Student

Teacher

AS STARTER PACK September 2016

City and Islington Sixth Form College

Mathematics Department

www.candimaths.uk

2

CONTENTS

INTRODUCTION 3

SUMMARY NOTES 4

WS CALCULUS 1 ~ Indices, powers and differentiation 6

WS CALCULUS 2 ~ Applications of differentiation 8

WS CURVE SKETCHING 1 ~ Introduction 10

WS CURVE SKETCHING 2 ~ Sketching quadratics 13

WS CURVE SKETCHING 3 ~ Equation of a straight line 15

WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations 18

WS QUADRATICS 2 ~ Completing the square 20

WS QUADRATICS 3 ~ Quadratic formula and discriminant 22

WS 9 TANGENTS AND NORMALS 25

WS SEQUENCES 1 ~ General sequences and summation notation 28

WS SEQUENCES 2 ~ Arithmetic sequences 31

WS INTEGRATION 33

IMPORTANT INFORMATION 36

3

INTRODUCTION Over the next 6 weeks you will be studying new topics in mathematics. Each of these new topics

builds upon GCSE work. At AS level you must ensure that you achieve a high standard of written

mathematics, which is clear, logical and fluent. You will need to think deeply about the concepts

and put in regular practice.

Homework: You will be given homework each week to support your learning. Some homework

will involve pre-learning that prepares you for the next lesson so it is very important that you

complete it. You will be expected to mark the homework yourself and your teachers will check the

working out and lay out of your work.

Week 2 Skills Test 1: This test is to check that you have not forgotten GCSE maths! If you do

not do well on this test then you will be given extra work to make sure you are ready for the A-

level mathematics.

Week 6 Skills Test 2: This test is to make sure you are ready for the harder parts of the course.

HW6: Practice Test in your C1 Homework Pack is an example of this test. You will need to work

hard - the pass mark for this test is 70%.

Lesson 1 Lesson 2 Lesson 3

Week 1 Induction 1 – Introduction

to course

Differentiation 1 Differentiation 2

Week 2 SKILLS TEST 1

Curves Sketching 1

Curve Sketching 2

(Quadratics)

Curve Sketching 3

(Equation of a straight line)

Week 3 Quadratics 1 Quadratics 2 Quadratics 3

Week 4 Tangents and Normals 1 Tangents and Normals 2 Induction 2 – ICT skills

Week 5 Sequences 1 Sequences 2 Induction 3 – Study skills

(folder check)

Week 6 SKILLS TEST 2 Integration Integration

Week 7 HALF TERM

Extra resources, links and digital copies of the booklets can be found at our website:

www.candimaths.uk

4

SUMMARY NOTES

Number

Algebra

2𝑥 + 3 = 11

2𝑥 = 8

𝑥 = 4

𝑥2 + 6𝑥 − 16 = 0 𝑥 + 8 𝑥 − 2 = 0

𝑥 = 2,−8

𝑥2 + 6𝑥 − 16 = 0 𝑥 + 3 2 − 9 − 16 = 0

𝑥 + 3 2 = 25

𝑥 = −3 ± 25

𝑥 = 2,−8

Linear

Quadratic

Complete the Square

Formula 𝑥 =−𝑏± 𝑏2−4𝑎𝑐

2𝑎

Completed Square Form

𝑦 = 𝑥2 + 8𝑥 + 21 𝑦 = 𝑥2 − 8𝑥 + 21

𝑦 = 𝑥 + 4 2 − 16 + 21 𝑦 = 𝑥 − 4 2 − 16 + 21

𝑦 = 𝑥 + 4 2 + 5 𝑦 = 𝑥 − 4 2 + 5

2𝑥2 + 11𝑥 + 15 = 𝑥 + 3 2𝑥 + 5

𝑥2 + 6𝑥 + 9 = 𝑥 + 3 2

𝑥2 − 6𝑥 + 9 = 𝑥 − 3 2

𝑥2 − 9 = 𝑥 + 3 𝑥 − 3

5𝑥2 + 19𝑥 + 12 = 5𝑥 + 4 𝑥 + 3

Factorising

3𝑥 + 5𝑦 = 20

𝑥2 + 𝑦2 = 5

Simultaneous Equations

𝑥 + 4𝑦 = 16 Elimination method

2𝑥 − 𝑦 = 4 Substitution method

ℕ Natural 1, 2, 3, .. [counting]

ℤ Integers -2, -1, 0, 1, 2,… [counting ±]

ℚ Rational 2

3, −

43

67, 86, 0 [fractions ±, all except Irrational]

ℝ Real [All including irrationals numbers eg 2,𝜋, 𝑒]

Irrational numbers cannot be written as fractions.

As decimals they are infinite and non-recurring.

