higher school certificate - harder hsc maths · 2016-11-17 · 5 preface in 2012 the format for the...

Higher School

Certificate

Mathematics

HSC Style Questions (Section 2)

FREE SAMPLE

J.P.Kinny-Lewis

1

Higher School

Certificate

Mathematics

HSC Style Questions

(Section 2)

J.P.Kinny-Lewis

2

First published by

John Kinny-Lewis in 2016

John Kinny-Lewis 2016

National Library of Australia

Cataloguing-in-publication data

ISBN: 978-0-9943347-0-1

This book is copyright. Apart from any fair dealing for the purposes of private study, research, criticism or

review as permitted under the Copyright Act 1968, no part may be reproduced, stored in a retrieval system, or

transmitted, in any form by any means, electronic, mechanical, photocopying, recording, or otherwise without

prior written permission.

Enquiries to be made to John Kinny-Lewis.

Copying for educational purposes.

Where copies of part or the whole of the book are made under Section 53B or Section 53D of the Copyright Act

1968, the law requires that records of such copying be kept. In such cases the copyright owner is entitled to

claim payment.

Typeset by John Kinny-Lewis

Edited by Sarah Murphy

Printed in Australia

3

Other Publications

HSC Preliminary Mathematics Multiple-Choice Questions by Term With Video Solutions HSC Preliminary Mathematics HSC Style Questions (Section 2) HSC Preliminary Mathematics HSC Style Questions (Section 2) Solutions HSC Mathematics Multiple-Choice Questions by Term With Video Solutions HSC Mathematics HSC Style Questions (Section 2) Solutions HSC Preliminary Extension 1 Mathematics Multiple-Choice Questions by Term With Video Solutions HSC Extension 1 Mathematics Multiple-Choice Questions by Term With Video Solutions

HSC Mathematics Harder Questions by Topic HSC Extension 1 Mathematics Harder Questions by Topic HSC Extension 2 Mathematics Harder Extension 1 Questions by Topic

4

CONTENTS

Preface ………………………………………………………………….. 5

How to Use this Book …………………………………………………... 5

Syllabus Reference (S R) ………………………………………………. 6

Term 1 Set 1…………………………………………………………….. 9

Term 1 Set 2…………………………………………………………….. 18

Term 1 Set 3…………………………………………………………….. 26

Term 1 Answers ………………………………………………………... 33

Term 2 Set 1…………………………………………………………….. 41

Term 2 Set 2…………………………………………………………….. 47

Term 2 Set 3…………………………………………………………….. 53

Term 2 Answers ………………………………………………………... 59

Term 3 Set 1…………………………………………………………….. 65

Term 3 Set 2…………………………………………………………….. 71

Term 3 Set 3…………………………………………………………….. 78

Term 3 Answers ………………………………………………………... 84

Term 4 Set 1…………………………………………………………….. 90

Term 4 Set 2…………………………………………………………….. 97

Term 4 Set 3…………………………………………………………….. 104

Term 4 Answers ………………………………………………………... 112

Revision Test 1 ………………………………………………………..... 118

Revision Test 2 ………………………………………………………..... 125

Revision Test 3 ………………………………………………………..... 132

Revision Tests Answers ……………………………………………….. 138

5

Preface

In 2012 the format for the Higher School Certificate (HSC) examination papers for Mathematics, Extension 1

Mathematics and Extension 2 Mathematics changed. The Mathematics paper began with 10 multiple choice

questions followed by six questions, each worth 15 marks. The 15 mark questions were numbered 11, 12, 13,

14, 15 and 16, the total being 100 marks.

This book follows the Board of Studies (BOSTES) recommendation for the HSC course syllabus topics. This

recommendation is not mandatory for schools, consequently schools may find a variation in their particular

program.

For an outline of the Preliminary HSC course you may go to the NSW Board of Studies website

(www.boardofstudies.nsw.edu.au) where a detailed document of the Preliminary Mathematics Syllabus may be

obtained.

How to use this book

This book sets out the questions by school terms.

