# Chancellor�s School

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```					Barnwell School

Department of Mathematics

GCSE – A/S
Bridging Pack

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To all prospective A/S Mathematics students:

You will appreciate that Mathematics is a subject which relies to a significant
extent on two elements:

I. Natural flair and aptitude;
II. Consolidation and practice of the topics within each course.

As you move from GCSE to A/S level, and hopefully beyond, it is absolutely
essential that your progress is not unduly hampered by weaknesses in basic
algebraic techniques for example.
In this document is a pack of six topics which you are required to work through
during the summer break. It represents some of the more challenging GCSE
topics, which are termed ‘assumed knowledge’ by the A/S examination
board. You will be tested on this work by the end of September. Students
who fail to reach a minimum standard will be required to do additional work
and their potential to continue the course will be reviewed. Parents will be
contacted.

LIST OF TOPICS

1.   Factorising
2.   Formulae
3.   Linear and Quadratic Equations
4.   Simultaneous Equations
5.   Simplifying, including indices
6.   The Sine Rule and Cosine Rule

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TOPIC 1 : Factorising
Examples:

1.   2 x 3  6 x 2  2 x 2 ( x  3)
2.   9a 2  4b 2  (3a  2b)(3a  2b)
3.   x 2  7 x  12  ( x  4)( x  3)
4.   6 x 2  11 x  4  (2 x  1)(3x  4)

Exercise:

Factorise completely:

1. n 2  np                                               6. 3x 2  75

2. h 2  25                                               7. 5h 2  8h  4

3. m 2  7m  10                                          8. 10 x 2  9 x  2

4. n 2  n  12                                           9. ab  5a  2b  10

5. 15  2b  b 2                                          10. ( x  3)( x  5)  ( x  3) 2

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TOPIC 2 : Formulae

A.        Re-arranging formulae

Examples:

In each case, re-arrange to find the letter in the bracket.

1.                                                        2.
y
x
7            x                                  y  4( x  3)       x 
2                                                     y
2 y  x  14                                                  x3
4
x  2 y  14                                                   y
x  3
4

3.                                                        4.

a 2  b 2  7ac             b                            3
 6  4n           m 
b 2  a 2  7ac                                            m
3  (6  4 n ) m
b   a 2  7ac
3
m
6  4n

Exercise:

Re-arrange each formula to make the letter in the bracket the subject.

1. y  3 x                              x 
2. 4 y  2 x  7                        x 
3. 3( y  2)  6  3( x  7)            x 
4. ab  cd  4e                         c 

5. 3a( x  y )  2b 2                   a 
6.
3y

x
x 
x     4z

7. bx  cy  d 2                        c 
8. k (l  m)  l (m  n)                l 

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B.       Substitution in formulae

Examples:

Find the value of x if a = 0.7, b = -3.5 and c = -2.15. Give answers to 3 sig.
figs.

1.

x  ab  c 2
x  0.7  (3.5)  (2.15) 2
x  7.07

2.

x  a(b 2  c)

x  0.7  3.5  2.15
2

x  2.66

Exercise:

Use the above values to find x in each case, correct to 3 sig. Figs.:

1. x  4bc  a 3

ac  b 2
2. x 
c
c
3. x  b 2 
a

4. 7 x  2a  3b  4c

5. ax  bx  c

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TOPIC 3 : Linear and Quadratic Equations
A.        Linear Equations

Examples:

Solve:
1.
5( x  3)  2  8
5 x  15  2  8
5 x  21
x  4.2

2.
4( x  1) 3(1  x)
          2
7           4
16( x  1)  21(1  x)  56
16 x  16  21  21x  56
37 x  93
19
x2
37

Exercise

Solve:

1. 3b  7   82b  3

3    4
2.     c
4    5

3. (3  x)  (3x  3)  30

1            1
4.     (2 x  1)  (9 x  10)  0
2            3

2( x  3) 4(1  2 x)
5.                       1
3         5

6. 7(3x  4)  8  4  2( x  3)

a a
7.     7
2 3

2( x  5) 3(2 x  3)
8.            
3         4

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Examples:

x 2  7 x  12  0
1.       Simple factors:
( x  3)(x  4)  0
x  3 or  4

6x 2  7 x  3  0
2.        Simple factors:
(2 x  3)(3x  1)  0
x  1 1 or 1
2    3

3.        Use of formula:                                  3x 2  7 x  2  0

 b  b 2  4ac                               7  49  4  3  2
x                                           x =
2a                                         6
7  73
x
6

x  2.59(2d . p.) or  0.26(2d . p.)

Exercise

Solve:                                                    Solve correct to 2d.p.:

1. ( x  6)(x  2)  0                                    7. x 2  6 x  4  0

2. x( x  1)  0                                          8. 2k 2  4k  3  0

3. x 2  5 x  6  0                                      9. 4 p 2  7 p  6

4. x 2  7 x  10                                                  3
10.          x
x 1
5. x 2  2 x

6. 2 x 2  3x  2  0

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TOPIC 4 : Simultaneous Equations

by elimination method

Examples:

Solve:

1.                                                         2.
x y 8                                                     5x  3 y  2
4 x  y  3 +                                              2x  y  0
5x  5                                                      5x  3 y  2
x 1                                                   x3   6x  3 y  0
y7                                                         x  2
y4

3.
8 p  7 q  13
3 p  2q  28

x2     16 p  14q  26
x7      21 p  14 g  196
37 p  222
p6
q5

Solve:

3 x  2 y  10
1.
x  2y  6

4x  2 y  1
2.
3x  4 y  5

2x  5 y  7
3.
3x  4 y  6

y  3x  9
4.
2 x  15  3 y  2

5. The sum of two numbers is 32 and their difference is 6. Find the two
numbers.
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TOPIC 5 : Simplifying, including indices

Examples:

1. a 6  a 3  a 9
2. m 5  m 3  m 2
3. ( x m ) n  x mm
a 3 (a 4  2ab)
4.
 a 7  2a 4 b
9a 2 b 27 ab 2 9a 2 b     c2      ac
5.             2
             2

c        c     c      27 ab     3b

Exercise:

Simplify as much as possible:                             Simplify as much as possible:
A.                                                        B.
1. a 4  a 3                                              1. a 3  a 4

2. a 4  a 3                                              2. x o  x 4

3. 3b 5  b 7                                             3. 18 a 2 b 2 c 2  2(ab) 1

4. a 2 (a 3  a )                                                     c 2  3  pa
1                2
4.      2         4

5. x 2 ( y 2  xy  z )                                   5.      3 2
4

6. (k 3 ) 2  k 4                                         6.       3 3  6 2
4

16 ab 2 ac                                           7. a 7  a 1
7.           2
c2    4b

 1 
1
mn 2 2m                                                8.

2
8.                                                                  4
4    n

a 3 (a 2  b 2 )                                     9. 15a 2 b 7  5ab 2
9.
a5

3x 2 ( y 2  x) x 4                                   10. (c 2 d 4 ) 2  c 1
1

10.                
9         4

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TOPIC 6 : The Sine and Cosine Rules

Remember: for any triangle:
B

c                     a

A                                            C
b

a     b    c
1.                 The Sine Rule
SinA SinB SinC

2. a 2  b 2  c 2  2bc cos A  The Cosine Rule

b2  c2  a2
3. Cos A =
2bc

Examples:

1. Find angle B and side BC.

A                                         7     8

SinB sin 53

7  sin 53
7m                              8m                      SinB               0.6988
8

53°                                                    B  44  (nearest deg ree)

C                                                            B
AngleA  180  44   53  83

 ( BC) 2  7 2  8 2  2  7  8  cos83

BC  99.3506
BC  10.0m(3sig .Figs.)

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Exercise:

B

57°
250m                       270m

D                                        A

2. Calculate BC

B

93°
250m

37°
C                                                                              D

3. Find x

x

8cm

110°
40°

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4. Find y

60°

5cm

10cm
y

5. In a triangle ABC, AB = 9cm, BC = 5cm and angle A = 24  .
I. Show that there are two possible values for angle C and calculate
each value;
II. Make a rough sketch to show both cases.

6. In a triangle ABC, AB = 5cm, AC = 8cm and angle A = 52  .

I. Find BC, correct to 3 significant figures;
II. Find angle ACB, correct to the nearest degree.

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Answers to the Exercises

Here are the answers to the above questions. You must ensure that you have
a large majority of these correct. You will benefit from identifying any
questions with which you need help as soon as possible in September.

Topic 1

1.   n(n – p)                                             6. 3(x + 5)(x – 5 )
2.   (h + 25)(h – 25)                                     7. (5h + 2)(h – 2)
3.   (m + 5)(m + 2)                                       8. (2x + 1)(5x + 2)
4.   (n – 4)(n + 3)                                       9. (b + 5)(a – 2)
5.   (5 + b)(3 – b)                                       10. 2(x + 3)(x + 4)

Topic 2

A.

y2
1. x 
3
4y  7
2.   x
2
3.   x   y  21
ab  4e
4.   c
d
2b 2
5.   a
3( x  y )

6. x   12 yz

d 2  bx
7. c 
y

km
8. l 
k mn

B.

1.   30.4
2.   -5.00
3.   3.91
4.   0.471
5.   0.768

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Topic 3

A.                                                        B.
45                                               1. x = 6 or -2
1. b 
13
16                                               2. x = 0 or -1
2.   c
15
3.   x  6                                               3. x = -3 or -2
17                                               4. x = 2 or 5
4.   x
24
 23                                             5. x = 0 or 2
5.   x
14
6.   x=2                                                           1
6. x   or 2
2
7. a = 43                                                 7. x = -5.24 or -0.76
8. x = 6.7                                                8. k = -2.58 or 0.58
9. p = -2.38 or 0.68
10. x = 1.30 or -2.30

Topic 4

1. x = 4, y = 1
2. x = 3.4, y = -1.3
2          2
3. x  , y  1
7          7
4. x  2, y  3
5. 19 and 13

Topic 5

A.                                                        B.
1. a 7                                                    1. a 1
2. a                                                      2. x 4
3                                                     3. 9abc 2
3. 2
b
4. a 5  a 3                                              4. 3/8 (pac)2
5. x 2 y 2  x 3 y  x 2 z                                5. 16/9
6. k 6  k 2                                              6. 16/243
4a 2                                                  7. a 8
7.
c
n3                                                    8. 2
8.
8
a2  b2        b2                                     9. 3ab9
9.           or1  2
a2          a
x ( y  x) x 6 y 2  x 7
6    2                                                           2

10.              or                                       10. (cd )
12           12
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Topic 6

1)   249m
2)   318m
3)   11.7m
4)   94.3

5)             (i)      132 or 47 

(ii)      Roughly (not to scale):

C

5cm

24°
A                                                                       B
9cm

C

5cm

24°
A                                                         B
9cm

We hope that you will benefit from working through this pack. Best wishes for
a successful course.

Mathematics Department, Barnwell School

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