HSC Mathematics Trial Examination Assessment Task Four 2014 Section I 10 marks Attempt Questions 1-10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1-10. __________________________________________________________________________ x 2 36 1. Which of the following is equal to ? x 6 (A) x 6 (B) x 6 (C) x 3 (D) x 3. 2. What are the solutions to 3 x 2 7 x 1 0 ? (A) x (B) x (C) x (D) x 7 37 6 7 61 6 7 37 6 7 61 6 . 3. What are the exact solutions of 2 cos x 11 (A) and 6 6 5 7 (B) and 6 6 5 (C) and 3 3 2 4 (D) . and 3 3 3 for 0 Page 1 of 11 x 2 ? HSC Mathematics Trial Examination 4. Which of the following define the domain and range of the function f ( x) log e x ? (A) Domain: all real x and Range: all real y. (B) Domain: x 0 and Range: y (C) Domain: all real x and Range: y (D) Domain: x 0 and Range: all real y. 5. What is the derivative of e3 x 1 (A) 2e 3 x e 3 x 1 3 (B) 2e 3 x e 3 x 1 1 (C) 6e 3 x e 3 x 1 3 (D) 6e 3 x e 3 x 1 1 2 0 0 ? . 6. What is the perpendicular distance of the point (4,5) from the line 3 x 2 y 10 0 ? (A) (B) (C) (D) 12 13 17 13 2 5 12 . 41 7. What is the solution of 5 x (A) log 4 5 (B) log 5 4 20 ? (C) 1 log 4 5 (D) 1 log 5 4 . Page 2 of 11 Assessment Task Four 2014 HSC Mathematics Trial Examination 8. A parabola has a focus (3,1) and directrix x = 5. What is the equation of the parabola? (A) y 1 2 (B) y 1 2 (C) x 3 2 8 y 3 (D) x 3 2 16 y 1 . 4 x 4 8 x 3 9. The diagram below shows the graph y f ( x) . Where is the function increasing, at a decreasing rate? (A) 2, 0 (B) 6, 1.8 (C) 10,1.8 (D) 14, 0 . 5 4x3 dx 10. What is the value of 3 (A) 192 (B) 408 (C) 544 (D) 706. Page 3 of 11 Assessment Task Four 2014 HSC Mathematics Trial Examination Assessment Task Four 2014 Section II 90 marks Attempt Questions 11-16 Allow about 2 hours and 45 minutes for this section Answer each question in the appropriate writing booklet. Extra writing booklets are available. In Question 11-16, your responses should include relevant mathematical reasoning and /or calculations. _____________________________________________________________________ Question 11 (15 marks) a) Evaluate ln 5 correct to three significant figures. 3 b) Evaluate lim x 3 x3 3x 2 x 3 c) Differentiate 1 tan x 2 4 2 d) Differentiate x ln x e) Find 2 2 4 xe x 1dx 2 f) Evaluate 0 3x x 2 1 1 2 dx g) Sketch the region defined by x 2 3 y 1 2 9 Page 4 of 11 3 HSC Mathematics Trial Examination Assessment Task Four 2014 Question 12 (15 marks) Start a new booklet. 4 kx 2dx 12 , and k is a constant, find the value of k. a) Given that 2 0 5, 2 and C 11,10 are two points on the number plane. M is the midpoint of AC and the perpendicular bisector of AC meets the x axis at D and the y axis at B. b) A i. Find the coordinates of M. 1 ii. Show that the equation of the perpendicular bisector of AC, i.e. line BMD, is 4 x 3 y 24 0 2 iii. Hence find the coordinates of the points B and D. 2 iv. Show that the quadrilateral ABCD is a rhombus. 2 c) Chairs are arranged in rows in front of a stage in a concert hall, so the row closest to the stage is the first row. Each row has two more chairs than the row in front of it. There are forty-two chairs in the tenth row. i. How many chairs are in the first row? 2 ii. The seating arrangement has a total of 680 chairs. How many rows of chairs are in the concert hall? 3 iii. How many chairs are in the last row? 1 Page 5 of 11 HSC Mathematics Trial Examination Assessment Task Four 2014 Question 13 (15 marks) Start a new booklet. a) The population P(t) of turtles in a conservation park is given by: P (t ) 200 75sin t 3 . where t is time in months. i. Find all times during the first 12 months when the population equals 275 turtles. ii. Sketch the graph of P(t) for 0 t 12. 2 2 b) The diagram shows the graphs of the function g ( x) 3 x and f ( x) 5 x3 5 x 2 27 x The graphs meet at O and T. i. ii. Find the x-coordinate of T. 1 Find the area of the shaded region between the graphs of the functions. 