There are 3 sets, each of 6 HSC style questions, each of 15 marks, for terms 1, 2, 3 and 4 of the Mathematics

course and 3 sets of revision questions. The answers are provided at the end of each term.

Each question has syllabus 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 incorrectly, then the syllabus reference topic gives direction to that specific topic, thus

enabling the student to focus on that particular area of concern.

http://www.boardofstudies.nsw.edu.au/

6

HSC Mathematics

Syllabus Reference Topics

2 Plane Geometry

2.5 Applications of geometrical properties

3 Probability

3.1 Random experiments, equally likely outcomes; probability of a given result.

3.2 Sum and product of results.

3.3 Experiments involving successive outcomes; tree diagrams.

6 Linear Functions and Lines

6.8 Coordinate methods in geometry

7 Series and Applications

7.1 Arithmetic series. Formulae for nth term and sum of n terms.

7.2 Geometric series. Formulae for nth term and sum of n terms.

7.3 Geometric series with ratio between 1 and 1.

The limit of nx ,as n , for x 1 , and the concept of limiting sum for a

geometric series.

7.4 Applications of arithmetic series.

Applications of geometric series: compound interest,

Simplified hire purchase and repayment problems.

Applications to recurring decimals.

7

10 Geometrical applications of differentiation

10.1 Significance of the sign of the derivative

10.2 Stationary points on curves

10.3 The second derivative. The notations

2

2

d yf ''(x), , y ''

dx

10.4 Geometrical significance of the second derivative.

10.5 The sketching of simple curves.

10.6 Problems on maxima and minima.

10.7 Tangents and normals to curves.

10.8 The primitive function and its geometrical interpretation.

11 Integration

11.1 The definite integral

11.2 The relation between the integral and the primitive function.

11.3 Approximate methods: trapezoidal rule and Simpson’s rule.

11.4 Applications of integration: areas and volumes of revolution.

12 Logarithmic and exponential functions

12.1 Review of index laws, and definition of ra for a 0 , where r is rational.

12.2 Definition of logarithm to the base a. Algebraic properties of logarithms and

exponents.

12.3 The functions xy a and ay log x for a 0 and real x. Change of base.

12.4 The derivatives of xy a and ay log x . Natural logarithms and exponential

function.

12.5 Differentiation and integration of simple composite functions involving

exponentials and logarithms.

8

13 Trigonometric Functions

13.1 Circular measure of angles. Angle, arc, sector.

13.2 The functions: sin x, cos x, tan x, cosec x, sec x, cot x and their graphs.

13.3 Periodicity and other simple properties of the functions: sin x, cos x and tan x.

13.4 Approximations to sin x,cos x and tan x, when x is small.

The result x 0

sin xlim 1

x

13.5 Differentiation of sin x, cos x and tan x

13.6 Primitive functions of 2sin x, cos x and sec x

13.7 Extension of 13.2-13.6 to functions of the form a sin(bx c)

14 Applications of calculus to the physical world

14.1 Rates of change as derivatives with respect to time. The notation . ..

x, x, etc

14.2 Exponential growth and decay; rate of change of population; the equation

dN

kN,dt

where k is the population growth constant.

14.3 Velocity and acceleration as time derivatives.

Applications involving:

(i) the determination of the velocity 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 velocity as a function of time together with

appropriate initial conditions.

9

HSC Mathematics

Term 1 (Set 1)

Question 11 (15 marks)

(Geometrical Applications of Differentiation)

2(a) For what values of x is the curve y 2x 6x 7 increasing ? 2

3 2

2

(b) Find the stationary points of y x 6x 9x 4 and determine their nature. 4

(c) The point (2,8) is a turning point on the parabola y ax bx 8.

3 2

3

Find the values of a and b.

(d) Find the equation of the tangent to the curve f(x) x 3x 7x 1 3

at the point where f '

3 2

'(x) 0.

(e) For the curve y x 3x 4, find the values of x where the curve is concave up. 3

10


(Integration)

(a) The derivative of a function f(x) is f '(x) 2x 1. 3

The line y 3x 1 is a tangent to the graph of f(x).

Find the function f(x).

