Mathexams 07 Student s Name. Teacher s Name. SPECILIST MTHEMTICS UNIT EXMINTION Paper : Multiple Choice and Extended nswer This exam consists of Section and Section November 07 Reading Time: 0 minutes Writing time: 80 minutes Instructions to students Section consists of 0 Multiple choice questions worth 0 marks. Section consists of 5 Extended answer questions worth 49 marks. There are 69 marks available. The multiple-choice questions should be answered on the separate answer sheet provided. The extended answer should be answered in the spaces provided. Express answers in exact form unless instructed otherwise. ll questions in both sections should be answered. sheet of formulas is provided for this examination. Students may bring ONE bound reference book to the exam. Students may use a CS Calculator. The use of electronic dictionaries is NOT permitted. SM Unit Paper - Technology active examination November 07
Mathexams 07 SM Unit Paper - Technology active examination November 07
Mathexams 07 SECTION : MULTIPLE CHOICE WRITE THE CORRECT RESPONSE ON THE NSWER SHEET PROVIDED. Let f ( x) x 6. Consider the function gx ( ). f( x) The graph of g(x) has an x-intercept m and y-intercept n. The values of m and n are: C D E 5 m and n 4 5 m and 5 m and 7 m and 7 m and 5 n 6 7 n 6 5 n 6 5 n 6. The vector 8i j. The point C lies on the line and divides it in the ratio 3:, such that C = 3C. The vector C i 6j i 3 j C i 3j D i 3j E i 3j is SM Unit Paper - Technology active examination November 07 3
Mathexams 07 3. Points, and C have position vectors a~ i~ j, b ~ ~ 3i ~ j k ~ ~ and c ~ 3 j k ~ ~ respectively. The cosine of angle C is equal to 5 6 0 C D E 7 6 3 6 3 7 6 6 3 4. The hyperbola with equation x x y has asymptotes with equations 9 36 y x and y x yx and y x C yx and y x D yx and y x E yx and y x SM Unit Paper - Technology active examination November 07 4
Mathexams 07 5. If the complex number z has modulus and argument 3, then z is equal to 4 C D E 8i 4i i i 4i Questions 6 to 8 refer to the following velocity time graph. 6. The magnitude of the acceleration is greatest between the points: and and C C C and D D D and E E E and F 7. The average velocity from to F is equal to: 4 m/s 4 m/s C 4.5 m/s D 4.5 m/s E 6 m/s 8. The average speed is equal to: 5.5 m/s 5.5 m/s C 4.5 m/s D 4.5 m/s E 6 m/s SM Unit Paper - Technology active examination November 07 5
Mathexams 07 Questions 9 to 0 refer to the following information. drag car travels a distance of 600 metres in seconds while accelerating uniformly from rest. 9. The acceleration of the drag car can be determined using the formula: v = u + at s ( u v ) t C C = r D s ut at E v = u + as 0. The speed, in km/h, after seconds is: 7 360 C 80 D 480 E 30. The polar co-ordinates for the point P with Cartesian co-ordinates 3, 3 are C D E 7 3, 6 5 3, 6 7 3, 6 3, 6 7 3, 6. polar equation for a spiral is r 3sin r 4 C 4 D r 3 E r sin cos SM Unit Paper - Technology active examination November 07 6
Mathexams 07 3. Consider the parametric equations x( t) 4cos( t) and y( t) sin( t). On Cartesian x-y axes, the set of points ( xywould, ) be C D E a parabola a circle a hyperbola a line an ellipse 4. If z = i, r = z and = rg z, then: r = and = 4 r = and = 4 C r = and = D r = 4 and = 4 E r = and = 4 5. If z = 6 cis 4 and z = cis, then zz is equal to: 3 8 cis 4 cis C cis D 8 cis 5 E cis 5 SM Unit Paper - Technology active examination November 07 7
Mathexams 07 6. 0 kg weight is resting on a smooth plane inclined at 5 to the horizontal and is prevented from slipping down the plane by a string as shown in the diagram. The magnitude of N is approximately N T 5 o 0 kg wt C D E 5 kg wt 9.66 kg wt 8.9 kg wt.59 kg wt 0 kg wt 7. If the following system of forces is in equilibrium, is: 6 kg wt 0 kg wt o 8 kg wt 90 0 C 35 D 0 E 45 SM Unit Paper - Technology active examination November 07 8