−2 + 3 − −4 = 5

+ −

Equivalent fractions

Fraction arithmetic

3

4+

5

6=

9

12+

10

12=

19

12 𝐿𝐶𝑀

3

4÷

6

5=

3

4×

5

6=

15

24=

5

8

Directed Numbers

BIDMAS

( ) 23 + 5 − 1 2 = 24

𝑥2 5 × 6 − 32 = 21

×÷ 10−4

2= 3

Indices (powers)

23 × 24 = 27 2−3 =1

23

25 ÷ 23 = 22 21 = 2

23 5 = 215 20 = 1

5

Geometry

Sequences

𝑦 = 𝑥2 + 9𝑥 − 22

𝑦 = 𝑥 + 11 𝑥 − 2

Graph Sketching

When

When

centre = 3, 1

𝑥,𝑦 𝑚 = 2 𝑟𝑎𝑑𝑖𝑢𝑠 = 4

𝑦 − 1 = 2 𝑥 − 3

𝑥 − 3 2 + 𝑦 − 1 2 = 16

Line

𝑦 = 2𝑥 − 5 or 2𝑥 − 𝑦 − 5 = 0

Circle

1, 3

5, 8

Normal (perpendicular line)

𝑚𝑖𝑑 𝑝𝑜𝑖𝑛𝑡 = 1 + 5

2,3 + 8

2

𝑑𝑖𝑠𝑡2 = 5 − 1 2 + 8 − 3 2

gradient 𝑚1 =8 − 3

5 − 1=

5

4

𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑛𝑜𝑟𝑚𝑎𝑙 𝑚2 =−4

5

Arithmetic Sequence

First term: 𝑎,

Common difference: 𝑑

Number of terms: 𝑛

𝑛th term: 𝑈𝑛 = 𝑎 + 𝑛 − 1 𝑑

Sum to 𝑛 terms: 𝑆𝑛 =𝑛

2 2𝑎 + 𝑛 − 1 𝑑

𝑆𝑛 =𝑛

2 𝑎 + 𝑙

6

WS CALCULUS 1 ~ Indices, powers and differentiation

Keywords BIDMAS, powers, indices, differentiation, evaluate

Exercise A

Simplify the following.

1. 3 × 3 × 4 2. 3 × 5 6 3. 20 7 ÷ 4 3 4. 7 3×2

3

5. 53

5 6. 2 3 7. 2 2 3 8. 2 2 3

9. 4 4 2 10. 3 2 4 2

Exercise B

1. Simplify the following, giving each term in the form , where and are constants.

(a) 2 (b) 3

2 (c)

1

3 3 (d)

1

3 3 − 2

(e) 2

3 (f)

3+

1

2 (g)

2 3

(h)

3 2−6

2

(i) 2 3 3

(j) 2 − + 2 (k) 3 2 2 + 2 (l) 3 − 2 (4 +

1

)

2. Use your answers to Q1 to differentiate each of the above expressions.

Exercise C – more challenging

1. Evaluate: (36

2 + 16

)

3

2. Solve to find : 3

2 = 2

Exam Questions

1. [C1 May 2014 Q2]

(a) Write down the value of 5

1

32 .

(1)

(b) Simplify fully 5

2

5 )32(

x .

(3)

7

2. [C1 Jan 2014 Q1]

Simplify fully

(a) 2

2 x

(1)

3. [C1 Jan 2014 Q2]

2 42 1y x

x

, x > 0

(a) Find d

d

y

x, giving each term in its simplest form.

(3)

Answers

EXA

1. 10 2. 15 7 3. 5 4 4. 14

3 5.

1

125

6. 6 7. 2 6 8. 8 6 9. 16 8 10. 12 2 8

EXB

1. (a) 2

2 (b) 3 −2 (c) 1

3 −3 (d)

1

3 4 −

2

3 3 (e) 2 −3

(f)

3 +1

2 −1 (g) 2 + 3 −1 (h)

3

2 − 3 −1 (i) 2

2 + 3

2 (j) 3 − 2 + 2

(k) 3 4 + 6 3 (l) 12 2 − 8 + 3 − 2 −1

2. (a) 1

(b) −

6

3 (c) −1

(d) 4

3 3 − 2 2 (e) −

6

(f) 1

3 √ 23 −1

2 2 (g) −3

2 (h) 3

2+

3

2 (i) 5 3 +3

2 (j) 3 2 − 2 + 2

(k) 12 3 + 18 2 (l) 24 − 8 +2

2

EXC

1. 2 2. 9

Exam Questions

1. (a) 2 (b) 1

4 2 2. 4 3. 4 +2

√ 3

8

WS CALCULUS 2 ~ Applications of differentiation

Exercise A - Complete the table

Equation of curve

Gradient Function

dy

dx

Gradient of the curve at these points

2x 1x 0x

Example = 4

= 4 3

4 2 3 = 32 4 −1 3 = −4 4 0 3 = 0

A = 5

B = 2 − 5

C = 3 − 9

D = 10 2

E = + 3 − 6

F = 2 + 3

G = 5 −

H = − 3 3

I = 19 + 4 2 − 3

J = 2 × 3

K = 1 − 3 + 4

L =1

2 2

M =1

4 + 2

N = 2 + 3 1 −

O =2

3 2

9

Exercise B - Complete the table

Equation of curve Gradient =

Solve for x

Example =1

3 3 −

5

2 2 + 8 2

= 2 − 5 + 8

⇒ 2 − 5 + 8 = 2 ⇒ 2 − 5 + 6 = 0 ⇒ − 3 − 2 = 0 ⇒ = 2,3

A = 2 + 2 4

B = 3 3

4

C = 3 + 8 2 + 9 − 2 4

Answers

EXA EXB

A 5 4 80 5 0 A = 2

B 2 4 -2 0 B = ±1

2

C 3 3 3 3

C = −1

3,−5

D 20 40 -20 0

E 2 − 3 1 -5 -3

F 1 + 2 5 -1 1

G 6 5 − 14 164 8 0

H −9 2 -36 -9 0

I 19 + 8 − 3 2 23 8 19

J 4 3 32 -4 0

K −3 + 4 3 29 -7 -3

L 2 -1 0

M 14⁄ 1

4⁄ 14⁄ 1

4⁄

N −2 − 2 -6 0 -2

O 43⁄ 8

3⁄ −43⁄ 0

10

WS CURVE SKETCHING 1 ~ Introduction

Keywords: curve, axes, intersection, maximum, minimum, linear, quadratic, cubic, reciprocal