(b)

3

3

1

1

1 xFind dx 2x

(c) Find x 2 x dx

3

(d)

In the figure above the straight line y 2x cuts the parabola y 2x 4 x

at the point A and the origin O.

(i) Show that the coordinates of A are (3,6).

2

1

(ii) Find the shaded area between the parabola and the straight line. 3

(e) The semicircle y 9 x is rotated about the x-axis. 3

Find the volume of the sphere so formed.

y

x O

A

11



4 3(a) Let f(x) x 2x

(i) Find the coordinates of the points where the curve crosses the axes. 2

(ii) Find the coordinates of th

e stationary points and determine their nature. 3

(iii) Find the coordinates of the points of inflexion. 2

(iv) Sketch the graph of f(x) clearly indicating the intercepts, stationary points 2

and points of inflexion.

(b) The diagram shows the graph of a function y f (x).

(i) For which values of x is the derivative f '(x) 0 ? 1

(ii) Sketch the graph of f '

2

(x). 3

dy(c) Consider the function whose derivative is given by x 2x 1 3x 1 . 2

dx

Determine t

he nature of the stationary point at x 0.

x

y

2 1 0

12


(Integration)

3

2

1(a) Find dx 2

2x

(b) (i) Differentiate y 4 x with respect to x.

2

2

3x (ii) Hence, or otherwise, find dx. 2

4 x

2

(c) In the figure above the shaded region bounded by the x-axis, the y-axis 4

and the parabola y x 1 is rotated about the y-axis to form a solid.

Find the exact v

olume of the solid so formed.

p

p

(d) The diagram shows the graph of y f (x). 1

What is the value of p, p 0, so that f (x) 4

1

1

y

x

2

3 x

y

2 1 0

13

(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 15 metres wide.

(i) Use the trapezoidal rule to find an approx

1

imate value for the area of the 2

cross-section.

(ii) Water flows through this section of the river at a speed of 0.4 ms . 2

Calculate the approximate volume of water that flows past this

section in one hour.

1.2 1.6

5 5 5 B A

14



B

A

o x

y

Not to Scale

3 2(a) The graph of y x 2x 4x 6 is sketched above. The points A and B

are the turning points.

(i) Find the coordinates of A and B.

3 2

3

(ii) For what values of x is the curve concave down ? 2

(iii) For what values of k does the equation x 2x 4x 6 k 2

have two real roots ?

15

(b) An open rectangular box has four sides and a base, but no lid, as in the figure above.

The dimensions of the base are: length 3x cm, width x cm and height y cm.

(i) Given that the surface 2

2

2

120 3xarea of the box is 120 cm show that y . 3

8x

x (ii) Show that the volume of the box is V 9x 5 . 18

(iii) He

nce determine the maximum volume of the box. 4

3x

x

y

16


(Integration)

3

1

(a) Find 4x 1 dx 2

(b) Given that 2x k dx 27. Find the value of k.

3

3

1(c) The region bounded by the curve y , the y-axis and the lines 3

x

y 1 and y 8 is revolved about the y-axis to form a solid.

Find the exact volume of the sol

id.

8

1

o x

y

17

3(d) The diagram above shows the curve f (x) 2 x . The shaded area A

is bound by the curve the x-axis and x 8.

(i) Calculate the exact area of A.

3

(ii) Use one application of Simpson's Rule to find the area of A. 2

(iii) Use two applications of the Trapezoidal Rule to find the area of A. 2

o 8 x

y

A

18

Term 1 (Set 2)



(a) Find a primitive of 2x 5. 2

4 2

(b) For the curve y x 6x 2x find the values of x where . 3

the curve is concave down.

(c)

3 2

4 2

Determine the values of x for which the curve y x 6x 9x 3 3

is increasing.

(d) Consider the curve y x 8x 16

dy (i) Show that 4x x 2 x 2

dx

2

(ii) Find the stationary points on the curve and determine their nature. 2

(iii) Sketch the curve showing all intercepts on the axes and stationary points. 2

(iv) Determine the values of x for which the curve is increasing. 1

19


(Integration)

4

1

4

1(a) Find x dx 3

x

(b) Find 1 2x dx

2

2

(c) The function f(x) passes through the point 1, 1 the derivative of

the function is f '(x) 3x 3. Find the function f(x).

