TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION 2014 MATHEMATICS EXTENSION 1

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1 Name: Class: TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION 04 MATHEMATICS EXTENSION General Instructions: Total Marks 70 Reading Time: 5 minutes. Section I: 0 marks Working Time: hours. Attempt Question 0. Write in black or blue pen. Answer on the Multiple Choice answer sheet provided. Board approved calculators & templates may be used Allow about 5 minutes for this section. A Standard Integral Sheet is provided. Section II: 60 Marks In Question - 6, show all relevant mathematical reasoning and/or calculations. Marks may not be awarded for careless or badly arranged working. Attempt Question - 4 Answer on blank paper unless otherwise instructed. Start a new page for each new question. Allow about hours & 45 minutes for this section. The answers to all questions are to be returned in separate stapled bundles clearly labelled Question, Question, etc. Each question must show your Candidate Number.

2 Section I 0 marks Attempt Questions 0 Allow about 5 minutes for this section Use the multiple-choice sheet for Questions -0. The focus of the parabola (x ) = 8y is (, ) (, ) (0, ) d) (, ). Find the acute angle between the lines x + y + 4 = 0 and x + y + = 0. Give your answer to the nearest minute. 8ᵒ5 78ᵒ4 54ᵒ8 d) 45. Find the coordinates of the point that divides the interval A (-, 7) B (, 0) externally in the ratio 4:. (-44, -8) (44, -8) (54, -) d) (54, ) 4. sin x equals to tan x +tan x tan x +tan x tan x tan x d) +tan x tan x 5. 8 people are to be seated around a circular table. If people wish to sit next to each other, how many different ways can this be done? d) 5040

3 6. Use the trapezoidal rule to find an approximation for log e x dx using subintervals d).5 7. Find the lim x 0 sin x cos x x d) 4 8. Which of the following is the graph of y = sin x? d) 9. The diagram below shows the path of a projectile fired with a horizontal velocity v from a cliff of height h. Which pair of the following values of v and h will give the greatest value of angle θ? v = 0ms h = 0m v = 0ms h = 50m v = 50ms h = 0m d) v = 0ms h = 50m 0. The solution to t + t > is t > t > or t < t > d) t > or t <

4 Section II 60 marks Attempt Questions 4 Allow about hour and 45 minutes for this section Answer each question in a SEPARATE page. Extra writing papers are available. In questions 4, your responses should include relevant mathematical reasoning and/or calculations. Question (5 marks) Start a new page. Find x (x + ) dx using the substitution u = x + The area bounded by the curve y = cosx, the x-axis and lines x = π and x = π is 4 4 rotated about the x-axis. Find the volume of the solid generated in exact form. There are 4 families, and each family has exactly 4 children. Assume that the probability of giving birth to a male and giving birth to a female are even. Determine the probability that exactly of the families will have exactly males and females as children. Question continues on page 5 4

5 d) Copy the diagram into your answer booklet. PAC, PBD are straight lines. EF is the tangent at P. e) Prove CD EF Prove by mathematical induction that for any positive integer n. End of Question 5

6 Question (5 marks) Start a new page. A body is in Simple Harmonic Motion and its position at a time t is given by the equation x = R cos(nt + α) + The period of motion is π seconds and 0 α π. Initially the body is at rest units to the left of the origin. i. Find the values of R, n and α. Find the velocity of the body when t = π 6 Show that the sum of the Arithmetic sequence is n log 0(x ) n+ log 0 (x ), log 0 (x ), log 0 (x ), log 0 (x ) n Andrew, whose height is metres, throws a ball from area A in the direction of the Cohen building which is 5 metres high. He throws the ball with an initial velocity u at angle α, and he is 0 metres from the base of the building. (Assume x = 0 and y = 0m/s ) i. Show that y = x tan α 5x u ( + tan α) +, at any time t. Hence, find between which two angles of projection must he throw the ball to ensure that it lands on the roof of the building, or over, given that u = 5m/s. (Answer to the nearest degrees). d) i. Differentiate x tan x Hence, or otherwise, find tan x dx End of Question 6

7 Question (5 marks) Start a new page. At any time t minutes, the rate of cooling of a body with temperature T, when the surrounding temperature is S, is given by the differential equation dt dt = k(t S) for some constant k. i. Show that T = S + Ae kt, for some constant A, satisfies this differential equation. A metal rod has an initial temperature of 90ᵒC and cools to 060ᵒC in 0 minutes when the surrounding temperature is 0ᵒC. Find how much longer it will take the rod to cool to 0ᵒC, giving your answer to the nearest minute. 4 Find the monic cubic equation whose roots are the squares of the roots of x + x + = 0 The acceleration of a particle P is given by the equation x = 8x + 7x + 9x Initially x = and the velocity, v = 6. i. Show that v = 9x ( + x) Hence, or otherwise, show that x(+x) dx = t + C, for some constant C i Find the derivative of log e ( + x ) iv. Using your result in part iii and the initial conditions, find x as a function of t. End of Question 7

8 Question 4 (5 marks) Start a new page. Show that x = 5sinθ and y = 5cosθ + satisfies the equation y + x y 4 = 0 In the diagram, PQ and SR are parallel railings which are metres apart. The points P and Q are fixed 4 metres apart on the lower railing. Two crossbars PR and QS intersect at T as shown in the diagram. The line through T Perpendicular to PQ intersects PQ at U and SR at V. The length of UT is y metres. i. Deduce that SR = y 4 Hence show that the total area A of PTQ and RTS is A = 4y + 8 y i Find the value of y that minimises A. Justify your answer. The coefficient of x k in ( + x) n, where n is a positive integer, is denoted by c k (so c k = n C k ) i. Show that c 0 + c + c + + (n + )c n = (n + ) n Find the sum, Showing all necessary working. c 0 c + c 4 + ( )n c n (n + )(n + ) End of paper 8

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