Exercise A

Match the following graphs to their equations

A B C

D E F

G H I

Equation Graph Equation Graph

= = − 2

= 2 = − 1

= − =1

= 3 = + 1 − 2

= 2

11

Exercise B [There is more than one possible answer for some of these questions!]

(a) On separate diagrams sketch the following the graphs. Make sure you label the axis correctly

and use a ruler where necessary.

(b) Try and write an equation for each graph (some are difficult!).

1. A linear graph that crosses the axes at 2, 0 and 0, 2 .

2. A quadratic graph that crosses the axes at 0, 3 only.

3. A cubic graph that crosses the axes at 0, −2 , −4, 0 , 1, 0 and 3, 0 .

4. A linear graph that has a gradient of 3 and crosses the -axis at 2.

5. A reciprocal graph that has asymptotes at = 3 and = 1.

6. A negative quadratic graph that passes through −1, 0 and the origin.

7. A cubic graph that crosses the origin and 2, 0 .

8. A reciprocal graph that passes through (0,1

2)

Exercise C

For each equation fill in the following table and use the results to sketch the curves.

0

0

Very big

Very small

1. = 3 + 1 2. = 2 + 1 3. =1

2 − 5

4. = + 1 − 4 5. = −2 + 2 6. = 3 − 1

Exam Question

1. [C1 Jan 2012 Q8]

The curve C1 has equation y = x2(x + 2).

(a) Find x

y

d

d.

(2)

(b) Sketch C1, showing the coordinates of the points where C1 meets the x-axis.

(3)

(c) Find the gradient of C1 at each point where C1 meets the x-axis.

(2)

The curve C2 has equation

y = (x − k)2(x − k + 2),

where k is a constant and k > 2.

(d) Sketch C2, showing the coordinates of the points where C2 meets the x and y axes.

(3)

12

Answers

EXA

EXB 1.

= − + 2 2.

= 2 + 3

3.

= −1

6 + 4 − 1 − 3

4.

= 3 − 6

5.

=1

− 3+ 1

6.

= + 3

7.

= − 2 2 8.

=1

+ 2

EXC 1.

2.

3.

4.

5.

6.

Exam question:

Exam question

(a) 3 2 + 4 (b) 0, 4 (c) crosses axes at: , 0 , − 2, 0 , 0, − 3 + 2

Equation Graph Equation Graph

= E = − 2 D

= 2 B = − 1 F

= − I =1

C

= 3 A = + 1 − 2 H

= 2 G

13

WS CURVE SKETCHING 2 ~ Sketching quadratics

Keywords: quadratic, negative, positive, factorise, intersection, axes, differentiation,

Exercise A

Factorise each quadratic and sketch each curve on a different set of axes, stating clearly the

coordinates of the points where the curve intersects the axes.

1. = 2 + 3 − 4 2. = 2 + − 6

3. = 2 − 2 + 1 4. = − 2 − 5 − 6

5. = 2 2 − 9 − 5 6. = 2 − 4

7. = −5 2 + 3 + 2 8. = 2 − 4

9. = 2 + 4 + 5

Exercise B

Using differentiation, find the minimum or maximum points of each curve in Exercise A and write

them on your diagrams.

Exercise C

Write down a possible equation for each of these curves.

1.

2.

3.

4.

5.

6.

7.

8.

9.

14

Answers

EXA/B 1.

= + 4 − 1

Crosses the axes at:

−4, 0 , 1, 0 and 0,−4

Minimum point:

(−3

2, −

25

4)

2.

= + 3 − 2

Crosses the axes at:

−3, 0 , 2, 0 and 0,−6

Minimum point:

(−1

2,

25

4)

3.

= − 1 2

Crosses the axes at:

1, 0 and 0, 1

Minimum point:

1, 0

4.

= − + 3 + 2

Crosses the axes at:

−3, 0 , −2, 0 and 0,−6

Maximum point:

(−5

2,

1

4)

5.

= 2 + 1 − 5

Crosses the axes at:

−1

2, 0 , 5, 0

and 0,−5

Minimum point:

(−9

4 −

121

8)

6.

= − 4

Crosses the axes at:

0, 0 , 4, 0

Minimum point: 2,−4

7.

= − 5 + 2 − 1

Crosses the axes at:

−2 5, 0 , 1, 0 and 0, 2

Maximum point:

(3

10,

49

10)

8.

= + 2 − 2

Crosses the axes at:

−2, 0 , 2, 0 and 0,−4

Minimum point:

0,−4

9.

Can’t be factorised!

Crosses the axes at:

0, 5

Minimum point:

−2, 1

EXC

1. = + 1 − 3 2. = + 2 2 3. = − + 1 − 2 4. = − + 1 − 2 5. = + 3 6. = − − 2 7. = 2 − 1 + 1 8. = − 2 9. = − 1 + 1 − 3

15

WS CURVE SKETCHING 3 ~ Equation of a straight line

Keywords: gradient,

=

=

=

2 − 1

2 − 1

Exercise A

Work out the gradient of each of these lines

1.

