Higher School Certificate · 2016-11-17 · 5 Preface In 2012 the format for the Higher School...

Higher School

Certificate

Preliminary

Mathematics

HSC Style Questions (Section 2)

Free Sample

J.P.Kinny-Lewis

1

Higher School

Certificate

Preliminary

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: 978-0-9943347-6-3

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 John Kinny-Lewis

3

Other Publications

HSC Preliminary Mathematics

Multiple-Choice Questions by Term

With Video Solutions



Solutions

HSC Mathematics



HSC Mathematics


HSC Mathematics


Solutions

HSC Preliminary Extension 1 Mathematics



HSC Extension 1 Mathematics



HSC Mathematics

Harder Questions by Topic


Harder Questions by Topic


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…………………………………………………………….. 16

Term 1 Set 3…………………………………………………………….. 22

Term 1 Answers ………………………………………………………... 28

Term 2 Set 1…………………………………………………………….. 34

Term 2 Set 2…………………………………………………………….. 42

Term 2 Set 3…………………………………………………………….. 50

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

Term 3 Set 1…………………………………………………………….. 64

Term 3 Set 2…………………………………………………………….. 73

Term 3 Set 3…………………………………………………………….. 82

Term 3 Answers ………………………………………………………... 91

Revision Test 1 ………………………………………………………..... 94

Revision Test 2 ………………………………………………………..... 102

Revision Test 3 ………………………………………………………..... 110

Revision Tests Answers ……………………………………………….. 119

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, and counted for a total of 100 marks.

This book follows the Board of Studies (BOSTES) recommendation for the Preliminary HSC

course syllabus topics. This recommendation is not mandatory for schools and 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, worth 15 marks, for terms 1, 2 and 3 of the

Preliminary Mathematics course and 3 sets of revision questions. The answers are provided at

the end of each term.

Each question has a 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

Preliminary Mathematics

Syllabus Reference Topics

The numbers on the right column under the heading ‘S R’ refer to the syllabus reference

numbers. A description of each topic is listed below:

1 Basic Arithmetic and Algebra

1.1 Arithmetical operations on rational numbers and quadratic surds.

1.2 Inequalities and absolute values.

1.3 Manipulation and substitution in algebraic expressions, factorization, and operations

on simple algebraic fractions.

1.4 Linear equations and inequalities. Quadratic equations. Simultaneous equations.

2 Plane Geometry

2.3 Tests for parallel lines. Angle sums of triangles, quadrilaterals and general polygons.

Exterior angle properties. Congruence of triangles and tests. Properties of special

triangles and quadrilaterals. Tests for special quadrilaterals. Properties of transversals

to parallel lines. Similarity of triangles and tests. Pythagoras Theorem and converse.

Area formulae.

2.4 Application of above properties to the solution of numerical exercises requiring one

or more steps of reasoning.

2.5 Application of above properties to simple theoretical problems requiring one or more

steps of reasoning.

7

4 Real Functions of a Real Variable and their Geometrical

Representation

4.1 Dependent and independent variables. Functional notation. Range and domain.

4.2 The graph of a function. Simple examples.

4.3 Algebraic representation of geometrical relationships. Locus Problems.

4.4 Region and inequality. Simple examples.

5 Trigonometric Ratios – Review and Some Preliminary Results

5.1 Review of the trigonometric ratios, using the unit circle.

5.2 Trigonometric ratios of: o o o, 90 , 180 , 360

5.3 The exact ratios.

5.4 Bearings and angles of elevation.

5.5 Sine and cosine rules for a triangle. Area of a triangle, given two sides and the

included angle.

6 Linear Functions and Lines

6.2 The straight line; equation of a line passing through a given point with a given slope;

equation of a line passing through two given points; the general equation

ax by c 0 ;

parallel lines, perpendicular lines.

6.3 Intersection of lines and the solution of two linear equations in two unknowns;

the equation of a line passing through the point of intersection of two given lines.

6.4 Regions determined by lines; linear inequalities.

6.5 The distance between two points and the (perpendicular) distance of a point from a

line.

