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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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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.
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