2.

3.

4.

5.

6.

Exercise B

Find the gradient of the straight line between the following points:

1. 12, 3 14, 2. −1, 5 2, 8

3. 0, 1 5,−9 4. 3,−2 −1, 4

5. 2, 3 0, 6. −3,−1 −1, 11

7. 20, 1 15, 8. −1,−5 −2,−8

9. 4,−3 13, −8 10. −20,−4 0,−12

Exercise C

Use the equation − 1 = − 1 to find the equation of the following lines in the form

= +

1. Passing through 2, 3 with gradient 4 2. Passing through 1, 5 with gradient −2

3. Passing through −1, 0 with gradient 3 4. Passing through 2, − with gradient 1

2

5. Passing through 12, 3 and 14, 6. Passing through −1, 5 and 2, 8

7. Passing through −1, 5 and 1, 9 8. Passing through 8, 0 and −2, 5

9. Passing through 2, 2 and 5, − 10. Passing through 20, 10 and 35, 5

16

Exercise D

Find the equations of the following lines in the form + + = 0 where , , and are

integers:

Question Working Out Equation of line

1. Gradient is 4 and intercept is -2

2. Gradient is 1

2 and

crosses axis at 5

3. Gradient is -6 and goes

through 0,2

4. Gradient is 3 and passes

through 1,2

5. Gradient is -1 and passes

through 4, 3

6. Gradient is 2

3 and

passes through 6,2

7. Line passes through

4,8 and 3,11

8. Line passes through

2,5 and 1,14

9. Line passes through

3,0 and is

perpendicular to

= 2 − 3

10. Line passes through

1,4 and is

perpendicular to

= − + 2

17

Answers

EXA

1. 2 2. -1 3. 12⁄ 4. -2 5. 3

4⁄ 6. −52⁄

EXB

1. 2 2. 1 3. -2 4. −32⁄ 5. -2

6. 6 7. −65⁄ 8. 3 9. −5

9⁄ 10. −25⁄

EXC

1. = 4 − 5 2. = −2 + 3. = 3 + 3 4. =1

2 − 8 5. = 2 − 21

6. = + 6 7. = 2 + 8. = −1

2 + 4 9. = −3 + 8 10. = −

1

3 +

50

3

EXD

1. 4 − − 2 = 0 2. − 2 + 10 = 0 3. 6 + − 2 = 0 4. 3 − + 5 = 0 5. + − 1 = 0

6. 2 − 3 − 6 = 0 7. 3 + − 20 = 0 8. 3 − + 11 = 0 9. + 2 − 3 = 0 10. − + 3 = 0

18

WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations

Key words Quadratic, Factorise, Simultaneous, Solve, Gradient function

Exercise A

Solve the following equation by factorisation.

1. 2 + 6 + 5 = 0 2. 2 + 2 − 8 = 0

3. 2 2 + 11 + 5 = 0 4. 2 2 + + 5 = 0

5. 2 2 − 9 − 5 = 0 6. 2 2 + 9 − 5 = 0

7. 3 2 + 40 + 13 = 0 8. 3 2 − 8 − 11 = 0

9. −2 2 − 9 + 5 = 0 10. 2 + 2 − 11 = 0

Exercise B

1. Solve the simultaneous equations:

(a) = 4 + 2 (b) = 2 2 + 5 + 10 (c) = 3 2 + 30 − 10

= 2 + 9 + 16 = 5 − 2 = 1 − 2

2. Solve each pair of simultaneous equations using substitution:

(a) 76

2

yx

xy (b)

104

1

xxy

xy (c)

234

52

yx

xy

(d) 10

2

22

yx

yx (e)

5

3

xx

y

xy

(f) 082

23

2

yxy

yx

(g) yx

yx

316

6

2

(h)

21

3

22

yxyx

xy (i)

043

52

2

xyy

xy

Exercise C

Find the minimum and maximum points of these curves by solving

= 0

1. = 3

3+

5 2

2+ 4 −

2. = 3

3−

7 2

2+ 12 −

3. =2 3

3−

23 2

2+ 11 + 3

4. =2 3

3−

13 2

2− + 350

5. 3 3

5−

2 2

5−

7

5− 10 = 0

19

Answers

EXA

1. = −1,−5 2. = 2,−4 3. = −1

2, −5 4. = −

5

2, −1

5. = −5

2, −1 6. =

1

2, −5 7. = −

1

3, −13 8. =

11

3, −3

9. =1

2, −5 10. = 1 ± 3

EXB

1. (a) = −3,−4 (b) = −1

2, −5 (c) =

1

3, −11

2. (a) )1,1)(49,7( (b) )6,5)(1,2( (c) )6,1)(5,(169

43

(d) )3,1)(1,3( (e) )4,1)(6,3( (f) ),4)(2,1(54

53

(g) )4,2)(5,1( (h) )4,1)(1,4( (i) )0,5)(4,3(

EXC

1. = −1,−4 2. = 3,4 3. =1

2, 11 4. = −

1

2,

5. =7

3, −1

20

WS QUADRATICS 2 ~ Completing the square

Keywords express, solve, completed square form, solution, roots

Exercise A

Multiply the brackets (revision – try this as a mental arithmetic exercise)

