) The sample size is too small to draw a conclusion about the company’s advertising.
QUESTION 6 NS — 1S 2y AA /1 1/ [ 1 dy _ x dx_ (B) A - o L -~ A A ~ - 7 P 7 - 7~ 7~ ~ = 7 NN IS RN NS s (A) The slope field that represents the differential equation 1 0 VNN NN D ) C (
QUESTION 7 Which of the following best describes an interval estimate of a population parameter? (A) the degree of uncertainty associated with an interval (B) the proportion of values expected to lie within an interval (C) aninterval that is calculated using the population parameter (D) an interval that is likely to include the unknown value of the population parameter
QUESTION 8 Given z = a + bi where a b € R a < 0 and b > 0 the Argand plane that represents a possible position of w where w = iZ7 is (A) (B) Im(z) Im(z) ° © 0 Re(z) 0 Re(z) (©) (D) Im(z) Im(z) 0 Re(z) 5 Re(z)
QUESTION 9 1 An antiderivative of ———=———F73x 15 V4 — x? cos‘l(E) (A) —In |cos‘1 g) (B) In |cos‘1 (g) (C) —2In |cos‘ (D) —2In |sin‘1 (g)
QUESTION 10 2 4 The value of f 0 m dx is X (A) itan‘1 G) (B) tan! G) (C) 2tan! G) (D) tan 1(2)
QUESTION 11 (9 marks) A parallelepiped is a three-dimensional figure defined by three vectors that form six faces that are parallelograms. F A D Letfizp m=q fi=r. a) Express the area of parallelogram ABGH in terms of p q and/or 7. [1 mark] b) Assuming A is the origin express the vector equation of the line passing through points B and D in terms of p q and/or 7. [2 marks] Two internal diagonals of the parallelepiped AF and BE are contained by the parallelogram ABFE. c) Express both AF and BE in terms of p qandr. [2 marks] d) Use the results from 11c¢) to prove that internal diagonals AF and BE bisect each other. [4 marks]
QUESTION 12 (10 marks) The diagram below shows a ramp inclined at an angle of 30° to the horizontal. A trolley of mass 20 kg is being pushed up the ramp by a worker with a net force parallel to the plane. Not drawn to scale a) Sketch and appropriately label all forces acting on the trolley in this situation on the diagram below. Assume there are negligible frictional forces and no air resistance. [2 marks] . \. Not drawn to scale b) Determine the minimum force required by the worker to prevent the trolley from moving down the ramp. [2 marks] c) Ifthe worker exerts a force of 178 N on the trolley determine the acceleration of the trolley up the ramp. [3 marks] A worker pulls the trolley up the same ramp with a net force parallel to the plane as shown in the diagram below. t The velocity v (m s~1) of the trolley at time t (s) is given by v = S m sTH(0<t<55). After 2 seconds the trolley is positioned 5 m from the bottom of the ramp. Not drawn to scale d) Determine the position of the trolley from the bottom of the ramp after 4 seconds. [3 marks]
QUESTION 13 (7 marks) Consider the following system of equations. x+2y+3z=4 3x —2y+3z=5 2x +4y+6z =k a) Use a Gaussian technique to determine the value of k for which there are infinitely many solutions. [3 marks] Consider the original system of equations when k = 3. b) Determine the number of solutions of this system of equations. Justify your conclusion. [2 marks] c) Use your result from 13b) to describe a geometric interpretation of this system of equations. [2 marks]
QUESTION 14 (6 marks) a) Determine the values of A and B in the following equation. 4x-1 A B (2x—1)2 (2x-1) (2x—1)2 [3 marks] b) Use the results from 14a) to determine the value of the following definite integral. dx J-Z 4x—-1 1 (2x—1)2 [3 marks]
QUESTION 15 (6 marks) Use mathematical induction to prove that 22 — 3n — 1 is divisible by 3 forn € Z*.
QUESTION 16 (8 marks) The general form of an ellipse in Cartesian form is (x —h) +(y—k) a? b2 =1 The path of an object with a position function of r(t) = cos?(t) i + sin(2t) j is stated to be elliptical. Evaluate the reasonableness of this statement.
