Higher School Certificate

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1 Higher School Certificate Mathematics HSC Stle Questions (Section ) FREE SAMPLE J.P.Kinn-Lewis

2 Higher School Certificate Mathematics HSC Stle Questions (Section ) J.P.Kinn-Lewis

3 First published b John Kinn-Lewis in 06 John Kinn-Lewis 06 National Librar of Australia Cataloguing-in-publication data ISBN: This book is copright. Apart from an fair dealing for the purposes of private stud, research, criticism or review as permitted under the Copright Act 968, no part ma be reproduced, stored in a retrieval sstem, or transmitted, in an form b an means, electronic, mechanical, photocoping, recording, or otherwise without prior written permission. Enquiries to be made to John Kinn-Lewis. Coping for educational purposes. Where copies of part or the whole of the book are made under Section 5B or Section 5D of the Copright Act 968, the law requires that records of such coping be kept. In such cases the copright owner is entitled to claim pament. Tpeset b John Kinn-Lewis Edited b Sarah Murph Printed in Australia

4 Other Publications HSC Preliminar Mathematics Multiple-Choice Questions b Term With Video Solutions HSC Preliminar Mathematics HSC Stle Questions (Section ) HSC Preliminar Mathematics HSC Stle Questions (Section ) Solutions HSC Mathematics Harder Questions b Topic HSC Etension Mathematics Harder Questions b Topic HSC Etension Mathematics Harder Etension Questions b Topic HSC Mathematics Multiple-Choice Questions b Term With Video Solutions HSC Mathematics HSC Stle Questions (Section ) Solutions HSC Preliminar Etension Mathematics Multiple-Choice Questions b Term With Video Solutions HSC Etension Mathematics Multiple-Choice Questions b Term With Video Solutions

5 CONTENTS Preface.. 5 How to Use this Book... 5 Sllabus Reference (S R). 6 Term Set.. 9 Term Set.. 8 Term Set.. 6 Term Answers... Term Set.. 4 Term Set.. 47 Term Set.. 5 Term Answers Term Set.. 65 Term Set.. 7 Term Set.. 78 Term Answers Term 4 Set.. 90 Term 4 Set.. 97 Term 4 Set.. 04 Term 4 Answers... Revision Test... 8 Revision Test... 5 Revision Test... Revision Tests Answers.. 8 4

6 Preface In 0 the format for the Higher School Certificate (HSC) eamination papers for Mathematics, Etension Mathematics and Etension Mathematics changed. The Mathematics paper began with 0 multiple choice questions followed b si questions, each worth 5 marks. The 5 mark questions were numbered,,, 4, 5 and 6, the total being 00 marks. This book follows the Board of Studies (BOSTES) recommendation for the HSC course sllabus topics. This recommendation is not mandator for schools, consequentl schools ma find a variation in their particular program. For an outline of the Preliminar HSC course ou ma go to the NSW Board of Studies website ( where a detailed document of the Preliminar Mathematics Sllabus ma be obtained. How to use this book This book sets out the questions b school terms. There are sets, each of 6 HSC stle questions, each of 5 marks, for terms,, and 4 of the Mathematics course and sets of revision questions. The answers are provided at the end of each term. Each question has sllabus reference topic that is at the beginning of each question. A list of these topics and reference numbers are in this book. If a question is answered incorrectl, then the sllabus reference topic gives direction to that specific topic, thus enabling the student to focus on that particular area of concern. 5

7 HSC Mathematics Sllabus Reference Topics Plane Geometr.5 Applications of geometrical properties Probabilit. Random eperiments, equall likel outcomes; probabilit of a given result.. Sum and product of results.. Eperiments involving successive outcomes; tree diagrams. 6 Linear Functions and Lines 6.8 Coordinate methods in geometr 7 Series and Applications 7. Arithmetic series. Formulae for nth term and sum of n terms. 7. Geometric series. Formulae for nth term and sum of n terms. 7. Geometric series with ratio between and. The limit of geometric series. n,as n, for, and the concept of limiting sum for a 7.4 Applications of arithmetic series. Applications of geometric series: compound interest, Simplified hire purchase and repament problems. Applications to recurring decimals. 6

8 0 Geometrical applications of differentiation 0. Significance of the sign of the derivative 0. Stationar points on curves 0. The second derivative. The notations f ''(), d d, '' 0.4 Geometrical significance of the second derivative. 0.5 The sketching of simple curves. 0.6 Problems on maima and minima. 0.7 Tangents and normals to curves. 0.8 The primitive function and its geometrical interpretation. Integration. The definite integral. The relation between the integral and the primitive function.. Approimate methods: trapezoidal rule and Simpson s rule..4 Applications of integration: areas and volumes of revolution. Logarithmic and eponential functions r. Review of inde laws, and definition of a for a 0, where r is rational.. Definition of logarithm to the base a. Algebraic properties of logarithms and eponents.. The functions a and loga for a 0 and real. Change of base. log.4 The derivatives of a and a. Natural logarithms and eponential function..5 Differentiation and integration of simple composite functions involving eponentials and logarithms. 7