1. )4)(4( xx 2. )4)(4( xx

3. 2)7( x 4. 2)6( x

5. 2)32( x 6. 2)53( x

7. Bill thinks that 222)( pxpx Is he correct?

Exercise B

Express in completed square form: qpxy 2)(

1. 162 xxy 2. 3122 xxy

3. 2682 xxy 4. 32102 xxy

5. 70162 xxy 6. 852 xxy

7. 25112 xxy 8. 12 xxy

*9. 18282 2 xxy *10. 1082 xxy

Exercise C

Solve by completing the square and leave your answers as fractions or surds

1. 046142 xx 2. 01382 xx

3. 022102 xx 4. 018102 xx

5. 030363 2 xx 6. 0652 xx

*7. 01882 2 xx *8. 06102 xx

Exercise D

1. Sketch the graphs of the functions below. Show the position of the vertex. [Hint: express in

completed form first]

a) 1362 xxy b) 23102 xxy

c) 1082 xxy d) 11155 2 xxy

2. Check some of your answers for EXC by substitution e.g. is 35x a solution for

022102 xx ?

3. Check that 54x is a solution for 01182 xx

4. Look up ‘completing the square’ on Wikipedia!

21

Answers

EXA

1. 1682 xx 2. 162 x 3. 49142 xx 4. 36122 xx

5. 9124 2 xx 6. 25309 2 xx 7. Not generally!

EXB

1. 8)3( 2 xy 2. 33)6( 2 xy 3. 10)4( 2 xy 4. 7)5( 2 xy

5. 6)8( 2 xy 6. 472

25)( xy 7. 4

212

211)( xy 8. 4

32

21)( xy

9. 80)7(2 2 xy 10. 26)4( 2 xy

EXC

1. 37x 2. 294x 3. 35x 4. 75x

5. 266x 6. 3,2x 7. 132 x 8. 315x

EXD

1. a) b)

c) d)

3. Hint: 022)35(10)35( 2 Expand and see if the LHS equals zero

x

y

)4

1,2

3(

)11,0(

x

y

)26,4(

)10,0(

x

y

)2,5(

)23,0(

x

y

)4,3(

)13,0(

22

WS QUADRATICS 3 ~ Quadratic formula and discriminant

Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions

Exercise A

For a quadratics equation in the form: = 2 + +

The quadratic formula: a

acbbx

2

42 can also be written

a

bx

acb

2

42

Example 01182 xx = 6 24, 1 6

Use the quadratic formulae to find the roots of the following equations (where possible).

1. (a) 01492 xx (b) 02452 xx

(c) 063162 xx (d) 025102 xx

(e) 01362 xx (f) 01072 xx

(g) 0362 2 xx (h) 0543 2 xx

2. Sketch the graphs for questions (b), (d), (e), (g)

Compare them with those of the person sitting next to you.

Exercise B

The discriminant: acb 42

Calculate the discriminant for each equation and state whether there are two real distinct roots,

one real root or no real roots.

1. 0322 xx 2. 0322 xx

3. 0962 xx 4. 0685 2 xx

5. 0342 2 xx 6. 09124 2 xx

Exercise C

Solve the following by first sketching the graph (it will help to factorise these)

1. 2 + 6 + 5 0 2. 2 + 2 − 8 0

3. 2 2 + 11 + 5 0 4. 2 2 + 15 − 1 0

5. − 2 − + 2 0 6. −2 2 + 13 − 11 0

23

Exam questions

1. [C1 May 2006 Q2]

Find the set of values of x for which x2 – 7x – 18 > 0.

(4) 2. [C1 Jan 2005 Q3]

Given that the equation kx2 + 12x + k = 0, where k is a positive constant, has equal roots, find

the value of k.

(4)

3. [C1 Jan 2007 Q5]

The equation 2x2 – 3x – (k + 1) = 0, where k is a constant, has no real roots.

Find the set of possible values of k.

(4)

4. [C1 May 2007 Q7]

The equation x2 + kx + (k + 3) = 0, where k is a constant, has different real roots.

(a) Show that 01242 kk . (2)

(b) Find the set of possible values of k. (4)

5. [C1 Solomon B Q10] Figure 1

Figure 1 shows the curve = 2 – 3 + 5 and the straight line = 2 + 1.

The curve and the line intersect at the points P and Q.

(a) Using algebra, show that P has coordinates (1, 3) and find the coordinates of Q. (4)

(b) Find an equation for the tangent to the curve at P. (4)

(c) Show that the tangent to the curve at Q has the equation = 5 − 11 (2)

(d) Find the coordinates of the point where the tangent to the curve at P intersects

the tangent to the curve at Q. (3)

Exercise D

1. Write an equation that has no real roots then sketch the graph to show the vertex.

2. Write out the proof of the quadratic formulae.

24

Answers

EXA

1. (a) -2, -7 (b) 8, -3 (c) 9, 7 (d) -5, -5 (e) no real roots

(f) 1.22, -8.22 (g) 2.37, 0.63 (h) 0.79, -2.12

2.