Mathexams 07 8. Given that cos = 3 9 where is an obtuse angle, the exact value of cos is C D E 3 6 4 3 3 9. The minimum value of cos (x) + 7 is 3 C 7 D 9 E 5 0. Let cosec x = 3, 3 < x. The exact value of tan x is C D E 0 4 4 SM Unit Paper - Technology active examination November 07 9
Mathexams 07 SECTION : EXTENDED NSWER Give your answers exactly or to two decimal places, unless stated otherwise. Question (8 marks) OC is a parallelogram where D is the midpoint of C. O and D intersect at point P. Let O a and OC c. a) Given that P D, write an expression for P in terms of,a and c. marks b) Given that OP O, write another expression for P in terms of,a and c. marks c) Hence deduce the values of and. marks SM Unit Paper - Technology active examination November 07 0
Mathexams 07 d) Find O P if OC is a square with side length d. 4 marks SM Unit Paper - Technology active examination November 07
Mathexams 07 Question (9 marks) Let u 6 i and w 3i where u, w C. a) Given that i) z u w u z, find iw ii) rg z 3 marks b) Mark uw, and z on the rgand diagram. Label the points as K, L and M respectively. mark c) Calculate the angle KLP. marks SM Unit Paper - Technology active examination November 07
Mathexams 07 d) Find u w in polar form. mark e) Given z z, i) write down z in polar form, mark z ii) find. z mark SM Unit Paper - Technology active examination November 07 3
Mathexams 07 Question 3 ( marks) Two ferries, and, travelling at constant velocities, have parametric equations given by: x ( t) 6 t y ( t) 3t 3 x ( t) t y ( t) t t 0. The distance is measured in kilometres and time in hours. a) Sketch the Cartesian paths of ferry and Ferry on the same set of axes. marks b) Show that the ferries will collide. marks... c) Determine the time of collision. mark... d) Determine the position of collision. mark... SM Unit Paper - Technology active examination November 07 4
Mathexams 07 toy boat s motion can be described by parametric equations: x( t) cos( t) y( t) cos( t) for t 0. e) Find the Cartesian equation of the toy boat s path. marks...... f) Sketch the path indicating the starting position and the initial direction of motion. 3 marks g) Describe the motion of the toy boat. mark... SM Unit Paper - Technology active examination November 07 5
Mathexams 07 Question 4 (8 marks) parachutist drops from an aeroplane so that the constant acceleration during free fall due to gravity and air resistance is 8 ms -. The parachute is released after 6 seconds, uniformly retarding the parachutist in 8 seconds to a constant speed of.5 ms -. This speed is maintained until the parachutist reaches the ground which is 0 metres below the point of release. a) Sketch the velocity time graph. marks b) How long is the parachutist in the air for? marks c) fter how long has the parachutist fallen half the distance? Give the answer correct to two decimal places. 4 marks SM Unit Paper - Technology active examination November 07 6
Mathexams 07 Question 5 ( marks) tower, 7 m high, stands on top of a building 9 m high. n observer at the bottom of the building notices that, as he walks away from the building, the angle which the tower subtends at his eyes seems to increase in size for a certain distance and then decrease. 7x a) Show that tan( ), x 0.. 3 marks x 44 SM Unit Paper - Technology active examination November 07 7
Mathexams 07 b) Sketch the graph of versus x for 0 x 30. Mark and label the maximum point with its coordinates. 3 marks 7x c) Differentiate the expression using calculus. Show all steps. x 44 3 marks... SM Unit Paper - Technology active examination November 07 8
Mathexams 07 d) Explain why will have its greatest value when tan is a maximum. mark e) Show algebraically that the maximum value of is tan. 4 7 3 marks......... End of Examination Paper SM Unit Paper - Technology active examination November 07 9