6.7 The mid-point of an interval.

8

8 The Tangent to a Curve and the Derivative of a Function

8.2 The notion of a limit of a function and the definition of continuity in terms of this

notion. Continuity of f + g, f – g, fg in terms of f and g.

8.4 Tangent as the limiting position of a secant. The gradient of the tangent. Equations of

tangent and normal at a given point of the curve y f (x).

8.6 The gradient or derivative as a function. Notations: dy d

f '(x), , f (x) , y'dx dx

8.7 Differentiation of nx for positive integral n. The tangent to y = f(x).

8.8 Differentiation of 1

12x and x from first principles. For the two functions u and v,

differentiation of Cu (C constant), u + v, u v, uv. The composite function rule.

Differentiation of u/v.

8.9 Differentiation of the general polynomial n

f (x) or f(x)/g(x), where f(x), g(x) are

polynomials.

9 The Quadratic Polynomial and the Parabola

9.1 The quadratic polynomial 2ax bx c . Graph of a quadratic function. Roots of a

quadratic equation. Quadratic Inequalities.

9.2 General theory of quadratic equations, relation between roots and coefficients.

The discriminant.

9.3 Classification of quadratic expressions; identity of two quadratic expressions.

9.4 Equations reducible to quadratics.

9.5 The parabola defined as a locus. The equation 2x 4Ay. Use of change of origin when

the vertex is not at (0,0).

9


Term 1 (Set 1)

Question 11 (15 marks)

(Basic Arithmetic and Algebra)

3234.6

(a) Evaluate correct to 3 significant figures 1 23.1 3.8

2

(b) Factorise 2x 7x 15 2

(c) Solve 2x 1 5

2

2

y 16(d) Simplify

2y 8

3x 1

2

(e) Solve 2 128 2

(f) Solve 3 2

x 11 2

6(g) Rationalize the denominator of .

7 5

2

2

Give your answer in the simplest form.

(h) Solve x 6x 2

10



4.6 5.9(a) Find correct to two decimal places the value of 1

6.3 5.1

1(b) Find integers a and b such that a b 10

10 3

2

(c) Solve 3x 1 8 2

4x 3(d) Solve 7

x

2

3 1(e) Simplify

n n 1

2

(f) Expand and simplify 2 1 3 2 2 2

(g) Solve simultaneously y 2x 5 and x 3y 6

3

2

(h) Factorise 2x 250 2

11


(Linear Functions and Lines)

(a) The points (5,2), ( 1,8) and (p,6) are collinear. 3

Find the value of p.

(b) Find the value of 'a' for the lines 2x 3y 4 and x a

y 6 3

to be perpendicular.

(c) Find the point of intersection of the lines 3

3x 2y 11 and 2x 3y 16

(d) Find the equation of the line that passes through the point of 3

intersection of the lines x 3y 6 0 and 2x y 5 0

and the

2 2 2

point (2,0).

(e) The line 3x 4y 5 0 is a tangent to the circle x y r . 3

Find the value of r.

12



(a) In the diagram ABCD is a rhombus. EDC is parallel to AB

and BC is parallel to the x axis.

(i) Find the length of AB 1

(ii) Find the coordinates of C 1

(iii) Show that the equation of EC is 4x 3y 8 0 2

(iv) Show that AC is perpendicular to BD

2

(v) Find the area of the trapezium ABCE 2

y

x

E

D

C B (3, 8)

A (0, 4)

13

(b) Find the distance between the parallel lines 5x 12y 5 0 3

and 5x 12y 8 0

(c) Sketch the region bounded by the inequalities

4

x y 1, y x 1, x y 1, y x 1

14


(Real Functions of a Real Variable and their Geometrical Representation)

(a) A function f(x) is defined as

1 x 1

f(x) x 1 x 1

1 x 1

(i) Sketch the graph of this function in the domain 3 x 3

3

(ii) Evaluate f ( 2) f (1) f (2) 2

(b) A region R is defined by the inequali

2 2ties x y 4 and y x

(i) Sketch this region R on the number plane. 3

(ii) Calculate the exact area of R.

