Preparation work for students intending to study a level

King Edward VI School Mathematics Department Preparation for students intending to study A level Mathematics.

Having chosen to study Mathematics you will need to make sure that you are confident with the core algebraic skills required. The A level course is much more rigorous than IGCSE and contains a lot of algebra; for this reason it is possible to gain an A* at IGCSE and still find the A level course challenging.

The following sheets form part of your preparation for a successful start to the A level course.

It is vital that you spend some time working through these questions prior to the start of the course in September and bring the work with you to your first Mathematics lesson.

There will be a test on this material during the second week of term. If you do not pass this test you will be expected to complete additional practice in order to bring your basic algebra skills up to the required standard. You will then be re-tested later in the term.

In the first Core Mathematics module calculators are not allowed and you should therefore attempt all of these questions without the use of a calculator.

Contents Resource sheet 1 – Basic algebra Resource sheet 2 - Equations Resource sheet 3 – Quadratic equations Resource sheet 4 – Simultaneous equations Resource sheet 5 - Inequalities Resource sheet 6 - Indices Practice Test Answers

Resource sheet 1

Basic algebra

1. Simplify the following expressions.

i. iii. v.

6 x  2 y  3 x  3y 8 x  2 y  z 3 y  2z

2 x  3 xy  2 y  4 yx

ii. 5 x  3 y  2 x  4 y iv. 4 x 2  3 y 2  2 x  2 y2 vi. 4 x 2 y  3 xy 2 1  2 yx2

2. Expand, and then simplify as far as possible, each of the following expressions. i.

3  x  2   5  x 1

ii. 4 5a  3  3  2a  5 

iii.

t  2 p 1  p  3t  2 

iv. 2  p  q   3q  p  3  p 3q 1

3. Factorise the following expressions. i. iii. v.

2x6y4 9a 2  3abc 5u 15uv 10u2

ii. 12a 3b  6c iv. 12 y 2  4 xy 8x vi. 6 xy  2 xy 2  4x 2 y2

4. In each of the following questions, multiply the terms and then simplify the answer. i.

3x  4xy

ii. 5a 2b  2ab2

iii.

4 pq  2 p2 q  3q3

iv. 3 x 3 z  2zy2  2 xz

5. Simplify each of the following as far as possible. i.

x 3 y2 xy

ii.

6a 3b4 3b

iii.

4 x  9 y2

2 3 iv. 10 p  3q  4q

3q 2

3 y 2x2

2p2

5p

6. Express each of the following as a single fraction. i. iii. v.

x x 3 4 23 p q x2 x 3 4

ii. iv. vi.

3 x  2x 5 3 3x x 2 3 x 1  2 x 1 3 2

Resource sheet 2 1.

i. iii. v. vii. ix.

Equations

Solve the following equations.

3 x 5 13 6 z 3  2 z 19 1 x  5  2x 3 5(đ?‘Ľ + 3) = 3(3đ?‘Ľ − 1) 3đ?‘?−1 2

+1=

đ?‘?

ii. 7 y  40  5 iv. 29 3a 17 9a vi. q  3  q  3

4 2đ?‘Ľ viii. 3(5 − 2đ?‘Ľ) = 3

x.

3

2đ?‘žâˆ’5 3

= 4(3đ?‘ž + 1)

2. For each of the following, make the indicated letter the subject of the formula. i. ii. V  ď °r 2 h y  3x  c [ x] [ h] (đ?‘˘+đ?‘Ł)đ?‘Ą

iii.

đ?‘ =

v.

V  ď °r h

2

2

[t]

E

đ?‘Ž2 2

+ đ?‘?2

3V T T 2

[ r]

vi.

[ E]

viii. K  H  2

4E3

vii. D 

iv. đ?‘? 2 =

H3

[a] [V ]

[H]

Resource sheet 3 Quadratic equations 1. Expand and then simplify each of the following expressions. i. iv.

 x  4  x  2   2x 1  x  3

ii. v.

 x  4  x  6   4x 1 3x  2

iii. vi.

 x  5  x  5 3  4x 3x 1

iii. vi.

x 2  8x 12 3x 2 14x  5

2. Factorise the following quadratic expressions. i. iv.

x 2  2x  3 x 2  x  20

ii. v.

x 2  5x  4 2x 2  5x  3

3. Use the method of factorising to solve each of the following quadratic equations. i. iv.

x 2  3x 10  0 x 2 11x 12  0

ii. v.

x 2 10x  21  0 2x 2  7x  5  0

iii. vi.

x 2  x 12  0 4x 2  8x  3  0

4. Use the quadratic formula to solve each of the following quadratic equations. Leave square roots in your answers where appropriate. i. iii.

x 2  5x  5  0 4x 2  2x 1  0

ii. x2  3x  2  0 iv.