(b)

(d)

(e)

(g)

Ex B

1. two real distinct roots 2. no real roots 3. one real root 4. no real roots

5. two real distinct roots 6. one real root

Ex C

1. −5 1 2. −4 2 3. −5 −1

2 4. −1

17

2

5. −2 1 6. 1 11

2

Exam questions

1. −2 or 9 2. = 6 3. −1

8 4. (b) – 2 or 6

5. (a) 4, 9 (b) = − + 4 (d) (15

6,3

2)

25

WS 9 TANGENTS AND NORMALS

Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions

Exercise A (Finding equations for the tangent and normal)

For each question follow these steps:

(a) Calculate the y coordinate (if necessary).

(b) Differentiate the function.

(c) Calculate the gradient at the point given.

(d) Write down the following , = =

(e) Find the equation of the tangent using − 1 = − 1 .

(f) Find the equation of the normal using − 1 = −1

− 1 as it has a perpendicular

gradient.

1. = 2 − + 12 5, 2

2. = − 2 + 4 + 5 = 3

3. = 3 − 2 − 6 = −1

4. =4

+ 3 − 5 2, 3

Exercise B (Sketching graphs)

For each question:

(a) Sketch the given curve.

(b) Calculate the tangent at the given point.

(c) Add this to your sketch (Remember sketches should include any points of axes intersection)

1. = 2 at 3, 9

2. = 2 2 − 11 + 5 at 2, −9

3. =1

when = 3

4. = 3 − 3 2 − 10 when = −3

5. = when = 4

Exercise C (Using completing the square)

Through completing the square, find the vertex of the following quadratics.

1. = 2 − 6 + 11 then calculate 3

2. = 2 + 4 − 1 then calculate −2 , what do you notice?

3. = 2 − 3 +13

4 Can you think of another way you could find this minimum point?

26

Exercise D (Working backwards to find coordinates)

Find the coordinates at which the following functions have their given gradient.

1. = 2 2 − 3 + 2

= −4

2. = −3 2 + − 5

= 13

3. = 3 − 2 2 + 5 − 1

= 9

4. = 2 3 +13

2 2 − + 2

= −2

5. =

=

1

3

Exercise E (Tangents, normals and simultaneous equations)

1. = 2 − 1

a) Sketch the quadratic function.

b) Find the equation of the normal to the quadratic when = 1

c) Find the coordinates where the normal to the quadratic intersects the curve again.

d) Add the normal line to your sketch, indicating the point of intersection.

2. = 3 + 3 2 − 4

a) Sketch the cubic function including all intersections with the coordinate axes.

b) Add on to your sketch the tangent to the curve at the origin.

c) Find

d) Find the tangent to the curve at the origin.

e) Find the other point that the tangent intersects the curve again.

3. = +3

a) Show that the point 1, 4 lies on the curve

b) Find

c) Show that the gradient of the tangent to the curve at is is −2

d) Find the equation of the normal to the curve at .

e) Find the point where the normal at intersects the curve again

Answers

EXA

tangent normal

1. = 3 − 13 = −1

3 +

11

3

2. = −2 + 14 − 2 + 13 = 0

3. = − + 3 = + 5

4. = 2 − 1 = −1

2 + 4

EXB

1. = 6 − 9 2. = −3 − 3 3. + 9 − 6 = 0

4. = 35 + 81 5. − 4 + 4 = 0

EXC

1. 3, 2 2. −2,−5 3. (−3

2, −1)

EXD

1. (−1

4,11

8) 2. −2,−19 3. (−

2

3, −

149

27) and 2,5 4. (

1

3,25

54) and (−

5

2,231

8)

5. (9

4,3

2)

EXE

1. Normal: + 2 − 1 = 0

Point of

intersection:(−3

2,5

4)

2. Tangent: = −4

Point of intersection: −3, 12 3. Normal: − 2 + = 0

Point of intersection:(6,13

2)

28

WS SEQUENCES 1 ~ General sequences and summation notation

Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,

decreasing, recurrence relation

Exercise A

1. Write down the first five terms in each sequence for = 1, 2, 3,

a) = 2n − 1 b) = 3n + 1 c) = n2

d) = −3 e) = 20 − 5n f) = 2n + 2

g) = 5 + −1 h) = 1

2 i) =

1

−

1

1

2. Write down the first five terms in each sequence:

a) 1 = + 3, 1 = 2 b) 1 = 3 , 1 = 4

c) 1 = 5 − 2, 1 = 3 d) 1 = 2 − 1, 1 = 2

e) 1 =1

, 1 = 5 f) 1 = + 1 , 1 = 1

g) 1 = − , 1 = 5 h) 2 = 1 + , 1 = 1, 2 = 1

Exercise B – exam style questions

1. A sequence of positive numbers is defined by

1 = √ 2 + 3 , 1 = 2

a) Find 2 and 3 in surd form.

b) Show that 5 = 4

2. A sequence 1, 2, 3 is defined by 1 = , 1 = 4 − where is a constant

a) Write down an expression for 2 in terms of .

b) Find 3 in terms of k, simplifying your answer.

Given that 3 = 13

c) Find the value of k.

3. A sequence is defined by 1 = 3 − 5, 1 =

a) Find 2 and 3 in terms of

Given that ∑ 3 1 = 92

b) Find .

29

4. A sequence is defined by 1 = + 2, 1 = 2

a) Find 2 and 3

Given that 3 = 6

b) Find the possible values of

5. A sequence is defined by 1 = − 3, 1 = 1, where is a constant

a) Find an expression for 2 in terms of a.

b) Show that 3 = 2 − 3 − 3

Given that 3 =

c) Find the possible values of .