3

2(d) A vase has a shape obtained by rotating part of a parabola y 4 x 1 3

about the y-axis as shown. The vase is 4 cm deep.

Find the volume of liquid that the vase will hold.

(e

2

1

22

1

1) (i) Differentiate y with respect to x. 2

4 x

8x (ii) Hence, or otherwise, evaluate dx 4 x

2

4

y

x

20



2(a) For the function p(x) x 2 x 1

(i) Find the coordinates of the turning points of p(x) and state their nature. 3

(ii) Draw a sketch of y p(x) in the domain 0 x 3.

3

(iii) Find the values of x for which p(x) is concave up. 2

2(b) A book is designed so that each page of print contains 216 cm ,surrounded

completely by borders as illustrated in the figure. Each page is to have a

border of 1 cm at the bottom and at ea

2

ch side, as well as a border of 2 cm

at the top. Let the width of a page be x cm and its length y cm.

216 (i) Show that the area A cm of one page is A x 3 .

x 2

3

(ii) Prove that the smallest print area possible will have dimensions 4

12 cm wide and 18 cm long.

x

y 1

2

1

1

21


(Integration)

(a) Find the equation of the curve y f (x) given that f ''(x) 6x 4 and there is 3

a minimum turning point at 1, 2 .

1(b) The area under the hyperbola y , for 1 x 2 is rotated abo

x

ut the x-axis. 3

Find the exact volume of revolution.

(c) The above diagram shows a sketch of the gradient function of the curve y f (x). 2

Draw a sketch of the function y f (x) given that f (0) 0.

2 1 0 x

y’

22

2(d) In the diagram above the parabola y 2x 8x and the line y 2x intersect

at the origin and the point A.

(i) Find the coordinates of the point A.

1

(ii) Calculate the shaded area enclosed by the parabola and the line. 3

(c) Consider the function xy 2

x 2 1 0 1 2

y

(i) Copy and complete the table above. 1

(ii) Using Simpson's Rule for five function values, find an estimate for 3

the area shaded in the diagram below.

O

y

x A

0

y

x 2

23



2

2 2

(a) Find a primitive of 3x 2x 5. 2

(b) Let f (x) x x 4

(i) Find the coordinates of the stationary points of

y f (x) and determine 3

their nature.

(ii) Hence sketch the graph of y f (x) showing all stationary points and 2

the x-intercepts.

(iii) Find the values of x for which the curve is concave up. 1

(c) A farmer wishes to construct two rectangular enclosures, as shown above.

Pen 2 is to be 5 times as wide as Pen 1. There is an existing wall (shaded)

that serves as a boundary fence as shown. All the other fences are to be

constructed from 48 metres of wire mesh.

(i) Let x be the length of both pens and y the width of Pen 1. 2

2

x Show that y 8

2

(ii) Hence show that the total area A m contained in the two enclosures is 2

x given by A 6x 8 .

2

(iii) Calculate the

maximum area of each pen. 3

x

y 5y

Pen 1 Pen 2

24


(Integration)

22

1

1(a) Find x dx 3

x

2

(b) The parabola y 1 4 1 x intersects the y-axis at A.

(i) Find the coordinates of A. 1

(ii)

Find the shaded area enclosed between the parabola 3

and the x and y axes.

2

2

(c) A bowl is formed by rotating the semicircle y 4 x and the parabola 4

y x 1 around the y-axis. The shaded area revolved is contained between

the x-axis and the two cur

ves as indicated on the diagram.

Find the exact volume of the solid so formed.

0

y

x

A

0

y

x

25

2(d) The table belows gives the five function values for y 4 x

in the domain 0 x 2

x 0 0.5 1 1.5 2

y 2.00 1.94 1.73 1.32 0

2

2

0

(i) Using Simpson's Rule find an approximate area for the quadrant 2

of the circle in the figure above.

