Heathcote School & Science College

DEPARTMENT OF MATHEMATICS Introduction to A level Maths and Further Maths

Transition Booklet 2017/18

Current GCSE Mathematics Grade ____ Target A Level Mathematics Grade ____

Introduction to A Level Maths at Heathcote

Welcome! And thank you for choosing to study A Level maths at Heathcote. Some students find the transition from GCSE to A level maths quite difficult. So in order to give you the best possible start, we have prepared this booklet for you. It is vitally important that you read through this book and then complete the questions. You will need to hand in the completed questions to your teacher at the start of the new school year. Some of the topics are grade 7/8/9 GCSE topics which you should have been introduced to already and which will appear in the course. You will need a good and secure knowledge of these topics before you start the course in September. In the booklet you will find diagnostic questions and worked solutions. Work carefully through these questions and try to show all your workings; this will show your knowledge and understanding. There is also a test yourself section with an accompanying mark scheme. You may need to research some topics beforehand to help you with completing the tasks. A list of websites has been provided on the next page should you wish to use them. At the beginning of the course you will be given a test to check how well you have understood the topics. If you do not pass the initial test you will be put on an intervention programme. This will attempt to bring your knowledge and understanding of the number and algebra skills up to the required standard. You will then be re-tested in October. To be successful at A Level Maths you will:

Need to have a good understanding of the fundamental Maths topics in this booklet

Be capable of working hard and independently in and out of maths lessons

Be able to show all your calculations i.e. full written solutions

Need to be involved in a feedback dialogue where you regularly discuss your work with a teacher and other students. This will help you develop a greater understanding of the maths

Need to be resilient; you will need to try all the problems, maybe attempt them a few times as Maths can at times be solved in many different ways

Benefit from wider reading, internet research and discussions. We hope that you will use this transition booklet wisely as it will give you a good start to your AS year. Hopefully it will help you answer some of your questions and give you a bit more confidence so that you can make the necessary progress and achieve your target A Level grades! Ms M Geldenhuys

Head of Mathematics

You may need to research some of the topics in books or via the internet to help you complete the questions to the best of your ability. The following book and internet sites might be of use:

AS Level Maths Head Start Published by CGP Workbooks ISBN: 987 1 84146 993 5 Cost: £4.95

A Level maths videos and tutorials www.hegartymaths.com

A Level textbook questions and worked solutions www.physicsandmathstutor.com

Step up! To A Level maths www.cimt.org.uk

Extra AS/A2 level resources http://www.schoolworkout.co.uk/a_level.htm

A Level past papers revision videos www.examsoiutions.net

Variety of A Level Maths resources www.mrbartonmaths.com

The jump is much bigger than expected so a lot more

‘independent’ work is required

Do not leave revision to the last few weeks

before the exam.

Do some extra work after school

on the topics covered in the lesson, as this

would benefit you

Use A level Maths websites

Although you might understand the concept behind a chapter, make

sure you are able to apply this in an exam style

question.

Try and read ahead (it really helps) and ALWAYS ask for help when you do not understand.

Some advice from the A Level maths students

You can expect a lot of Home work!!

Summary of overlap (Bold) and transition between Higher GCSE Maths and A Level Maths

Higher tier GCSE A Level

Nu

mb

er

Percentages Fractions Accuracy Exponential growth & decay Index Laws Surds (including rationalising) Direct & Inverse proportion

Manipulating Surds Laws of indices Factorising quadratics Solving quadratic equations (by factorising, completing the square & quadratic formula) Discriminant of a quadratic function Solving Simultaneous equations (including one quadratic equation) Solving inequalities (including quadratic inequalities) Equation of a circle & circle geometry Algebraic manipulation Division of polynomials Remainder & factor theorem Differentiation of polynomials & applications Integration of Polynomials & applications

Alg

ebra

Changing subject of a formula Plotting non-linear graphs (quadratic, cubic, reciprocal, exponential) Sequences (nth term) Expanding & factorising algebraic expressions Solving linear inequalities Solving Simultaneous equations Solving quadratics (by factorising & difference of two squares) Equation of a circle Transformations of graphs

Sketching graphs of functions Exponential graphs Transformations of graphs Sequences & series Laws of logarithms Binomial expansion Sine & Cosine rule Radians, arc length and sector area Graphs of trigonometric equations Solving trigonometric equations Differentiation of functions involving fractional and negative indices Integration of functions involving fractional and negative indices Trapezium rule to estimate area under a curve G

eom

etry

&

Mea

sure

Dimensions of formulae Surface area/volume of 3D shapes Transformations of 2D shapes Vector geometry Properties of circles (including arcs, sectors & segment) Pythagoras Rule (including 3D) Trigonometry of right-angled triangles (Including 3D) Sine & cosine rule (including 3D) Graphs of trig. functions