QUESTION 17 (7 marks) The curve shown below represents the solution to the differential equation ay _ ¥ dx X +1 x>0 < A(e? 0) Use y = wx (where w is a function of x) to determine the equation of the curve in simplest form.
QUESTION 18 (7 marks) The following complex equations share at least one common solution. z3—8=0 kz* + 2kz3 — 2kz®> — 6z — 24k =0 Given k € R determine the value of k.
QUESTION 1 Letz = cis (Z?n) Im (z*%) is (A) —0.95 (B) 0.09 (C) 031 (D) 0.81
QUESTION 2 The line [ 1s described by the vector equation X 0 4 (y)=<7>+k(16) Z 13 —2 Which of the following points lies on line [? A) (0 7 11) (B) (2 15 12) (C) (416 11) (D) (4 23 15)
QUESTION 3 Use implicit differentiation to determine the value of the gradient of 2e* — 3y? + 372 = 2e at (1 7). (A) —24.17 (B) —13.41 (C) 0.29 (D) 2.85
QUESTION 4 An object lies on a horizontal plane as shown in the diagram below. When a force of 350 N is applied at an angle of 20° the object remains stationary. @ Not drawn to scale The frictional force acting on the object correct to the nearest unit is (A) 119N (B) 120N (C) 328N (D) 329N
QUESTION 5 When using proof by mathematical induction to show that (1 + i)* = (—4) where n € Z* the inductive step requires the proof of A) A+d*=(D (B) (1+D* = (-4 (C) (14 )H+1 = (—4)k+1 (D) (1 + 0)*+4 = (—4)k+1
QUESTION 6 1 The vector (—2) is normal to the plane containing the points (0 0 0) (3 4 5) and (2 k 4). 1 The value of k is (A) —6.5 (B) -3 © 3 (D) 6.5
QUESTION 7 Consider the line that passes through the origin as shown in the complex plane below. Im(z) Re(z) Not drawn to scale The subset of the complex plane that best defines this line is (A) |z—3+2i| =36 (B) |z+3-2i| =36 (C) arg(z—(3—2i))=123.7° (D) arg(z—3+ 2i) = 146.3°
QUESTION 8 Which of the following is a solution of the equation z2 + 2iz — 5 = 0? (A) 1-—2i (B) 1+ 2i C) 2—i (D) 2+
QUESTION 9 The win/loss results after a soccer competition involving four teams are represented in the matrix M shown below. Losing teams A B C D o A 0 2 1 0 = M = § B 0 0 0 | EJ 2 C ] 2 0 | 5 = D 2 ] 1 0 Key: Team A defeated Team B drew with Team C and lost to Team D. Given the model M + M? + M3 to rank the teams determine their final positions from first to fourth. (A) C A D B (B) C D A B (C) D A C B (D) D C A B
QUESTION 10 The velocity v (m s~ 1) of a particle at time ¢t (s) is given by v = sin 1(¢) (0 <t < 15). The acceleration of the particle when t = 0.2 s is (A) 0.20ms™2 (B) 1.02ms™? (C) 1.04ms™2 (D) 1.37ms™2
QUESTION 11 (6 marks) a) Use the binomial theorem to expand (cos(8) + i sin(@))3. Express your result in simplified form. [2 marks] b) Use De Moivre’s theorem to expand (cos(8) + isin(6))3. [1 mark] ¢) Use your results from 11a) and 11b) to prove cos(38) = 4 cos>(0) — 3 cos(8). [3 marks]
QUESTION 12 (6 marks) Consider the semicircle shown below. a) Determine the equation of the semicircle in the form y = f(x). b) Determine the volume formed by rotating the semicircle about the x-axis. c) Evaluate the reasonableness of your answer to 12b). [2 marks] [2 marks] [2 marks]
QUESTION 13 (5 marks) The heights of a sample of male and female students in a secondary school are summarised below. Mean height (cm) Standard deviation (cm) a) Determine the approximate 95% confidence interval for the population mean of the heights of the female students. [2 marks] The confidence interval (175.377 cm 177.223 cm) was calculated for the population mean of the heights of the male students. b) Determine the confidence level that was used to calculate this interval. Give your answer correct to 0.1%. [3 marks]
QUESTION 14 (5 marks) An exponential random variable X with parameter A > 0 has a probability density function given by _ [ re™™ x>0 1) {O otherwise 1 The mean of X 1s ;\ The length of time that car owners keep their first car is an exponentially distributed random variable with a mean of 4 years. a) Determine the probability that an individual keeps their first car for between 3 and 5 years. [3 marks] 50% of all individuals keep their first car for at most k years. b) Determine the value of k. [2 marks]