6. A sequence 1, 2, 3 is defined by 1 = 1, 1 = + 5 where is a constant.

a) Write down an expression for 2 and 3 in terms of

b) Given that 3 = 41, find the possible values of

7. The sequence of positive numbers 1, 2, 3 is given by 1 = − 3 2, 1 = 1

a) Find 2, 3 and 4

b) Write down the value of 500.

8. A sequence is given by 1 = + , 1 = 1 where is a constant and 0

a) Show that 3 = 1 + 3 + 2 2

Given that 3 = 1,

b) Find the value of .

c) Write down the value of 2014.

Exercise C

1. UKMT Maths Challenge Question:

A sequence 1, 2, 3 is defined for positive integer values of by

3 = 2 + 1 −

Where 1 = 0, 2 = 2, and 3 = 1.

What is the sum of the first 100 terms of the sequence?

2. Calculate the first 10 terms of the following sequence 1 = 1, 2 = 1 and = −1 + −2. What is the name of this sequence and why is it so famous?

3. Research the Mandelbrot Set - this is all done using sequences.

30

http://www.youtube.com/watch?v=G_GBwuYuOOs

4. Calculate the first few terms of the following infinity series, where n is an integer.

a)

b)

c)

What do they value to they tend towards? Why?

Answers

1.a) -1, 1, 3, 5, 7 b) 4,7,10,13,16 c) 1, 4, 9, 16, 25 d) -3, 9, -27, 81, -243 e) 15, 10, 5, 0, -5

f) 3, 8, 14, 24, 42 g) 4, 6, 4, 6, 4 h) 1, 3, 6, 10, 15 i)1/2, 1/6,

1/12, 1/20,

1/30

2a) 2, 5, 8, 11, 14 b) 4, 12, 36, 108, 324 c) 3, 13, 63, 313, 1563 d) 2, 3, 8, 63, 3968

e) 5, 1/5, 5, 1/5, 5 f) 1, 2, 6, 42, 1806 g) 5, -5, 5, -5, 5 h) 1, 1, 2, 3, 5

3a) , 10 4

4a) 2 = 4 − b) 3 = 16 − 35 c) = 3

5a) 2 = 3 − 5 3 = 9 − 20 b) = 9

6a) 2 = 2 + 2 3 = 2 2 + 2 + 2 b) = 1,−2

7a) 2 = − 3 c) = 5,−2

8a) 2 = + 5 3 = 2 + 5 + 5 b) = 4,−9

9a) 2 = 4, 3 = 1, 4 = 4 b) 500 = 4

10b) = −3

2 c) 2014 = −

1

2

31

WS SEQUENCES ~ Arithmetic sequences

Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,

decreasing, recurrence relation

= + − 1 =1

2 + =

1

2 2 + − 1

Exercise A

Write an expression, in terms of and , for each of these statements:

1. 15th term is 104 2. 27th term is -5

3. 3rd term is 78 4. 109th term is 10

5. 11th term is 20 6. Sum of the first 6 terms is 34

7 Sum of the first 40 terms is 109 8. Sum of the first 17 terms is 80

Exercise B

Use either of the formulae at the top to calculate the missing quantity.

Note you can use = + − 1 directly 8 = +

1. = 3, = 5 Write out 1, 2, 3, 4 then calculate 4

2. = , = −2 Write out 1, 2, 3, 4, 5, 6 then calculate 6

Use and appropriate formulae to calculate the missing quantity.

3. = 3 4. = 5. = = 5 = 5 = 3 = 10 = 12 = 10 = 12 = 4 = 31

6. = 4 7. = 8. = 8 = = 3 = = 12 = 8 = 10 12 = 8 = 164 10 = −55

9. = 10. + 4 = 24 11. 4 = 15 = 3 + 9 = 38 11 = 1 = = = 140 =

32

Exercise C – Exam Questions

1. [C1 Jan 2012 Q9]

A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive.

Scheme 1: Salary in Year 1 is £P.

Salary increases by £(2T ) each year, forming an arithmetic sequence.

Scheme 2: Salary in Year 1 is £(P + 1800).

Salary increases by £T each year, forming an arithmetic sequence.

(a) Show that the total earned under Salary Scheme 1 for the 10-year period is

£(10P + 90T ).

(2) For the 10-year period, the total earned is the same for both salary schemes.

(b) Find the value of T.

(4) For this value of T, the salary in Year 10 under Salary Scheme 2 is £29 850.

(c) Find the value of P.

(3)

2. [C1 Jun 2013 Q7]

A company, which is making 200 mobile phones each week, plans to increase its production.

The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to

220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week N.

(a) Find the value of N.

(2) The company then plans to continue to make 600 mobile phones each week.

(b) Find the total number of mobile phones that will be made in the first 52 weeks starting from

and including week 1.

(5)

Answers

Exercise A 1. a + 14d = 104 2. a + 26d = -5 3. a + 2d = 78 4.a + 108d = 10 5.a + 10d = 20 6. (1/2)(6) (2a + 5d) = 34 7. (1/2)(40) (2a + 39d) = 109

8. (1/2)(17) (2a + 16d) = 80

Exercise B 1) 3, 8, 13, 18 42 2) 7, 5, 3, 1, -1, -3 12 3) 48 4) -8 5) 9 6) 510 7) 10 8) -3

9) 8 10) 11, 3 11) 21, -2

Exercise C 1. (b) T = 400 (c) P = £24 450 2. (a) N = 21 (b) 27 000

33

WS INTEGRATION

Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing,

decreasing, recurrence relation

=

⇒ =

1

+ 1 1 + , −1

→

←

34

Exercise A

Write the following in a form that they can be integrated.