Dat

a &

P

rob

abili

ty

Probability & tree diagrams Addition & multiplication of probabilities Cumulative frequency Correlation & lines of best fit Histograms & frequency density Comparing distributions Sampling methods

Discrete & continuous data Sampling Measures of average (mode, median & mean from grouped data) Measures of spread (range, interquartile range, standard deviation & variance) Normal distribution and confidence intervals Binomial distribution Probability Measuring correlation & describing regression

DIAGNOSTIC QUESTIONS AND WORKED SOLUTIONS

1 Solving quadratic equations

Question 1

Solve x2 + 6x + 8 = 0 (2)

Question 2

Solve the equation y2 – 7y + 12 = 0

Hence solve the equation x4 – 7x

2 + 12 = 0

(4)

Question 3

(i) Express x2 – 6x + 2 in the form (x-a)

2 – b

(3)

(ii) State the coordinates of the minimum value on the graph of y = x2 – 6x + 2

(1)

Total / 10

Question 1

Solve x2 + 6x + 8 = 0

(x + 2)(x + 4) = 0

x = –2 or –4

(2)

Question 2

Solve the equation y2 – 7y + 12 = 0

Hence solve the equation x4 – 7x

2 + 12 = 0

(4)

Question 3

(i) Express x2 – 6x + 2 in the form (x-a)

2 – b

(3)

(ii) State the coordinates of the minimum value on the graph of y = x2 – 6x + 2

(1)

2 Changing the subject

Question 1

Make v the subject of the formula E = mv2

(3)

Question 2

Make r the subject of the formula V = Π r2

(3)

Question 3

Make c the subject of the formula P =

(4)

Total / 10

Question 1

Make v the subject of the formula E = mv2

(3)

Question 2

Make r the subject of the formula V = Π r2

(3)

Question 3

Make c the subject of the formula P =

(4)

3 Simultaneous equations

Question 1

Find the coordinates of the point of intersection of the lines y = 3x + 1 and x + 3y = 6

(3)

Question 2

Find the coordinates of the point of intersection of the lines 5x + 2y = 20 and y = 5 - x

(3)

Question 3

Solve the simultaneous equations

x2 + y

2 = 5

y = 3x + 1

(4)

Total / 10

Question 1

Find the coordinates of the point of intersection of the lines y = 3x + 1 and x + 3y = 6

(3)

Question 2

Find the coordinates of the point of intersection of the lines 5x + 2y = 20 and y = 5 – x

(3)

Question 3

Solve the simultaneous equations

x2 + y

2 = 5 y = 3x + 1

(4)

4 Surds

Question 1

(i) Simplify (3 + )(3 - )

(2)

(ii) Express in the form a + b where a and b are rational

(3)

Question 2

(i) Simplify 5 + . Express your answer in the form a where a and b are integers and b is as small as possible.

(2)

(ii) Express in the form p + q where p and q are rational

(3)

Total/10

Question 1

(i) Simplify (3 + )(3 - )

(2)

(ii) Express in the form a + b where a and b are rational

(3)

Question 2

(i) Simplify 5 + 4 . Express your answer in the form a where a and b are integers and b is as small as possible.

(2)

(ii) Express in the form p + q where p and q are rational

(3)

5 Indices

Question 1

Simplify the following

(i) a0

(1)

(ii) a6 ÷ a

-2

(1)

(iii) (9a6b

2)-0.5

(3)

Question 2

(i) Find the value of ( ) -0.5

(2)

(ii) Simplify

(3)

Total / 10

Question 1

Simplify the following

(i) a0

(1)

(ii) a6 ÷ a

-2

(1)

(iii) (9a6b

2)-0.5

(3)

( = )

Question 2

(i) Find the value of ( ) -0.5

(2)

(ii) Simplify

(3)

6 Properties of Lines

Question 1

A (0,2), B (7,9) and C (6,10) are three points.

(i) Show that AB and BC are perpendicular

(3)

(ii) Find the length of AC

(2)

Question 2

Find, in the form y = mx + c, the equation of the line passing through A (3,7) and B (5,-1).

Show that the midpoint of AB lies on the line x + 2y = 10

(5)

Total / 10

Question 1

A (0,2), B (7,9) and C (6,10) are three points.

