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1 04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators ma be used A table of standard integrals is provided at the back of this paper In Questions 6, show relevant mathematical reasoning and/or calculations Total marks 00 Section I Pages 4 0 marks Attempt Questions 0 Allow about 5 minutes for this section Section II Pages marks Attempt Questions 6 Allow about hours and 45 minutes for this section 60

2 Section I 0 marks Attempt Questions 0 Allow about 5 minutes for this section Use the multiple-choice answer sheet for Questions 0. What is the value of π, correct to significant figures? 6 (A).64 (B).65 (C).644 (D).645 Which graph best represents = ( )? (A) (B) (C) (D) What is the solution to the equation log ( ) = 8? (A) 4 (B) 7 (C) 65 (D) 57

3 4 Which epression is equal to e d? (A) e + c (B) e + c (C) (D) e + c + e + c + 5 Which equation represents the line perpendicular to = 8, passing through the point (, 0)? (A) + = 4 (B) + = 6 (C) = 4 (D) = 6 6 Which epression is a factorisation of 8 + 7? (A) ( )(4 + 9) (B) ( + )(4 + 9) (C) ( )( ) (D) ( + )( ) 7 How man solutions of the equation (sin )(tan + ) = 0 lie between 0 and π? (A) (B) (C) (D) 4

4 8 Which epression is a term of the geometric series 6 +? (A) 07 0 (B) 07 0 (C) 07 (D) 07 9 The graph shows the displacement of a particle moving along a straight line as a function of time t. P O t Which statement describes the motion of the particle at the point P? (A) The velocit is negative and the acceleration is positive. (B) The velocit is negative and the acceleration is negative. (C) The velocit is positive and the acceleration is positive. (D) The velocit is positive and the acceleration is negative. 0 Three runners compete in a race. The probabilities that the three runners finish the race in under 0 seconds are, and respectivel What is the probabilit that at least one of the three runners will finish the race in under 0 seconds? (A) (B) (C) (D)

5 Section II 90 marks Attempt Questions 6 Allow about hours and 45 minutes for this section Answer each question in the appropriate writing booklet. Etra writing booklets are available. In Questions 6, our responses should include relevant mathematical reasoning and/or calculations. Question (5 marks) Use the Question Writing Booklet. (a) Rationalise the denominator of 5. (b) Factorise +. (c) Differentiate. + (d) Find ( +) d. (e) Evaluate π 0 sin d. Question continues on page 6 5

6 Question (continued) (f) The gradient function of a curve = ƒ ( ) is given b ƒ ( ) = 4 5. The curve passes through the point (, ). Find the equation of the curve. π (g) The angle of a sector in a circle of radius 8 cm is radians, as shown in the 7 diagram. p 7 8 cm NOT TO SCALE Find the eact value of the perimeter of the sector. End of Question 6

7 Question (5 marks) Use the Question Writing Booklet. (a) Evaluate the arithmetic series (b) The points A(0, 4), B(, 0) and C(6, ) form a triangle, as shown in the diagram. A(0, 4) NOT TO SCALE C(6, ) O B(, 0) (i) Show that the equation of AC is + 8 = 0. (ii) Find the perpendicular distance from B to AC. (iii) Hence, or otherwise, find the area of ABC. (c) A packet of lollies contains 5 red lollies and 4 green lollies. Two lollies are selected at random without replacement. (i) Draw a tree diagram to show the possible outcomes. Include the probabilit on each branch. (ii) What is the probabilit that the two lollies are of different colours? (d) The parabola = + 8 and the line = intersect at the origin and at the point A. A = 4 = + 8 (i) Find the -coordinate of the point A. (ii) Calculate the area enclosed b the parabola and the line. 7

8 Question (5 marks) Use the Question Writing Booklet. (a) (i) Differentiate + sin. (ii) Hence, or otherwise, find cos + sin. d (b) A quantit of radioactive material decas according to the equation dm dt = km, where M is the mass of the material in kg, t is the time in ears and k is a constant. (i) Show that M = Ae kt is a solution to the equation, where A is a constant. (ii) The time for half of the material to deca is 00 ears. If the initial amount of material is 0 kg, find the amount remaining after 000 ears. (c) The displacement of a particle moving along the -ais is given b = t, + t where is the displacement from the origin in metres, t is the time in seconds, and t 0. (i) Show that the acceleration of the particle is alwas negative. (ii) What value does the velocit approach as t increases indefinitel? Question continues on page 9 8