3x 2  x  4

Resource sheet 4 Simultaneous Equations 1.

Solve each of the following pairs of simultaneous equations using elimination.

i.

3 x  4 y 10 2x5y9

ii.

4 x 3 y 10 5 x  2 y 1

iii.

9x7y6 5x4y3

iv.

6x3y9 2 x  2 y 1

2.

Solve each of the following pairs of simultaneous equations using substitution.

i.

3 x  2 y 16 y  4 x 3

ii.

2 x 3 y  21  0 3xy4

3.

Solve the following one linear, one quadratic simultaneous equations.

i.

đ?‘Ś = đ?‘Ľ2 đ?‘Ś =đ?‘Ľ+2

ii.

đ?‘Ľ 2 + đ?‘Ś 2 = 29 đ?‘Ľ + 2đ?‘Ś = 12

Resource sheet 5

Inequalities

1.

Solve each of the following inequalities.

i.

3x  4  20

ii. 2  5x  27

7x 15

v. 4  3 1 2x   4  x 1

iv.

3

 4x



iii. 4  2x 1  18  x

Resource sheet 6

Indices

1. Simplify the following: i. ii. iii. iv.

đ?‘? Ă— đ?‘?5 3đ?‘? 2 Ă— 2đ?‘? 5 đ?‘?đ?‘? 2 Ă— đ?‘?đ?‘? 3 2đ?‘›6 Ă— (−6đ?‘›)2

v. 8�8 á 2�3 vi. � 9 á �11 vii. (�3 )2

2. Find the value of: i.

41/2

2 vi. ( )−2

ii.

271/3

vii. 8−2/3

iii.

( )1/2

viii. (0.04)1/2

iv.

5−2

ix. ( )2/3

v.

180

x. ( )

1 9

3

8

27 1 −3/2

16

Practice Test (Time allowed 30 minutes) You may not use a calculator.

If đ?‘Žđ?‘Ľ 2 + đ?‘?đ?‘Ľ + đ?‘? = 0 đ?‘Ąâ„Žđ?‘’đ?‘› đ?‘Ľ =

−đ?‘?Âąâˆšđ?‘?2 −4đ?‘Žđ?‘?

2đ?‘Ž

1. Expand and simplify

(a) (2đ?‘Ľ + 3)(2đ?‘Ľ − 1)

(b) (đ?‘Ž + 3)2

(c) 4đ?‘Ľ(3đ?‘Ľ − 2) − đ?‘Ľ(2đ?‘Ľ + 5)

(b) đ?‘Ś 2 − 64

(c) 2đ?‘Ľ 2 + 5đ?‘Ľ − 3

2. Factorise

(a) đ?‘Ľ 2 − 7đ?‘Ľ (d) 6đ?‘Ą 2 − 13đ?‘Ą + 5

3. Simplify 4đ?‘Ľ 3 đ?‘Ś

(a) 8đ?‘Ľ 2 đ?‘Ś 3

(b)

3đ?‘Ľ+2 3

+

4đ?‘Ľâˆ’1 6

4. Solve the following equations

(a)

ℎ−1 4

(d) đ?‘Ľ −

+

3â„Ž

5 đ?‘Ľâˆ’2 3

=4

(b) đ?‘Ľ 2 − 8đ?‘Ľ = 0

(c) đ?‘?2 + 4đ?‘? = 12

= 6đ?‘Ľ − 10

5. Simplify the following expression

2đ?‘Ľ + 1 đ?‘Ľ − 5 − 3 2

6. Write each of the following as a single power of each letter 1

(a) đ?‘Ľ 4

(b) (đ?‘Ľ 2 đ?‘Ś)3

(d) (đ?‘Ľ −2 đ?‘Ś 4 )−1

đ?‘Ľ5

(c) đ?‘Ľ −2

7. Work out the values of the following giving your answers as fractions.

(a) 4−2

(b) 100

2

(d) (3)−3

8. Solve the simultaneous equations.

(a) 3đ?‘Ľ − 5đ?‘Ś = −11 (a) 5đ?‘Ľ − 2đ?‘Ś = 7 (b)

8

(c) (27)1/3

(b) đ?‘Ľ 2 + đ?‘Ś 2 = 26 đ?‘Ľâˆ’đ?‘Ś =4

9. Rearrange the following equations to make đ?‘Ľ the subject.

(a) � 2 = �2 + 2��

1

(b) đ?‘‰ = 3 đ?œ‹đ?‘Ľ 2 â„Ž

đ?‘Ľ+2

(c) đ?‘Ś = đ?‘Ľ+1

10. Solve 5đ?‘Ľ 2 − đ?‘Ľ − 1 = 0 giving your solutions in surd form.

Now mark your work.