(i) Show that AB and BC are perpendicular

Grad of AB = = 1

Grad of BC = = -1

Product of gradients = 1 x -1 = -1 → AB and BC perpendicular

(3)

(ii) Find the length of AC

(6-0)2 + (10 – 2)

2 = AC

2

AC = 10

(2)

Question 2

Find, in the form y = mx + c, the equation of the line passing through A (3,7) and B (5,-1).

Show that the midpoint of AB lies on the line x + 2y = 10

(5)

7 Sketching curves

Question 1

In the cubic polynomial f(x), the coefficient of x3 is 1. The roots of f(x) = 0 are -1, 2 and 5.

Sketch the graph of y = f(x)

(3)

Question 2

Sketch the graph of y = 9 – x2

(3)

Question 3

The graph below shows the graph of y =

On the same axes plot the graph of y = x2 – 5x + 5 for 0 ≤ x ≤ 5

(4)

Total/10

Question 1

In the cubic polynomial f(x), the coefficient of x3 is 1. The roots of f(x) = 0 are -1, 2 and 5.

Sketch the graph of y = f(x)

(3)

Question 2

Sketch the graph of y = 9 – x2

9

-3 3

(3)

Question 3

The graph below shows the graph of y =

On the same axes plot the graph of y = x2 – 5x + 5 for 0 ≤ x ≤ 5

(4)

8 Transformation of functions

Question 1

The curve y = x2 – 4 is translated by ( )

Write down an equation for the translated curve. You need not simplify your answer.

(2)

Question 2

This diagram shows graphs A and B.

(i) State the transformation which maps graph A onto graph B

(2)

(ii) The equation of graph A is y = f(x).

Which one of the following is the equation of graph B ?

y = f(x) + 2 y = f(x) – 2 y = f(x+2) y = f(x-2)

y = 2f(x) y = f(x+3) y = f(x-3) y = 3f(x)

(2)

Question 3

(i) Describe the transformation which maps the curve y = x2 onto the curve y = (x+4)

2

(2)

(ii) Sketch the graph of y = x2

– 4

(2)

Total / 10

Question 1

The curve y = x2 – 4 is translated by ( )

Write down an equation for the translated curve. You need not simplify your answer.

(2)

Question 2

This diagram shows graphs A and B.

(i) State the transformation which maps graph A onto graph B

(2)

(ii) The equation of graph A is y = f(x).

Which one of the following is the equation of graph B ?

y = f(x) + 2 y = f(x) – 2 y = f(x+2) y = f(x-2)

y = 2f(x) y = f(x+3) y = f(x-3) y = 3f(x)

Answer f(x-2) (2)

Question 3 (i) Describe the transformation which maps the curve y = x

2 onto the curve y = (x+4)

2

(2)

(ii) Sketch the graph of y = x2

– 4

(2)

9 Trigonometric ratios

Question 1

Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall.

The ladder is 16 feet long.

The foot of the ladder is 4 feet from the base of the wall.

(i) Work out how high up the wall the ladder reaches. Give your answer to 3 significant figures.

(2)

(ii) Work out the angle the base of the ladder makes with the ground. Give your answer to 3 significant figures

(2)

Question 2

Given that cos Ɵ = and Ɵ is acute, find the exact value of tan Ɵ

(3)

Question 3

Sketch the graph of y = cos x for

(3)

Total / 10

Question 1

Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall.

The ladder is 16 feet long.

The foot of the ladder is 4 feet from the base of the wall.

(i) Work out how high up the wall the ladder reaches. Give your answer to 3 significant figures.

√162 - 4

2

√256 -16 correct substitution (M1)

√240

15.49

15.5 (3sf) (A1) (2)

(ii) Work out the angle the base of the ladder makes with the ground. Give your answer to 3 sig fig

cos x = correct ratio and substitution (M1)

cos x = 0.25

x = 75.522

x = 75.5° (A1) (2)

Question 2

Given that cos Ɵ = and Ɵ is acute, find the exact value of tan Ɵ

(3)

Question 3

Sketch the graph of y = cos x for

(3)

10 Sine / Cosine Rule

Question 1

For triangle ABC, calculate

(i) the length of BC

(3)

(ii) the area of triangle ABC

(3)

Question 2

The course for a yacht race is a triangle as shown in the diagram below. The yachts start at A, then travel to B, then to C and finally

back to A.

Calculate the total length of the course for this race.

(4)

Total / 10

Question 1

For triangle ABC, calculate

(i) the length of BC

(3)

(ii) the area of triangle ABC

(3)

Question 2

The course for a yacht race is a triangle as shown in the diagram below. The yachts start at A, then travel to B, then to C and finally

back to A.