9 Trigonometric Functions. Circular measure of angles. Angle, arc, sector.. The functions: sin, cos, tan, cosec, sec, cot and their graphs.. Periodicit and other simple properties of the functions: sin, cos and tan..4 Approimations to sin,cos and tan, when is small. The result sin lim 0.5 Differentiation of sin, cos and tan.6 Primitive functions of sin, cos and sec.7 Etension of.-.6 to functions of the form a sin(b c) 4 Applications of calculus to the phsical world 4. Rates of change as derivatives with respect to time. The notation...,, etc 4. Eponential growth and deca; rate of change of population; the equation dn dt kn, where k is the population growth constant. 4. Velocit and acceleration as time derivatives. Applications involving: (i) the determination of the velocit and acceleration of a particle given its distance from a point as a function of time; (ii) the determination of the distance of a particle from a given point, given its acceleration or velocit as a function of time together with appropriate initial conditions. 8

10 HSC Mathematics Term (Set ) Question (5 marks) (Geometrical Applications of Differentiation) (a) For what values of is the curve 6 7 increasing? (b) Find the stationar points of and determine their nature. 4 (c) The point (,8) is a turning point on the parabola a b 8. Find the values of a and b. (d) Find the equation of the tangent to the curve f() 7 at the point where f ''() 0. (e) For the curve 4, find the values of where the curve is concave up. 9

11 Question (5 marks) (Integration) (a) The derivative of a function f() is f '(). The line is a tangent to the graph of f(). Find the function f(). (b) Find d (c) Find d (d) A O In the figure above the straight line cuts the parabola 4 at the point A and the origin O. (i) Show that the coordinates of A are (,6). (ii) Find the shaded area between the parabola and the straight line. (e) The semicircle 9 is rotated about the -ais. Find the volume of the sphere so formed. 0

12 Question (5 marks) (Geometrical Applications of Differentiation) (a) Let f() 4 (i) Find the coordinates of the points where the curve crosses the aes. (ii) Find the coordinates of the stationar points and determine their nature. (iii) Find the coordinates of the points of infleion. (iv) Sketch the graph of f() clearl indicating the intercepts, stationar points and points of infleion. 0 (b) The diagram shows the graph of a function f (). (i) For which values of is the derivative f '() 0? (ii) Sketch the graph of f '(). d d Determine the nature of the stationar point at 0. (c) Consider the function whose derivative is given b.

13 Question 4 (5 marks) (Integration) (a) Find d (b) (i) Differentiate 4 with respect to. (ii) Hence, or otherwise, find d. 4 (c) In the figure above the shaded region bounded b the -ais, the -ais 4 and the parabola is rotated about the -ais to form a solid. Find the eact volume of the solid so formed. 0 (d) The diagram shows the graph of f (). What is the value of p, p 0, so that f () 4 p p

14 A B..6 (e) The diagram shows the cross-section of a river, with depths of the river shown in metres, at 5 metre intervals. The river is 5 metres wide. (i) Use the trapezoidal rule to find an approimate value for the area of the cross-section. (ii) Water flows through this section of the river at a speed of 0.4 ms. Calculate the approimate volume of water that flows past this section in one hour.

15 Question 5 (5 marks) (Geometrical Applications of Differentiation) A Not to Scale o B (a) The graph of 4 6 is sketched above. The points A and B are the turning points. (i) Find the coordinates of A and B. (ii) For what values of is the curve concave down? (iii) For what values of k does the equation have two real roots? 4 6 k 4

16 (b) An open rectangular bo has four sides and a base, but no lid, as in the figure above. The dimensions of the base are: length cm, width cm and height cm. 0 (i) Given that the surface area of the bo is 0 cm show that. 8 (ii) Show that the volume of the bo is V (iii) Hence determine the maimum volume of the bo. 4 5

17 Question 6 (5 marks) (Integration) (a) Find 4 d (b) Given that k d 7. Find the value of k. 8 o (c) The region bounded b the curve, the -ais and the lines and 8 is revolved about the -ais to form a solid. Find the eact volume of the solid. 6

18 A o 8 (d) The diagram above shows the curve f (). The shaded area A is bound b the curve the -ais and 8. (i) Calculate the eact area of A. (ii) Use one application of Simpson's Rule to find the area of A. (iii) Use two applications of the Trapezoidal Rule to find the area of A. 7