3 c) Tina borrows $5000 at 1.5% per month reducible interest and pays the loan off in equal monthly instalments. Tina is to repay the loan in 3 years. Calculate the value of each monthly instalment. Page 6 of 11 3 HSC Mathematics Trial Examination Assessment Task Four 2014 d) The diagram shows two quadrants, centre O. OA = 3 cm, OD = 2 cm, AOP = radians. Diagram is NOT drawn to scale. 5 2 i. Show that ii. If the area of the shaded region APQC is Find the size of iii. is an expression for the area of the shaded region APQC. 1 5 squared centimetres. 6 AOP. 1 Hence find the exact area of shaded sector OQD. Page 7 of 11 2 HSC Mathematics Trial Examination Assessment Task Four 2014 Question 14 (15 marks) Start a new booklet. a) A particle travels so that its displacement (x metres), after t seconds is given by: x 12t 3t 2 . i. Where is the particle 3 seconds after it starts? 1 ii. When does the particle turn around? 1 iii. How far does the particle travel during the first 5 seconds? 2 iv. Find the greatest speed during the first 5 seconds. 1 b) A cylinder is to be cut from a solid sphere. The diagram below shows a cross section of the sphere and cylinder. The sphere has a diameter of 8 cm. The cylinder has a height of h cm and a radius of r cm. Diagram is NOT drawn to scale. i. Show that the volume (V) of the cylinder is given by: V ii. 2 64 h 2 h 4 Find the value of h such that the volume of the cylinder is a maximum. c) On an island, the population P after t years is given by: P 3 P0 e kt . The initial population of the island is halved in 25 years. ln 0.5 i. Show that k 25 ii. How long will it take for the population to reduce from 5000 people to 2000 people ? 1 iii. 2 What percentage of the original population will be present after 75 years? Page 8 of 11 2 HSC Mathematics Trial Examination Assessment Task Four 2014 Question 15 (15 marks) Start a new booklet. a) i. Copy this table and complete it, leave your answers as fractions. x 1 2 3 1 4 5 2 ( + 1) ii. Use the 5 functional values from part i, and Simpson’s rule, to find an approximation to 5 1 b) 2 dx . x x 1 2 x 2 Write your approximation using two decimal places. 2 2 . x x 1 iii. Show that iv. Deduce the value of the integral in part ii, correct to two decimal places. x 1 2 In the diagram, ABCD is a quadrilateral and BD is a diagonal. CB = 8 cm, AB = 9 cm, AD = 6 cm and BD = 12 cm. DAB = 2 CBD. Diagram is NOT drawn to scale. i. Prove triangle ABD and BDC are similar. 2 ii. Find the length of CD. 2 Prove that AB and CD are parallel 1 iii. Page 9 of 11 HSC Mathematics Trial Examination c) The graphs of y sin x and y 1 cos x are shown intersecting at x Assessment Task Four 2014 and x 3 is rotated 3 2 Calculate the total area of the two shaded regions. Question 16 (15 marks) Start a new booklet. a) The region bounded by the curve y sec x , the lines x 4 and x 3 through one complete revolution about the x axis. Find the volume of the solid of revolution. Give your answer in exact form. b) The acceleration of a particle is given by: x 12e 2t where x is displacement in metres and t is time in seconds. Initially its velocity is 7 ms -1 and its displacement is 4 m. i. Show that the velocity of the particle is given by: x 6e 2t 1 2 ii. Graph the velocity with respect to time. 2 iii. Find the displacement when t = 3 seconds. 2 Page 10 of 11 HSC Mathematics Trial Examination c) Consider the function y 1 3 x x 3 , for 2 Assessment Task Four 2014 x 3. i. Find all stationary points and determine their nature. 3 ii. Find the point of inflexion. 1 iii. Sketch the curve for 2 x 3. Do not find the x- intercepts. iv. What is the minimum value for the curve over the stated domain? END OF THE EXAMINATION Page 11 of 11 1 1

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