QUESTION 15 (6 marks) A research student has collected fertility and survival rates for a certain endangered species over a number of years. A small group of this species has been moved into a secure property. The Leslie matrix for the survival rates of the species is 0 0.1 0.8 0.5 095 O 0 0 0 083 0 0 0 0 064 0 The female population of the species moved into the property at the start of Year 1 is 5 & 50 tl2s 20 a) Explain the meaning of the number 0.8 in the Leslie matrix. [1 mark] b) Calculate the expected total female population in the property at the start of Year 2. [2 marks] c) Determine the expected total population in the property at the start of Year 11 based on 55% of the population being female. [3 marks]
QUESTION 16 (4 marks) A plane passes through the points A(0 0 1) B(1 2 3) and C(—1 0 —2). a) Determine a vector normal to the plane. [2 marks] b) Use your result from 16a) to determine the Cartesian equation of the plane. [2 marks]
QUESTION 17 (7 marks) At a honey bottling factory the honey dispensing machine is designed to fill each bottle with masses that are normally distributed with a mean of 503.5 grams and a standard deviation of 17.2 grams. The company randomly samples bottles of honey so that the probability of the mean mass being greater than 500 grams is 0.9. a) Determine the sample size that the company used. [5 marks] A worker at the factory claims that the results of the sample from 17a) would not be valid as the mass of honey dispensed into each bottle cannot be assumed to be normally distributed. b) Comment on the reasonableness of this claim. [2 marks]
QUESTION 18 (7 marks) An object of mass 5 kg moves in a straight line while being acted on by a net force F (N) in terms of its position from a reference point x (5 m < x < 10 m) such that 60 + 5x T 12x— 7%+ 3 Given the object was at rest when x = 5 m determine the position of the object from the reference point when its speed is 0.5 m s™.
QUESTION 19 (7 marks) Two objects A and B move in three-dimensional space such that their positions over time (t = 0) are described by the following vectors until they collide. r (t) = cos(at) i + 2 sin(at) j — V3 cos(at) k bt V3t rp(t) = i +/3btj +—k a) Given —t < a <0 determine when and where the objects collide. [5 marks] b) Evaluate the reasonableness of your result from 19a). [2 marks]
QUESTION 20 (7 marks) A cubic function of the form y = ax3® + bx? + cx passes through the points (2 2) (3 1) and (5 5). a) Use matrix algebra to determine the values of a b and c. [5 marks] b) Evaluate the reasonableness of your result from 20a). [2 marks]
QUESTION 1 The position of a particle is given by r= (t+ 2)i + tj for t>0. Determine the corresponding Cartesian equation. (A) yax-4 (B) y=x°+4 (C) y=x —4x+4 (D) y=x*+4x+4
QUESTION 2 Consider the proof of the following proposition using mathematical induction. 3 rr el) =Sn(n +1 +2) VneZ* r=l An appropriate assumption statement within the proof is k (A) Yo k(k+)= kk +k +2) r=l k (B) > k(k+1) = 5n(n+ (+2) r=l k (C) Dr(r+)= = kk +k +2) r=] k 1 (D) Yir(r+) =3n(n+ n+ 2) r=1
QUESTION 3 One solution of z? — z7 — 7z-2=0 isz=-2. Which equation could be used to determine the remaining solutions? (A) z7-3z-1=0 (B) z7-3z+1=0 (C) z7-z-1=0 (D) z>-z+1=0
QUESTION 4 The age-specific population distribution of a particular species of animal is shown. see eae) redngrae | 0) Sunivalewe | 06 | os) oe The Leslie matrix based on this data is 94 82 37 6 (Ay [06 0 0 0 0 08 0 0 L0 0 04 0 1 2 3 4 (B) |13 0 0 0 09 0 0 0 0 02 0 0 09 0 oo Oo O&O a (D) 0 08 0 0 0.6 0.8 0.4 (C) 13 0 O 0 1.3 0.9 0.2 06 0 O 0
QUESTION 5 A confidence interval for a parameter is a range of values within which the (A) sample estimate of the parameter always lies. (B) sample estimate of the parameter never lies. (C) parameter always lies. (D) parameter never lies.