1

1 f (a h) f (a)(c) Given that f (x) and g(x)

x h

(i) find g(x) in the simplest form.

2

(ii) Hence find g(x) when h 0. 1

(d) A function is d

2efined by f (x) x 4 find

(i) the domain 2

(ii) the range

1

15



(a) Two functions are defined as f (x) x 1 and g(x) 2x 3

(i) Sketch the graphs of these functions in the domain 1 x

3

(ii) Find the coordinates of the point of intersection of f (x) and g(x) 2

(iii) Hence, or otherwise solve the inequality x 1 2x 3

2 2

1

(b) The equation of the locus of a circle is given by x y 4x 2y 1 0. Find

(i) the coordinates of the centre

2

(ii) the length of the radius 1

(c) A point P(x, y) moves such that it is equidistant from the lines 3

3x 4y 5 0 and 3x 4y 9 0. Find the equation of the locus of P.

(d) A point P(x, y) moves such that it is

equidistant from the points 3

A( 1, 2) and B(3, 4). Find the equation of the locus of P.

16

Term 1 (Set 2)



3(a) Evaluate 4 correct to 2 significant figures ? 1

3

(b) Factorise 8x 27 2

(c) Solve 1 2x

2

2

4 y y

2

x 2x 1(d) Simplify 2

x 1

(e) Simplify 6 2

3 y3 2

(f) Solve 4(x 5) 2 2(x 1) 2

1 1(g) Simplify 2

1 2 1 2

2(h) Solve x 1

x

2

17



2(3.25)(a) Find correct to two decimal places the value of 1

5.98 3.67

(b) Simplify the expression 5x 3(x 4)

2

(c) Solve x 1 x 3 2

x x 1(d) Solve 1

2 3

2

2

(e) If A 2 rh 2 r ,find h correct to two decimal places when

2

2

r 4.25 and A 120.

5 1(f) Simplify 2

5 1

(g) Solve simultaneously y x and x

2 2

y 6 0 2

(h) Factorise x y x y 2

18



(a) M(6,2) is the midpoint of the interval AB, where A has the coordinates ( 2,8) 3

Find the coordinates of B.

(b) Find the equation of the line that is perpendicular to the line

2x 3y 4 3

and passes through the point ( 1,3)

(c) Find the acute angle between the lines x y 1 and x 1 3

(d) Find t

he equation of the line that has a gradient of 2 and passes through 3

the point of intersection of the lines x y 6 0 and 2x y 4 0.

(e) Find the exact distance fro

m the point (2, 3) and the line y 2x 1 3

19



(a) In the diagram A, B and C are the points (2,6), (8,8) and (6,2) respectively.

(i) Find the exact length of OB 1

(ii) Find the midpoint of OB 1

(iii) Find the gradient of AC 1

(iv) Show that AC OB 2

(v) Find the midpoint of AC and hence explain why OABC is a rhombus 2

(vi) Hence, or otherwise, find the area of OABC 2

2 2(b) Show that the line 4x 3y 20 0 is a tangent to the circle x y 16 3

(c) Sketch the region x y 1

3

y

x O

A(2,6)

B (8, 8)

C (6, 2)

20



x


2 x

a f(x) 0 x

x

xx 4

b

(i) Given that f ( 1) f (1) f (5) find the values of a and b.

3

(ii) Hence evaluate f ( 2) f (1) f (5) 2

(b) A region R is defined by the inequalit

2

2

ies y x and y 1 x

Sketch this region R on the number plane. 4

f (x) f (2)(c) Given that f (x) x x and g(x)

x 2

(i) find

g(x) in the simplest form. 2

(ii) Hence find g(x) when x 2.01

2

1

(d) A function is defined by f (x) 25 x find :

(i) the domain

2

(ii) the range 1

21



3(a) For the function f (x) x x

(i) Show that f (x) is an odd function. 2

(ii) Sketch the graph of f (x) in the domain 2 x 2

x(b) For the function f (x)

x 1

x 1 (i) Show that = 1

x 1 x 1

2

(ii) Hence, or otherwise, determine the asymptotes of the function. 1

(c) A point P(x, y) moves such that it is 2 units from the point ( 1, 2) 3

Find the equation of the locus of P.