19 Term (Set ) Question (5 marks) (Geometrical Applications of Differentiation) (a) Find a primitive of 5. 4 (b) For the curve 6 find the values of where. the curve is concave down. (c) Determine the values of for which the curve 6 9 is increasing. 4 (d) Consider the curve 8 6 d (i) Show that 4 d (ii) Find the stationar points on the curve and determine their nature. (iii) Sketch the curve showing all intercepts on the aes and stationar points. (iv) Determine the values of for which the curve is increasing. 8

20 Question (5 marks) (Integration) 4 (a) Find d (b) Find d 4 (c) The function f() passes through the point, the derivative of the function is f '(). Find the function f(). 4 (d) A vase has a shape obtained b rotating part of a parabola 4 about the -ais as shown. The vase is 4 cm deep. Find the volume of liquid that the vase will hold. (e) (i) Differentiate with respect to. 4 (ii) Hence, or otherwise, evaluate 8 d 4 9

21 Question (5 marks) (Geometrical Applications of Differentiation) (a) For the function p() (i) Find the coordinates of the turning points of p() and state their nature. (ii) Draw a sketch of p() in the domain 0. (iii) Find the values of for which p() is concave up. (b) A book is designed so that each page of print contains 6 cm,surrounded completel b borders as illustrated in the figure. Each page is to have a border of cm at the bottom and at each side, as well as a border of cm at the top. Let the width of a page be cm and its length cm. 6 (i) Show that the area A cm of one page is A. (ii) Prove that the smallest print area possible will have dimensions 4 cm wide and 8 cm long. 0

22 Question 4 (5 marks) (Integration) (a) Find the equation of the curve f () given that f ''() 6 4 and there is a minimum turning point at,. (b) The area under the hperbola, for is rotated about the -ais. Find the eact volume of revolution. 0 (c) The above diagram shows a sketch of the gradient function of the curve f (). Draw a sketch of the function f () given that f (0) 0.

23 O A (d) In the diagram above the parabola 8 and the line intersect at the origin and the point A. (i) Find the coordinates of the point A. (ii) Calculate the shaded area enclosed b the parabola and the line. (c) Consider the function 0 (i) Cop and complete the table above. (ii) Using Simpson's Rule for five function values, find an estimate for the area shaded in the diagram below. 0

24 Question 5 (5 marks) (Geometrical Applications of Differentiation) (a) Find a primitive of 5. (b) Let f () 4 (i) Find the coordinates of the stationar points of f () and determine their nature. (ii) Hence sketch the graph of f () showing all stationar points and the -intercepts. (iii) Find the values of for which the curve is concave up. 5 Pen Pen (c) A farmer wishes to construct two rectangular enclosures, as shown above. Pen is to be 5 times as wide as Pen. There is an eisting wall (shaded) that serves as a boundar fence as shown. All the other fences are to be constructed from 48 metres of wire mesh. (i) Let be the length of both pens and the width of Pen. Show that 8 (ii) Hence show that the total area A m contained in the two enclosures is given b A 6 8. (iii) Calculate the maimum area of each pen.

25 Question 6 (5 marks) (Integration) (a) Find d A 0 (b) The parabola 4 intersects the -ais at A. (i) Find the coordinates of A. (ii) Find the shaded area enclosed between the parabola and the and aes. 0 (c) A bowl is formed b rotating the semicircle 4 and the parabola 4 around the -ais. The shaded area revolved is contained between the -ais and the two curves as indicated on the diagram. Find the eact volume of the solid so formed. 4

26 (d) The table belows gives the five function values for 4 in the domain (i) Using Simpson's Rule find an approimate area for the quadrant of the circle in the figure above. (ii) Find the eact value of 4 d 0 (iii) Using parts (i) and (ii) determine an approimate value for. 5

27 Term (Set ) Question (5 marks) (Geometrical Applications of Differentiation) (a) Find a primitive of a b c. (b) For what values of is the curve 8 decreasing. (c) Find the equation of the normal to the curve 6 0 at the point 4 d where 0. d 4 (d) Consider the curve given b (i) Find the coordinates of the stationar points. d (ii) Find all values of for which 0. d (iii) Determine the nature of the stationar points. (iv) Sketch the curve in the domain 6

28 Question (5 marks) (Integration) Not to Scale A 0 B (a) The parabola and the parabola 4 intersect at the points A and B. (i) Show that the coordinates of A and B are respectivel, and,0 (ii) Find the shaded area enclosed b the parabolas. (b) Evaluate d 4 5 (c) (i) Differentiate with respect to. (ii) Hence, or otherwise, find 4 4 d 0 (d) The circle 4 is rotated about the -ais. The shaded area is bound b the line and the circle. Find the eact volume of the spherical cap so formed. 7