QUESTION 6 The shaded region defined as {2 :|z+2-i|< 5} a {z : Re(z)< hz € C is best represented by (A) (B) (D)
QUESTION 7 The differential equation for which the solution is a logistic equation of the form y = and C are constants is —— where a b b+Ce dy A) ~—=0.25(1—0.01¢ (a) =0.25(1-0.011 (B) ® _.25(1-0.01y) dt (C) ® _0.251(1-0.011) ame (D) “ = 0.25y(1—0.01y)
QUESTION 8 Point A is the centre of a sphere and point B lies on its surface as shown. The equation of the sphere is (A) x7 -2x + y74274+27=23 (B) x7 +2x4+y?4+27-27=23 (C) x? -2x4+y? 4727422 =25 (D) x7 4+2x+y?427-27=25 @ ici 0 -1) @ B(1 -3 -5) Z KRr-c-cccoo oo \
QUESTION 9 The geometric interpretation of a certain system of three equations with no solution is shown. Given two of the equations are x + y—z=0.5 and x —y —z=0.5 the third equation could be (A) 2x-2y-2z=1 (B) 2x+2y-—2z=1 (C) 2x-2y+2z=3 (D) 2x+2y-—2z=3
QUESTION 10 A random variable is drawn from a population with the distribution shown in the histogram. Frequency 0 4 8 12 16 20 A number of samples of size 10 were randomly selected from this distribution and the sample means xX were recorded. The histogram that most likely represents the distribution of the sample means is (A) (B) 1 Frequency Frequency T TT 0 4 8 12 16 20 0 4 8 12 16 20 (C) (D) > > S S oO oO ; 3 3 3 fo fa 0 4 8 12 16 20 0 4 8 12 16 20
QUESTION 11 (5 marks) Determine the following definite integrals. a) b) ty 1+x 5 AX oer; 2sin? (x) dx oes [2 marks] [3 marks]
QUESTION 12 (5 marks) 1 -2 0 2 -1 -l Given A= f >} B= t ‘ and C -( 0 3} determine X in the matrix equation XA—-XC=B.
QUESTION 13 (5 marks) z| Given z € C where z # 0 prove —— = | zl ZZ
QUESTION 14 (6 marks) Consider a cube with three edges positioned along the x- y- and z-axes on the Cartesian plane as shown. Points O A and B are vertices of the cube. Not to scale 0% A a) Given OA =2i determine OB. Express your answer in terms of j and k [1 mark] b) Calculate OA x OB. [1 mark] Consider the triangle formed by joining points O A and B. c) Use the result from Question 14b) to determine the area of the triangle. [2 marks] Let points M and N be the midpoints of the triangle’s sides OA and OB respectively. d) Determine MN. [1 mark] e) Use the result from Question 14d) to show that the length of AB is twice the length of MN. [7 mark]
QUESTION 15 (5 marks) The sum of a geometric progression with n terms where the first term is | and the common ratio is 7 is given by nN ltertrPers...gr = + (for r #1). r- Prove that this rule is true Vn € Z* using mathematical induction by completing the steps of the proof as indicated. a) Initial statement: [1 mark] Assuming the rule is true for n = k rk] l4¢r¢r°4¢re4+.i¢r%t = 1 (r 41) b) Inductive step: [3 marks] c) Conclusion: [1 mark]
QUESTION 16 (5 marks) A curve is defined by the parametric equations x = 2 tan(@) and y =3sin(26) where 0< 0 < > Given that dy can be expressed in the form acos* (0)+ bcos” (0) where a b € R determine the values he of a and b.