(ii) Find the exact value of 4 x dx 1

(iii) Using parts (i) and (ii) determine an approximate value for . 1

1 0

y

x 2

26

Term 1 (Set 3)



2

3 2

(a) Find a primitive of 3ax 2bx c. 2

(b) For what values of x is the curve y x x 8x 1 decreasing.

3 2

2

2

2

(c) Find the equation of the normal to the curve y x 6x 10 at the point 4

d y where 0.

dx

4 3 2(d) Consider the curve given by y 3x 12x 12x 2

(i) Find the coordinates of the stationary points. 2

(ii) Find all values of

2

2

d y x for which 0. 1

dx

(iii) Determine the nature of the stationary points.

2

(iv) Sketch the curve in the domain 1 x 3 2

27


(Integration)

2(a) The parabola y x x 2 and the parabola y 4 x intersect at the

points A and B.

(i) Show that the coordinates of A and B are respectively 1,3 and 2,0 2

(ii) Find t

3 3

2

he shaded area enclosed by the parabolas. 3

x 1(b) Evaluate dx

x 1

54

43 4

3

(c) (i) Differentiate y x 1 with respect to x. 2

(ii) Hence, or otherwise, find x x 1

dx 1

2 2(d) The circle x y 4 is rotated about the x-axis. The shaded area 3

is bound by the line x 1 and the circle.

Find the exact volume of the spherical cap so

formed.

B

0

y

x

A

Not to Scale

2 0

y

x

28



2(a) The parabola y ax bx c touches the line y x at the origin and has 3

a maximum turning point where x 1. Find the values of a,b and c.

(b) The point (2,0) is a turn

3 2ing point on the parabola y x ax b. 3

Find the values of a and b.

(c) The diagram shows the graph of a function y f (x).

(i) For which values of x is the derivative f '(x) 0 ? 1

(ii) Sketch the graph of f '

(x). 3

1(d) For the curve y x

x

(i) Find the stationary points and determine their nature.

3

(ii) Explain why there are no points of inflexion. 2

x

y

2 1 0

29


(Integration)

2

(a) The shaded area is bound by the lines x 0, the parabola y x 2 1,

the straight line y 1 x and the x-axis, as in the diagram .

(i) The straight line and the parabola meet at A and the parabo

la intersects 2

the x-axis at B. Show that the coordinates of A and B are respectively

1,0 and 3,0 .

(ii) Find the area of the shaded region. 3

2 2

(b) The shaded area bounded by the parabolas y x 1 , y x 1 and

the x-axis, is revolved around the x-axis. Find the exact volume generated. 3

1 0

y

x

Not to Scale

B A x

y

0

Not to Scale

30

2 2

3

(c) The semicircle x a y is rotated about the y-axis. Show that the 3

4 volume of the sphere so formed is given by the formula V a .3

2

2

2

22

2

x(d) (i) Differentiate y with respect to x. 2

x 1

x (ii) Hence, or otherwise, evaluate dx

x 1

2

31



2 3(a) For the function f (x) 12 9x 3x x

(i) Find the coordinates of the turning points of f(x) and state their nature. 3

(ii) Find the coordinates of the point of inflex

ion and show that it is a 2

point of inflexion.

(iii) Draw a sketch of y f (x) in the domain 2 x 4. 3

(b) Bill and Ben set out for a town. Bill is 6 km West of the town and walking at a constant

3 km /hour.

Ben is 4 km South of the town and walking at a constant rate of 4 km /hour.

(i) 2 2 Show that their distance apart after t hours is given by D =25t 68t 52. 3

(ii) Hence find how long it takes them to reach their minimum distance apart. 2

(iii) Find their minimum distance apart. 2

4 km

6 km

32


(Integration)

2

2

1

(a) Given that ax 2 9 . Find the value of a. 3

(b) The diagram above shows a sketch of the gradient function of the curve y f (x). 2

Draw a sketch of the function y f (x) given that f (0) 1.

1 x(c) Evaluate dx

x

4

1

3

2 2(d) The parabola y x intersects the parabola x y at the origin and the point A.

(i) Find the coordinates of A.