Calculate the total length of the course for this race.

Total length = 384 + 650 = 1034m (4)

TEST YOURSELF

1 Solving quadratic equations

Question 1

Find the real roots of the equation x4 – 5x

2 – 36 = 0 by considering it as a quadratic equation in x

2

(4)

Question 2

(i) Write 4x2 - 24x + 27 in the form of a(x - b)

2 + c

(4)

(ii) State the coordinates of the minimum point on the curve y = 4x2 - 24x + 27.

(2)

Total / 10

2 Changing the Subject

Question 1

Make t the subject of the formula s = at2

(3)

Question 2

Make x the subject of 3x – 5y = y - mx

(3)

Question 3

Make x the subject of the equation y =

(4)

Total / 10

3 Simultaneous equations

Question 1

Find the coordinates of the point of intersection of the lines x + 2y = 5 and y = 5x - 1

(3)

Question 2

The lines y =5x – a and y = 2x + 18 meet at the point (7,b).

Find the values of a and b.

(3)

Question 3

A line and a curve has the following equations :

3x + 2y = 7 y = x2 – 2x + 3

Find the coordinates of the points of intersection of the line and the curve by solving these simultaneous equations algebraically

(4)

Total / 10

4 Surds

Question 1

(i) Simplify +

(2)

(ii) Express in the form a + b , where a and b are integers.

(3)

Question 2

(i) Simplify 6 x 5 -

(2)

(ii) Express ( 2 - 3 )2 in the form a + b , where a and b are integers.

(3)

Total / 10

5 Indices

Question 1

Find the value of the following.

(i) ( ) -2

(2)

(ii)

(2)

Question 2

(i) Find a, given that a3 = 64x

12y

3

(2)

(ii) ( ) -5

(2)

Question 3

Simplify

(2)

Total / 10

6 Properties of Lines

Question 1

The points A (-1,6), B (1,0) and C (13,4) are joined by straight lines. Prove that AB and BC are perpendicular.

(2)

Question 2

A and B are points with coordinates (-1,4) and (7,8) respectively. Find the coordinates of the midpoint, M, of AB.

(1)

Question 3

A line has gradient -4 and passes through the point (2,-6). Find the coordinates of its points of intersection with the axes.

(4)

Question 4

Find the equation of the line which is parallel to y = 3x + 1 and which passes through the point with coordinates (4,5).

(3)

Total / 10

7 Sketching curves

Question 1

You are given that f(x) = (x + 1)(x – 2)(x – 4)

Sketch the graph of y = f(x)

(3)

Question 2

Sketch the graph of y = x(x - 3)2

(3)

Question 3

This diagram shows a sketch of the graph of y =

Sketch the graph of y = , showing clearly any points where it crosses the axes.

(3)

Question 4

This curve has equation y = x (10 - x). State the value of x at the point A.

Total / 10

8 Transformation of functions

Question 1

The graph of y = x2 – 8x + 25 is translated by ( ). State an equation for the resultant graph.

(1)

Question 2

f(x) = x3 – 5x + 2

Show that f(x – 3) = x3 – 9x

2 + 22x - 10

(4)

Question 3

You are given that f(x) = 2x3 + 7x

2 – 7x – 12

Show that f(x – 4) = 2x3 – 17x

2 + 33x

(3)

Question 4

You are given that f(x) = (x + 1)(x – 2)(x – 4).

The graph of y = f(x) is translated by ( ).

State an equation for the resulting graph. You need not simplify your answer.

(2)

Total / 10

9 Trigonometric ratios

Question 1

AP is a telephone pole. The angle of elevation of the top of the pole from the point R on the ground is 42°as seen in the diagram.

Calculate the height of the pole. Give your answer to 3 significant figures. (3)

Question 2

Given that sin Ɵ = , find in surd form the possible values of cos Ɵ.

(3)

Question 3

The graph of y = sin x for is shown below.

What are the coordinates of the 4 points labelled on the graph?

(………, ………)

(………, ………)

(………, ………)

(………, ………)

(4)

Total / 10

10 Sine / Cosine Rule

Question 1

This diagram shows a village green which is bordered by 3 straight roads AB, BC and AC. The road AC runs due North and the

measurements are shown in metres.

(i) Calculate the bearing of B from C, giving your answer to the nearest 0.1o

(4)

(ii) Calculate the area of the village green.

(2)

Question 2

This diagram shows a logo ABCD. It is symmetrical about AC.