QUESTION 17 (7 marks) An object of mass 2 kg is moving with a constant velocity (m sg!) of v=3i +k. At an instant two forces (N) F = Stj- 3k and F = -tj+ k act simultaneously on the object for t seconds where 0<¢ <2. Determine the magnitude of the momentum of the object when ¢t=1.
QUESTION 18 (6 marks) ; ; a: ee A particular solution to the differential equation dx (¥ n 1) tan( y) through the origin. Determine this solution in the form x = f(y). Leave your answer in simplified form. where x = 0 and = < y <0 passes
QUESTION 19 (6 marks) Object A is released from the origin with constant velocity v such that its position after t seconds is given by r =2V3tit+3tj+2tk t>0. At a later time object B is released from point P (3V3 6 0) and travels towards point 0(5V3 8 4) with constant velocity vp such that Iv al =/2 lv Al: Given that objects A and B collide determine the time between the release of the two objects. Assume all positions are given in metres and all velocities are given in metres per second.
QUESTION 1 The acceleration (m s) of an object moving with simple harmonic motion is modelled by a = —2.95x where x is its displacement (m) from the origin. Determine the period of the motion in seconds. (A) 1.72 (B) 2.13 (C) 2.95 (D) 3.66
QUESTION 2 The standard deviation for the scores of 1000 students completing an entry test at a certain university is 13. A researcher takes repeated random samples of the test results with each sample comprising 40 scores and calculates the mean score for each sample. Determine the standard deviation of the distribution of the sample mean scores. (A) 3.08 (B) 2.06 (C) 0.41 (D) 0.33
QUESTION 3 Given that 2i is a root of z’- pz-—q=9 where p g € R determine the values of p and g. (A) p=-4 and q=-4 (B) p=-4 andq=4 (C) p=0 and gq=-—4 (D) p=0 and q=4
QUESTION 4 The position of a particle can be modelled using r = cos(t)i—2sin(t) j 20. Which curve best represents the path of the particle? (A) y (B) (C) y (D)
QUESTION 5 A plane contains the origin and the points (1 2 3) and (3 2 1). A vector normal to the plane is 4 (A) | -8
QUESTION 6 Two coplanar forces of magnitudes 12 N and 10 N act on an object in the directions shown. Not to scale y 22° 10N 12N Determine the magnitude of the resultant force acting on the object. (A) 12.41 N (B) 15.55N (C) 15.69N (D) 18.27N
QUESTION 7 Matrix N represents the results for a competition involving four teams. Losing teams P QR S_ a P 0 01 1 Nae wo 2/1 0 0 0 Ss =RiO 1 0 0 & = S10 1 1 0. Key: Team P lost to team Q but won against teams R and S. Using the ranking model N + 0.5N7 the teams that placed first second and third respectively are (A) PS andQ. (B) P SandR. (C) S PandQ. (D) S PandR.