29 Question (5 marks) (Geometrical Applications of Differentiation) (a) The parabola a b c touches the line at the origin and has a maimum turning point where. Find the values of a,b and c. (b) The point (,0) is a turning point on the parabola a b. Find the values of a and b. 0 (c) The diagram shows the graph of a function f (). (i) For which values of is the derivative f '() 0? (ii) Sketch the graph of f '(). (d) For the curve (i) Find the stationar points and determine their nature. (ii) Eplain wh there are no points of infleion. 8

30 Question 4 (5 marks) (Integration) Not to Scale 0 A B (a) The shaded area is bound b the lines 0, the parabola, the straight line and the -ais, as in the diagram. (i) The straight line and the parabola meet at A and the parabola intersects the -ais at B. Show that the coordinates of A and B are respectivel,0 and,0. (ii) Find the area of the shaded region. Not to Scale 0 (b) The shaded area bounded b the parabolas, and the -ais, is revolved around the -ais. Find the eact volume generated. 9

31 (c) The semicircle a is rotated about the -ais. Show that the volume of the sphere so formed is given b the formula V a. 4 (d) (i) Differentiate with respect to. (ii) Hence, or otherwise, evaluate d 0

32 Question 5 (5 marks) (Geometrical Applications of Differentiation) (a) For the function f () 9 (i) Find the coordinates of the turning points of f() and state their nature. (ii) Find the coordinates of the point of infleion and show that it is a point of infleion. (iii) Draw a sketch of f () in the domain 4. 6 km 4 km (b) Bill and Ben set out for a town. Bill is 6 km West of the town and walking at a constant km /hour. Ben is 4 km South of the town and walking at a constant rate of 4 km /hour. (i) Show that their distance apart after t hours is given b D =5t 68t 5. (ii) Hence find how long it takes them to reach their minimum distance apart. (iii) Find their minimum distance apart.

33 Question 6 (5 marks) (Integration) (a) Given that a 9. Find the value of a. Not to Scale o (b) The diagram above shows a sketch of the gradient function of the curve f (). Draw a sketch of the function f () given that f (0). (c) Evaluate 4 d Not to Scale A 0 (d) The parabola intersects the parabola at the origin and the point A. (i) Find the coordinates of A. (ii) Find the shaded area enclosed between the two parabolas. (iii) The shaded area is rotated about the -ais, find the eact volume generated.

34 Term Answers Set Question (a).5 (b),0 ma,, 4 min (c) a 4, b 6 (d) 0 (e) Question (a) f() (b) c (c) (d) (ii) 9 units (e) V 6 units Question 7 (a) (i) 0,0,,0 (ii) 0,0 horizontal point of infleion,, min 6 (iii) 0,0,, (iv) o

35 (b) (i), (ii) o (c) horizontal point of infleion Question (a) c (b) (i) ' (ii) 4 c 0 (c) units (d) p (e) (i) 4 m (ii) 060 kl Question 5 4 (a) (i) 4,4,,4 (ii) (iii) k 4 or k (b) (iii) 0 0 units Question 6 (a) 4 c (b) k 9.5 (c) units 6 (d) (i) units (ii) units (iii) units 4

36 Set Question (a) 5 c (b) (c), (d) (ii),0 min, 0,6 ma,,0 min (iii) (0,6) (,0) (iv) 0, and Question 5 (a) 5.5 (b) c (c) f() (d) (e) (i) ' (ii)

37 Question (a) (i), min,, ma 7 (ii) (,0) 0 (iii) Question 4 (a) f() (b) units (c) 0 6

38 (d) (i) A, 6 (ii) 9 units (e) (i) (ii) Area 5 5 Question 5 (a) 5 c (b) (i), 4 min, 0,0 ma,, 4 min. (ii) (0,0) (,0) (iii) 6, 6 (c) (iii) Area Pen m,area Pen 60 m Question 6 (a) 4 5 (b) (i) A 0, (ii) units (c) V 9 units (d) (i).08 (ii) (iii).08 7

39 Set Question (a) a b c d (b) (c) (d) (i) 0,,,5,, (ii) (iii) 0, min,,5 ma,, min (iv) (,9) (,5) (0,) (,) 0 Question (a) (ii) 9 units (b) 4 5 (c) (i) 0 (ii) c (d) units 8

40 Question (a) a, b, c 0 (b) a, b 4 (c) (i) 0, (ii) 0 (d) (i), min,, ma (ii) " 0 Question (a) (ii) units (b) units (d) (i) (ii) 0 Question 5 (a) (i),5 ma,, 7 min (ii), (iii) (,5 ) 0 (4,8) (b) (ii).6 hours (iii).4 km 9

41 Question 6 (a) a (b) 0 (c) 6 (d) (i) A, (ii) units (iii) units 0 40

STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE MATHEMATICS. Time allowed Three hours (Plus 5 minutes reading time)

STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE 00 MATHEMATICS Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of