90% - very good.

80% - go over the questions you answered incorrectly and try again.

70% - your algebra skills need some work; go through the questions in the pack again.

Any score less than 70% indicates that your algebra skills are weak and you may struggle with A level Mathematics. The following book may help: Head Start to AS-Level Maths Published by CGP Workbooks ISBN: 978 1 84146 993 5

Answers Resource sheet 1 – Basic algebra 1. i. 9𝑥 + 5𝑦

ii. 3𝑥 + 7𝑦

iii. 8𝑥 − 𝑦 − 𝑧

v. 2𝑥 − 2𝑦 + 7𝑥𝑦

vi. 6𝑥 2 𝑦 + 3𝑥𝑦 2 + 1

2. i. 8𝑥 + 1 iv. 𝑝 − 7𝑞

ii. 14𝑎 + 27

iii. 5𝑝𝑡 + 𝑡 − 2𝑝

3. i. 2(𝑥 + 3𝑦 + 2) iv. 4(3𝑦 2 + 𝑥𝑦 + 2𝑥)

ii. 3(4𝑎 − 𝑏 + 2𝑐) v. 5𝑢(1 + 3𝑣 + 2𝑢)

iii. 3𝑎(3𝑎 + 𝑏𝑐) vi. 2𝑥𝑦(3 + 𝑦 − 2𝑥𝑦)

4. i. 12𝑥 2 𝑦 iv. 12𝑥 4 𝑦 2 𝑧 3

ii. 10𝑎3 𝑏3

iii. 24𝑝3 𝑞 5

5. i. 𝑥 2 𝑦

ii. 2𝑎3 𝑏 3

iii.

iv. 4𝑥 2 + 2𝑥 + 𝑦 2

iv.

6. i.

4𝑞 3

𝑥

𝑝2

7𝑥

ii.

12 𝑥 2 +6

iv.

6𝑦

v.

2𝑥

−𝑥

iii.

15 7𝑥+8

vi.

12

3𝑝+2𝑞 𝑝𝑞 12𝑥+1 6

Resource sheet 2 – Equations 1. i. 𝑥 = 6 iv. 𝑎 = −2 1 vii. 𝑥 = 4 2 1

x. 𝑞 = − 2

ii. 𝑦 = −5 v. 𝑥 = −3 1 viii. 𝑥 = 2 4

iii. 𝑧 = 4 vi. 𝑞 = 8 3 ix. 𝑝 = − 7

2. i. 𝑥 =

𝑦−𝑐

𝑉

2𝑠

ii. ℎ = 𝜋𝑟 2

3

iii. 𝑡 = 𝑢+𝑣

𝑉

iv. 𝑎 = √2(𝑐 2 − 𝑏 2

v. 𝑟 = √𝜋ℎ

3

vii. 𝐸 = 𝐷−4

viii. 𝐻 =

vi. 𝑉 =

𝑇 3

3−2𝐾 𝐾−1

Resource sheet 3 – Quadratic equations 1. i. 𝑥 2 + 6𝑥 + 8 iv. 2𝑥 2 + 7𝑥 + 3

ii. 𝑥 2 − 10𝑥 + 24 v. 12𝑥 2 + 5𝑥 − 2

iii. 𝑥 2 − 25 vi. −12𝑥 2 + 13𝑥 − 3

2. i. (𝑥 − 1)(𝑥 + 3) iv. (𝑥 − 4)(𝑥 + 5)

ii. (𝑥 − 1)(𝑥 − 4) v. (2𝑥 + 3)(𝑥 + 1)

iii. (𝑥 + 2)(𝑥 + 6) vi. (3𝑥 + 1)(𝑥 − 5)