(d) A point P(x, y) moves such that PA 2PB where

A and B are 3

the points ( 3,0) and (3,0) respectively.

(i) Find the equation of the locus of P.

3

(ii) Show that this locus is a circle and find the coordinates of the centre and 2

the length of the radius of this circle.

22

Term 1 (Set 3)



(a) The diameter of the Earth is 12756 km. Write this number 1

in scientific notation, correct to two significant figures.

3 2

(b) Factorise fully x x x 1

3

2

2

(c) Solve 2x 1 1 x 2

m 8(d) Simplify

m 2m 4

n 1 n

2

5 5(e) Simplify

4

2 2

2

x 1 x 2(f) Solve < 7 2

2 3

(g) Simplify 2 2 1 2 2 1

2

2

(h) Solve x 2x 1 2

23



3

2

987(a) Find correct to two decimal places the value of 1

2

1 1 1(b) Simplify the expression

n n 1 n 1

2

(c) Solve x 1 2x 1 2

x 2x 1(d) Solve

x 1 2x

2

(e) A manufacturer increases the price of a car by 25% to a new 2

selling price of $30 000.

What was the selling price before this increase?

(f) Solve 2 1 x 3 2

x(g) Evaluate ,where x 0 2

x

(h) Express 63 112 in the form a 7 and find a

2

24



(a) The three lines 3x y 4 0, 2x y 6 0 and x 0 enclose a triangle. 3

Find the area of the triangle.

(b) Find the equation of the line that is parallel to the line 3x 5y 4

0 4

and passes through the point (2, 1)

(c) For the points A(3, 2), B(7,1) and C(1,9)

(i)

Show that they are the vertices of a right-angled triangle 2

(ii) Find the area of triangle ABC. 2

(d) The points A(1, 3), B(2,3), C(6,5) and a fourth point D are the vertices 3

of a parallelogram. Find the coordinates of D given th

at D lies on

the line x 2y 7 0.

25



(a) In the diagram A, B and O are the points (2,6), (4,0) and (0,0) respectively.

L, M and N are the midpoints of OA,AB and OB respectively.

Given that L and N have the coordinates (1,3) and (2,0) respectively:

(i) Find the coordinates of M 1

(ii) Find the equations of OM and AN 3

(iii) Find C the point of intersection of OM and AN 2

(iv) Find the exact distance OC 2

(v) Given that the distance CM is 2 units, 1

show that the ratio OC : CM 2 :1

(vi) Show that L,C and B are collinear.

2

(b) Sketch the region bound by the inequalities 4

2x y 2 0, 2x y 2 0 and y 0

L

N

C M

y

x O

A(2,6)

B (4, 0)

26



2


1 x 1

f(x) 2x 3 1 x 0

1 x x 0

(i) Sketch f (x) in the domain 2 x 2.

3

(ii) Hence evaluate 2f ( 2) f (2) 2

(b) A region R is defined by the inequal

2ities y 1 1 x and y 1

(i) Sketch this region R on the number plane. 3

(ii) Calculate the area of R.

1

1 f (x) f (1)(c) Given that f (x) and g(x)

x x 1

(i) Find g(x) in the simplest form.

2

(ii) Hence find g(x) when x 1.01 1

(d) A function is defined by f (x)

2

1 find

1 x

(i) the domain 1

(ii) the range

2

27



4(a) For the function f (x) x 1

(i) Show that f (x) is an even function. 2

(ii) Sketch the graph of f (x) in the domain 2 x 2

Clearly show all x and y intercepts.

(b) A point P(x, y) moves

such that PA PB where A and B

are the points ( 1, 2) and (3,6) respectively.

(i) Find the equation of the locus of P.

3

(ii) Hence, or otherwise, show that the locus of P is a circle 2

and state the coordinates of the centre and the length of

the radius of this circle.

(c) A point P(x, y) moves such that it is equidistant from the lines 3

3x 4y 0 and 4x 3y 2 0. Find the eq uations of the locus of P.