1

(ii) Find the shaded area enclosed between the two parabolas. 3

(iii) The shaded area is rotated about the x-axis, find the exact volume generated. 3

o x

Not to Scale

A

0

y

x

Not to Scale

33

Term 1 Answers

Set 1

Question 11

(a) x 1.5 (b) 1,0 max, 3, 4 min (c) a 4, b 16

(d) y 10x (e) x 1

2 2

2

3

Question 12

1 2(a) f (x) x x 2 (b) x c (c) (d) (ii) 9 units

32x

(e) V 36 units

Question 13

273(a) (i) 0,0 , 2,0 (ii) 0,0 horizontal point of inflexion, , min

2 16

(iii) 0,0 , 1, 1

(iv)

2

y

o x

34

(b) (i) x 1, x 1

(ii)

(c) horizontal point of inflexion

2

2 2

3 2

Question 14

1 x(a) c (b) (i) y' (ii) 3 4 x c

4x 4 x

10(c) units (d) p 2 (e) (i) 14 m (ii) 20160 kl

3

3

Question 15

2 14 2 14(a) (i) 2,14 , ,4 (ii) x (iii) k 4 or k 14

3 327 27

(b) (iii) 20 30 units

3

32

2 2 2

Question 16

1(a) 4x 1 c (b) k 9.5 (c) 3 units

6

(d) (i) 32 units (ii) 32 units (iii) 32 units

y’

1 o x

35

Set 2

2

Question 11

(a) x 5x c (b) 1 x 1 (c) x 1, x 3

(d) (ii) 2,0 min, 0,16 max, 2,0 min

(iii)

(iv) 2 0, and 2 x x

5 3

22

Question 12

1(a) 5.5 (b) 1 2x c (c) f (x) x 3x 5 10

2x412(d) (e) (i) y' (ii) 0

15 4 x

(0,16)

y

x (2,0)

36

Question 13

1 23(a) (i) 1, 2 min, , 1 max

273

2(iii) x

3

3 2 3

Question 14

(a) f (x) x 2x x 2 (b) units2

(c)

(ii)

y

0 2

x

(3,10)

2

y

0

x

37

2(d) (i) A 3, 6 (ii) 9 units

(e) (i)

x -2 1 0 1 2

y 4 2 1 0.5

0.25

5(ii) Area 512

3 2

Question 15

(a) x x 5x c (b) (i) 2, 4 min, 0,0 max, 2, 4 min.

(ii)

6 6(iii) ,

3 3 x x

2 2(c) (iii) Area Pen 1 32m ,Area Pen 2 160 m

2 3

Question 16

295 1(a) 4 (b) (i) A 0,3 (ii) 2 units (c) V units 6 4 6

(d) (i) 3.08 (ii) (iii) 3.08

(0,0)

(2,0) x

y

38

Set 3

3 2

Question 11

4(a) ax bx cx d (b) x 2 (c) x 12y 74 0 3

3 3(d) (i) 0,2 , 1,5 , 2,2 (ii) x

3

(iii) 0,2 min, 1,5 max, 2,2 min

(iv)

4 5

2 3 4 4

3

Question 12

15(a) (ii) 9 units (b) 4 (c) (i) 20x x 1 (ii) x 1 c 6 20

5(d) units

3

(3,29)

0

(1,5)

(0,2) (2,2)

x

y

39

Question 13

1(a) a , b 1, c 0 (b) a 3, b 4 (c) (i) 1 x 0, x 2

2

(ii)

(d) (i) 1,2 min, 1, 2 max (ii) y" 0

2 3

22

Question 14

2x25(a) (ii) 1 units (b) units (d) (i) (ii) 0

6 5 x 1

Question 15

(a) (i) 3,15 max, 1, 17 min (ii) 1, 1

(iii)

(b) (ii) 1.36 hours (iii) 2.4 km

(4,8)

(3,15

)

0 x

y

2 1 0 x

y

40

Question 16

(a) a 3

(b)

2 32 1 3(c) 6 (d) (i) A 1,1 (ii) units (iii) units

3 3 10

0 x

y

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