3. i. 𝑥 = 2, 𝑥 = −5 iv. 𝑥 = −1, 𝑥 = 12

ii. 𝑥 = 7, 𝑥 = 3 5 v. 𝑥 = − 2 , 𝑥 = −1

iii. 𝑥 = 3, 𝑥 = −4 1 3 vi. 𝑥 = − 2 , 𝑥 = − 2

4. i. 𝑥 =

−5±√5

iii. 𝑥 =

2 −2±√20 8

ii. 𝑥 = or 𝑥 =

−1±√5 4

iv. 𝑥 =

3±√17 2 1±√49 6

4

or 𝑥 = 3 , 𝑥 = −1

Resource sheet 4 – Simultaneous equations 1. i. 𝑥 = 2, 𝑦 = 1 7 iv. 𝑥 = 2 , 𝑦 = −4

ii. 𝑥 = 1, 𝑦 = −2

2. i. 𝑥 = 2, 𝑦 = 5

ii. 𝑥 = 3, 𝑦 = −5

iii. 𝑥 = 3, 𝑦 = 3

3. i. 𝑥 2 = 𝑥 + 2; 𝑥 = 2, 𝑦 = 4 𝑜𝑟 𝑥 = −1, 𝑦 = 1 ii. (12 − 2𝑦)2 + 𝑦 2 = 29; 𝑦 = 5, 𝑥 = 2 𝑜𝑟 𝑦 =

23 5

,𝑥 =

14 5

Resource sheet 5 – Inequalities 1. i. 𝑥 < 8 iv. 𝑥 < −3

ii. 𝑥 > −5 1 v. 𝑥 ≤ 2

iii. 𝑥 ≥ 2

Resource sheet 6 – Indices 1. i. 𝑏 6 ii. 6𝑐 7 iii. 𝑏 2 𝑐 5 iv. 72𝑛8

v. 4𝑛5 vi. 𝑑 −2 vii. 𝑎6

2. i. √4 = 2

vi. 4

9

1

3

ii. √27 = 3

vii. 4

1

iii. 3

viii. 0.2

iv. 25

ix.

1

4 9

x. 64

v. 1

Practice Test 1. (a) 4𝑥 2 + 4𝑥 − 3 (b) 𝑎2 + 6𝑎 + 9 (c) 12𝑥 2 − 8𝑥 − 2𝑥 2 − 5𝑥 10𝑥 2 − 13𝑥

[1] [1] [1] [1]

2. (a) 𝑥(𝑥 − 7) (b) (𝑦 − 8)(𝑦 + 8) (c) (2𝑥 − 1)(𝑥 + 3) (d) (3𝑡 − 5)(2𝑡 − 1)

[1] [1] [1] [1]

𝑥

3. (a) 2𝑦 2 (b)

6𝑥+4

[1] +

6 10𝑥+3

4𝑥−1 6

6

4. (a) 5(ℎ − 1) + 12ℎ = 80 17ℎ − 5 = 80 ℎ=5

[1] [1]

[1] [1] [1]

(b) 𝑥(𝑥 − 8) = 0 𝑥 = 0, 𝑥 = 8

[1] [1]

(c) (𝑝 − 2)(𝑝 + 6) = 0 𝑝 = 2, 𝑝 = −6

[1] [1]

(d) 3𝑥 − 𝑥 + 2 = 18𝑥 − 10 16𝑥 = 32 𝑥=2

5.

4𝑥+2 6

−

3𝑥−15

𝑥+17

6

1

7. (a) 16 2

(c) 3

[1] [1]

6

6. (a) 𝑥 −4 (c) 𝑥 7

[1] [1] [1]

[1] [1]

(b) 𝑥 6 𝑦 3 (d) 𝑥 2 𝑦 −4

[1] [1]

[1]

(b) 1

[1]

[1]

(d)

27 8

8. (a) Matching coefficients of 𝑥 𝑜𝑟 𝑦 Showing a substitution to calculate 2nd value 𝑥 = 3, 𝑦 = 4 (b) 𝑦 = 𝑥 − 4 [1] 2 2 𝑥 + (𝑥 − 4) = 26 [1] 2𝑥 2 − 8𝑥 + 16 = 26 [1] (𝑥 − 5)(𝑥 + 1) = 0 𝑜𝑟 (𝑥 − 5)(2𝑥 + 2) = 0 (or correct use of quadratic formula) 𝑥 = 5, 𝑦 = 1 𝑜𝑟 𝑥 = −1, 𝑦 = −5

9. (a) 𝑥 =

𝑣 2 −𝑢2 2𝑎 3𝑉

(b) 𝑥 2 = 𝜋ℎ

3𝑉

𝑥 = √𝜋ℎ (c) 𝑦(𝑥 + 1) = 𝑥 + 2 𝑦𝑥 + 𝑦 = 𝑥 + 2 𝑦𝑥 − 𝑥 = 2 − 𝑦 𝑥(𝑦 − 1) = 2 − 𝑦 2−𝑦 𝑥= 𝑦−1

10.

𝑥=

1±√21 10

[1]

[1] [1] [2]

[1] [2]

[1] [1]

[1] [1] [1] [1] [1] [1]

[1]

TOTAL MARKS = 50