(d) A point P(x, y) moves such that it is equidistant from the point (0, 2) 3

and the x axis. Find the equation of the locus of P.

28

Term 1 Answers

Set 1

Question 11

y 4(a) 1.39 (b) (2x 3)(x 5) (c) x 2 or x 3 (d)

2

(e) x 2 (f) x (g) 3 7+ 5 (h) x 0, x 6

2

Question 12

1(a) 2.38 (b) a 3, b 1 (c) x 3, x 2 (d) x 1

3

4n 3(e) (f) 4 2 (g) x 3, y 1 (h) 2 x 5 x 5x 25

n n 1

Question 13

2(a) p 1 (b) a (c) x 5, y 2 (d) x y 2 0 (e) r 1

3

Question 14

2(a) (i) AB 5 units (ii) C(8,8) (v) 36 units (b) distance 1 unit

3

(c)

1

29

Question 15

(a) (i)

(ii) 1

(b) (i)

(ii) 2 units

2

1 1(c) (i) g(x) (ii) g(0) (d) (i) x 2 or x 2 (ii) f (x) 0

a a h a

Question 16

(a) (i)

(ii) (2,1)

(iii) x 2

(b) (i) ( 2,1)

(ii) r 2

(c) 3x 4y 7 0

(d) 2x y 5 0

2 2

( 3,1)

(3, 1) 1

1

1 1 3 3

( 1, 2)

( 1, 5)

(2,1)

(3,2)

(3,3)

2 1 1 3

30

Set 2

2

Question 11

x 1(a) 5.9 (b) (2x 3)(4x 6x 9) (c) 3 x 4 (d)

x 1

(e) 48 (f) x (g) 2 2 (h) x 2, x 1

Question 12

3 5(a) 4.57 (b) 2x 12 (c) x 1 (d) x 8 (e) 0.24 (f)

2

(g) x 3, y 9 and x 2, y 4 (h) x y x y 1

o

Question 13

8 5(a) B 14, 4 (b) 3x 2y 9 0 (c) 45 (d) 2x y 4 0 (e)

5

2

Question 14

(a) (i) OB 8 2 units (ii) M(4,4) (v) (4,4) (vi) 32 unit

(c)

1

1

31

Question 15

1 1(a) (i) a , b 10 (ii)

2 4

(b)

(c) (i) g(x) x 3 (ii) g(2.01) 5.01 (d) (i) 5 x (ii) 0 f (x) 5

Question 16

(a) (ii)

1 1

2 2

2 2

(b) (ii) x 1, f (x) 1 (c) x 1 + y 2 =4

(d) (i) x y 10x 9 0 (ii) 5,0 , r 4

( 2, 6)

(2,6)

1 1

32

Set 3

4 2

n

Question 11

(a) 1.3 10 (b) (x 1)(x 1) (c) x 0, x 2 (d) m 2

(e) 5 (f) x 7 (g) 8 2 (h) x 1 2

2

Question 12

2n 1(a) 5.40 (b) (c) x 0, x 2 (d) x 1 (e) $24 000

n n 1

(f) 2 x 3 (g) x 1 if x 0 and x 1 if x 0 (h) a 7

2 2

Question 13

(a) 10 units (b) 3x 5y 1 0 (c) (ii) 25 units (d) 5, 1

Question 14

(a) (i) M(3,3) (ii) OM: y x, AN: x 2 (iii) C (2,2) (iv) OC 2 2

(b)

2

1 1

33

Question 15

(a) (i)

(ii) 1

(b) (i)

(ii) 2

units2

1001(c) (i) g(x) (ii) g(1.01) (d) (i) all real x (ii) 0 f (x) 1 x 101

Question 16

(a) (ii)

2 2

2

(b) (i) x 1 + y 4 8 (ii) C (1,4), r 2 2

(c) x 7y 2 0, 7x y 2 0 (d) x 4 y 1

( 2,1)

1 1

(2, 3)

1

3

1 1 2 2

( 2,15) (2,15)

1 1 1

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Transcript of Higher School Certificate · 2016-11-17 · 5 Preface In 2012 the format for the Higher School...