9 Question (continued) (d) Chris leaves island A in a boat and sails 4 km on a bearing of 078 to island B. Chris then sails on a bearing of 9 for 0 km to island C, as shown in the diagram. N N 4 km B 9 A 78 0 km NOT TO SCALE (i) (ii) C Show that the distance from island C to island A is approimatel 0 km. Chris wants to sail from island C directl to island A. On what bearing should Chris sail? Give our answer correct to the nearest degree. End of Question 9

10 Question 4 (5 marks) Use the Question 4 Writing Booklet. (a) Find the coordinates of the stationar point on the graph = e e, and determine its nature. (b) The roots of the quadratic equation k = 0 are α and β. (i) Find the value of α + β. (ii) Given that α β + αβ = 6, find the value of k. (c) The region bounded b the curve = + and the -ais between = 0 and = 4 is rotated about the -ais to form a solid. = + O 4 Find the volume of the solid. (d) At the beginning of ever 8-hour period, a patient is given 0 ml of a particular drug. During each of these 8-hour periods, the patient s bod partiall breaks down the drug. Onl of the total amount of the drug present in the patient s bod at the beginning of each 8-hour period remains at the end of that period. (i) How much of the drug is in the patient s bod immediatel after the second dose is given? (ii) Show that the total amount of the drug in the patient s bod never eceeds 5 ml. Question 4 continues on page 0

11 Question 4 (continued) (e) The diagram shows the graph of a function ƒ. ( ) The graph has a horizontal point of infleion at A, a point of infleion at B and a maimum turning point at C. B C A O 4 5 Sketch the graph of the derivative ƒ ( ). End of Question 4

12 Question 5 (5 marks) Use the Question 5 Writing Booklet. (a) Find all solutions of sin + cos = 0, where 0 π. (b) In DEF, a point S is chosen on the side DE. The length of DS is, and the length of ES is. The line through S parallel to DF meets EF at Q. The line through S parallel to EF meets DF at R. F R Q D S E The area of DEF is A. The areas of DSR and SEQ are A and A respectivel. (i) Show that DEF is similar to DSR. (ii) Eplain wh DR =. DF + (iii) Show that A =. A + (iv) Using the result from part (iii) and a similar epression for deduce that A = A + A. A, A Question 5 continues on page

13 Question 5 (continued) (c) The line = m is a tangent to the curve = e at a point P. (i) Sketch the line and the curve on one diagram. (ii) Find the coordinates of P. (iii) Find the value of m. End of Question 5 Please turn over

14 Question 6 (5 marks) Use the Question 6 Writing Booklet. (a) Use Simpson s Rule with five function values to show that π π π sec d (b) At the start of a month, Jo opens a bank account and makes a deposit of \$500. At the start of each subsequent month, Jo makes a deposit which is % more than the previous deposit. At the end of ever month, the bank pas interest of 0.% (per month) on the balance of the account. (i) Eplain wh the balance of the account at the end of the second month is \$500 (.00) + \$500 (.0)(.00). (ii) Find the balance of the account at the end of the 60th month, correct to the nearest dollar. Question 6 continues on page 5 4

15 Question 6 (continued) (c) The diagram shows a window consisting of two sections. The top section is a semicircle of diameter m. The bottom section is a rectangle of width m and height m. The entire frame of the window, including the piece that separates the two sections, is made using 0 m of thin metal. m m The semicircular section is made of coloured glass and the rectangular section is made of clear glass. Under test conditions the amount of light coming through one square metre of the coloured glass is unit and the amount of light coming through one square metre of the clear glass is units. The total amount of light coming through the window under test conditions is L units. π (i) Show that = (ii) Show that 5π L = (iii) Find the values of and that maimise the amount of light coming through the window under test conditions. End of paper 5

16 STANDARD INTEGRALS n n+ d =, n ; 0, n + if n < 0 d = ln, > 0 a a e d = a e, a 0 cosa d = sina, a 0 a sin a d = cosa, a 0 a sec a d = tana, a 0 a seca tana d = seca, a 0 a d = tan, a 0 a + a a d = sin, a > 0, a < < a a a d = ln( ) + a, > a > 0 a d = ln( ) + + a + a NOTE : ln = log, > 0 e 6 04 Board of Studies, Teaching and Educational Standards NSW

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