1. = 2 =1

3 3. =

5

2

4. =1

5. =

4

6. =

1

2

7. = 3 + 2 2 8. = 2 − 1 3 9. = + 1

10. = − 2 − 3 11. = 2 1 +2

*12. =

23 2 3

Exercise B

Now integrate the expressions in Exercise A (don’t forget the constant of integration!).

Exercise C (Finding the constant of integration)

1. A function passes through 1, 8 and has gradient function

= 5 , find the equation of the curve.

2. A function passes through 1, 9 and has gradient function

= 6 2 + 5 , find the equation of the

curve.

3. A function passes through 2, 1 and has gradient function

=

4

2 , find the equation of the curve.

4. Solve the differential

= ( +

1

)2

given the point 1, 3 .

Exercise D - Exam Questions

1. [C1 May 2012 Q1]

Find

x

xx d5

26

2

2, giving each term in its simplest form.

(4) 2. [C1 Jan 2012 Q7]

A curve with equation y = f(x) passes through the point (2, 10). Given that

f ′(x) = 3x2 − 3x + 5,

find the value of f(1).

(5)

3. [C1 May 2011 Q6]

Given that x

xx

2

5

36 can be written in the form 6x p + 3xq,

(a) write down the value of p and the value of q.

(2)

Given that x

y

d

d =

x

xx

2

5

36 and that y = 90 when x = 4,

(b) find y in terms of x, simplifying the coefficient of each term.

(5)

35

Answers

EXA

1. =

2 2. =1

3 −1 3. = 5 −2 4. = −

2 5. = 4 −1 6. =1

2 −

2 7. = 9 2 +

12 + 4

8. = 8 3 − 12 2 + 6 − 1 9. = 3

2 +

2 10. = − 3 + 3 2 11. = 2 + 4

2 12.

= 2

2 + 3

2

EXB

1. 2

3

3

2 + 2. 3.−5 −1 + 4. 2

2 + 5. 6.

2

7. 3 3 + 6 2 + 4 + 8. 2 4 − 4 3 + 3 2 − + 9. 2

5

2 +2

3

3

2 + 10. −

4+ 3 +

11. 2 +8

3

3

2 +

12. 24

29

2

2 +36

17

2 +

EXC

1. = 5 + 3 2. = 2 3 + 5 + 2 3. = 3 −4

4. =

1

3 3 + 2 −

1

+

5

3

EXD

1. 2 3 − 2 −1 + 5 + 2. 1 =5

2 3. (a) =

1

2, = 2 (b) = 4

3

2 + 3 − 6

36

CORE 3 FORMULA SHEET

Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and

C2.

Logarithms and exponentials

xax alne

Trigonometric identities

BABABA sincoscossin)(sin

BABABA sinsincoscos)(cos

))(( tantan1

tantan)(tan

21

kBA

BA

BABA

2cos

2sin2sinsin

BABABA

2sin

2cos2sinsin

BABABA

2cos

2cos2coscos

BABABA

2sin

2sin2coscos

BABABA

Differentiation

f(x) f (x)

tan kx k sec2 kx

sec x sec x tan x

cot x –cosec2 x

cosec x –cosec x cot x

IMPORTANT INFORMATION

Maths and Computer Science Teachers: room email

Ceinwen Hilton 232 ceinwen.hilton@candi.ac.uk

Elliot Henchy 232 elliot.henchy@candi.ac.uk

Flo Oakley 232 flo.oakley@candi.ac.uk

Greg Jefferys 218 greg.jefferys@candi.ac.uk

Dan Nelson 214 daniel.nelson@candi.ac.uk

Najm Anwar 214 najm.anwar@candi.ac.uk

Nadya de Villiers 214 nadya.devilliers@candi.ac.uk

Vijay Goswami 214 vijay.goswami@candi.ac.uk

Website

Please take some time to visit our website: www.candimaths.uk

Homework

Work outside lessons should take 4-5 hours. You will be set homework on all the main topics.

Complete the set work thoughtfully; it is for your benefit. Remember to check and mark your

answers, write any comments or questions to the teacher on your work and submit it on time.

You should also review notes, revise for future tests and plan ahead as part of your homework.

Support – to help you succeed The department runs several support workshops at lunchtimes and after college where you can get extra help. This is

also an opportunity for you to get to know other teachers and students.

Expectations

Students take increasing responsibility for their learning at the Sixth Form. Do join in the classes,

volunteer answers and ask questions. Spend time at home organising your equipment, notes and

learning. Learning demands, courage, determination and resourcefulness. Use other text books,

YouTube, websites, work with other students and talk with teachers.

Other Links www.examsolutions.net Most popular site with past exam papers and video solutions.

Also clear explanations of topics

www.physicsandmathstutor.com Exam revision site

www.numberphile.com Short video clips of popular maths

www.nrichmaths.org Problem solving challenges

www.geogebra.org Geometry, graphs and animations

www.mathscareers.org.uk/ Careers linked to mathematics

www.supermathsworld.com Multiple choice practice with cartoons