Find the length of AB and hence find the area of the logo

(4)

Total/10

TEST YOURSELF MARK SCHEME Section Question Answer Marks Notes

1 1 x = ± 3 M2

M1

A1

Use of quadratic formula (M1)

in x2

(M1)

x2 = 9

cao

2(i) 4 (x – 3)2 - 9 B1

B1

M1A1

a = 4

b = 3

c = -9

2(ii) (3,-9) B2 B1 for each coordinate

2 1

B3 B2 for t omitted

M1 for constructive first step

M1 for finding square root of their ‘t2’

2

M1

M1

A1

for 3x + mx = y + 5y oe

for x(3 + m) or ft sign error

3

M1

M1

M1

M1

for multiplying by x-2

for expanding brackets

for cllecting x and ‘other’ terms

for factorising and dividing

Award all four marks only if fuly correct

3 1 x = y = oe www

B3 B2 for one coordinate correct, or correct

solution not erxpressed as coordinates

(or) M1 for substitution or elimination of

one variable oe

2 a = 3

b = 32

M1

A1A1

Equating 5x – a and 2x + 18 and

substituting x = 7

3 x = -0.5 or 1

y = 4.25 or 2

M1

M1

A1

A1

for 7-3x = 2(x2 – 2x + 3) oe

for quadratic in x (2x2 – x – 1 = 0 oe)

x

y

4 1(i) M1

A1 for oe seen

1(ii) 10 + M1

M1

A1

for attempt to multiply num and denom

by 5 +

for 18 or 25 – 7 seen

2(i) M1

A1 for or oe

2(ii) 49 - 12 B2

B1

for 49

for 12

If B0, award M1 for 3 correct terms of 4

- 6 - 6 + 45

5 1(i) 9 M1 A1

for 32 oe

1(ii) 8 (condone -8 or ±8) M1 A1

for 160.25 = 2

2(i) 4x4y M1 A1

for two elements correct

2(ii) 32 M1 A1

for 25 oe

3

B1 B1

numerator denominator

6 1 Grad of AB = -3

Grad of BC =

product of gradients = -1

B1 B1

either gradient product of gradients need to equal -1

2 (3,6) B1

3 Coordinates (0,2) (0.5,0)

M1 M1 A1A1

for y = -4x + c for y = -4x + 14 one mark for each set of coordinates

4 y = 3x - 7 M1 M1 A1

Gradient = 3 Subst in (4,5) into their ‘y = mx + c’

7 1 Cubic the correct way up x-axis cuts at -1, 2, 4 shown y-axis cuts at 8 shown

G1 G1 G1

2 Sketch of cubic correct way up Curve through (0,0) Curve touches x-axis at x=3

G1 G1 G1

3 Correct graph with clear asymptote at x = 2 (0, -0.5) shown

G2 G1

(G1 for only one branch correct0

4 10 B1

8 1 y = x2 – 8x + 5 B1

2 f(x – 3) = (x – 3)3 –5(x – 3) + 2 (x2 – 6x + 9)(x – 3) f(x – 3) = x3 – 3x2 – 6x2 + 18x + 9x – 27 – 5x + 15 + 2 = x3 – 9x2 + 22x - 10

B1 B1 A1 B1

Substitution Partial expansion of cubic term All correct unsimplified Correct consolidation

3 f(x-4) = 2(x-4)3 + 7(x-4)2 – 7(x-4) – 12 2x3 – 17x2 + 33x

M1 M1 M1

Substitution Correct expansion of one pair of brackets correct completion to given answer

4 (x + 1 – 3)(x – 2 – 3)(x – 4 – 3) ie (x -2)(x – 5)(x – 7)

M1 A1

Allow one slip Oe

9 1 Tan 42° =

0.9004 =

13.5(06) m = height of pole

M1 M1 A1

2 ± B3 B2 for either - or or ± oe

or M1 for seen

3 (0, 0) ( 90, 1)

(270, -1) (360, 0)

B1 B1 B1 B1

10 1(i) C = 141.1…..

Bearing = 038. 8 (accept 038.9)

M1 M1 A1 A1

Correct attempt at cosine rule Correct full method for C C Bearing

1(ii) 3030 to 3050 acceptable M1 A1

Correct use of 0.5xaxbxsinC

2

AB = 7.80 (or better, 7.799…)

Area = 52.2 to 52.3

M1 A1 M1 A1

Correct use of sine rule AB 2 x 0.5 x ’their AB’ x 11.4 x sin 36 Area