QUESTION 8 Given f(x) = tan! (2x) determine (3). (A) 0.05 (B) 0.15 (C) 2.17 (D) 3.10
QUESTION 9 The time in minutes between the arrival of customers at a certain shop is assumed to be a random variable X with an exponential distribution that has the probability density function 0.1207 0-12 x>0 0 otherwise f(x) = | A customer arrives at the shop. The probability that the next customer arrives within 30 to 60 seconds to the nearest percent is (A) 3% (B) 5% (C) 7% (D) 11%
QUESTION 10 The Argand diagram that represents the solutions to z* =16 cis 3 (CS zeECis (A) (B) Im(z) (C) (D)
QUESTION 11 (4 marks) The bounded region between the graphs of the functions y = —1+sec =| and y = 0. 1x” over a certain domain is shaded as shown. The two functions intersect at the origin and point A. 7 A 6 x a) Determine the coordinates of point A. [1 mark] b) Calculate the area of the shaded region. [1 mark] The shaded region is rotated about the x-axis to form a solid of revolution. c) Determine the volume of the solid formed. [2 marks]
QUESTION 12 (7 marks) Consider the complex number z = -—3 + 2i. a) Determine z using the binomial theorem. Leave your answer in the form a+bi where a be R. b) Convert z into the form of rcis(@) where —z <0 < z. c) Use the result from Question 12b) to determine z using De Moivre’s theorem. Leave your answer in the form of rcis(@) where -a <0 < a. d) Evaluate the reasonableness of your results from Questions 12a) and 12c) noting that the two methods to determine z* should produce the same result. [2 marks] [1 mark] [2 marks] [2 marks]
QUESTION 13 (4 marks) The wait time for customers put on hold when calling complaint departments is assumed to be normally distributed. A company claims that the mean wait time for their customers is 7.6 minutes. The following data represents the wait time (minutes) from a random sample of 12 customers who called the complaint department of this company. a) Determine the mean of this data. [1 mark] The standard deviation of this data is calculated to be 2.384 minutes. b) Use an approximate 95% confidence interval for the mean to evaluate the reasonableness of the company’s claim. Justify your decision using mathematical reasoning. [3 marks]
QUESTION 14 (4 marks) At a certain location a biologist measures the width of a river to be 12 m. She also records the depth of the river at regular 2 m interval widths as shown. wien) | oo | 2 | 4 | 6 | s | wo | 2 The biologist estimates the cross-sectional area of the river at this location to be 15 m”. Use Simpson’s rule to evaluate the reasonableness of this estimation. Justify your area calculation and decision regarding reasonableness using mathematical reasoning.
QUESTION 15 (7 marks) The travel time for students attending a certain university is assumed to be normally distributed with a population mean of 25.2 minutes and standard deviation of 4.7 minutes. Travel times are collected from a random sample of 120 of these students and used to calculate a sample mean X | 1N minutes. a) Determine P(X i< 25). [2 marks] b) Given P(X 1> k) = (0.9 determine the value of k. [1 mark] Travel times are collected from a second random sample of the university’s students and used to calculate a second sample mean X’> in minutes. c) Given P(X < 25) 0.4 determine the number of students in the second sample. [4 marks] 2
QUESTION 16 (6 marks) A curve modelled by the relation xy —y+ cos /(2x) = 1 where —0.35<x<0.27 andO<y<l intersects the y-axis at point A. Determine the equation of the tangent to the curve at point A.
QUESTION 17 (6 marks) An object is projected upwards from ground level with an initial velocity of 15 m sat an angle of 54° to the horizontal. The object just passes over a drone hovering in the air. An observer is positioned directly below the drone and at a horizontal distance of 20 m from where the object is projected. The observer commented that: ¢ it took the object around 2 to 2.5 seconds after its projection to reach the drone ¢ the object was still moving in an upwards direction as it passed the drone. Assuming that air resistance is negligible use a vector calculus approach to evaluate the reasonableness of the observer’s comments.
QUESTION 18 (5 marks) Consider the complex solutions to the following equation where 0 < arg(z) < z. (z+1)(2!4-2 47) —zila 474_73 +2°-z)=1-2 Let w be the solution with the maximum possible real part and w be the solution with the maximum possible imaginary part. 4 Show that Mi EZ. WwW
QUESTION 19 (7 marks) The height of Year 9 students at a school is assumed to be normally distributed with a population mean height of uw cm. A teacher at the school measured the height of all the students in her Year 9 class. This data was used to calculate an approximate 95% confidence interval for u of (163.7 166.9) cm. The teacher repeated the procedure using data from another Year 9 class. Although this class had the same number of students its data produced an approximate 95% confidence interval for uw of (167.8 172.4) cm. Using the same data the teacher recalculated the approximate confidence intervals for for each class using a confidence level of x%. She observed that the upper bound of the confidence interval from her Year 9 class now equalled the lower bound of the confidence interval from the other Year 9 class. Determine the value of x. Give your answer rounded to one decimal place.
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