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Level 2 Certificate
FURTHER MATHEMATICS
Paper 2 Calculator

Time allowed: 1 hour 45 minutes

Materials For Examiner’s Use
For this paper you must have:
• a calculator Pages Mark
• mathematical instruments.
2–3

Instructions 4–5
• Use black ink or black ball-point pen. Draw diagrams in pencil.
6–7
• Fill in the boxes at the top of this page.
• Answer all questions. 8–9
• You must answer the questions in the spaces provided. Do not write
10–11
outside the box around each page or on blank pages.
• If you need extra space for your answer(s), use the lined pages at the end of 12–13
this book. Write the question number against your answer(s).
14–15
• Do all rough work in this book. Cross through any work you do not want
to be marked. 16–17
• In all calculations, show clearly how you work out your answer.
18–19

Information TOTAL
• The marks for questions are shown in brackets.
• The maximum mark for this paper is 80.
• You may ask for more graph paper and tracing paper.
These must be tagged securely to this answer book.
• The use of a calculator is expected but calculators with a facility for symbolic
algebra must not be used.

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Do not write
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Answer all questions in the spaces provided. box

1 Expand and simplify 5(2x – 1) + 4(11 – x)

Give your answer in the form a(bx + c) where a, b and c are integers greater than 1
[3 marks]

Answer

2 (a) 5m is decreased by 40%

The answer is (m + 1)

Work out the value of m.

[2 marks]

Answer

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2 (b) 3
Solve 2w – 10 = 18
[2 marks]

w=

3 The rectangle and triangle shown have equal areas.

d
Work out the value of
e
Give your answer in its simplest form.
[3 marks]

Answer 10

Turn over ►

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4
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4 The equations of the two circles shown are
x2 + y2 = 100 and x2 + y2 = 36

Work out the shaded area.

Give your answer as an integer multiple of π.
[3 marks]

Answer units2

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5
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5 SQR is a right-angled triangle.
P is a point on SQ.
Angle SPR = 45°
M is the midpoint of QR.
k is a constant.

Work out the coordinates of M.

[3 marks]

Answer ( , )

Turn over ►

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6
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x + 2w box
6 Rearrange y= to make w the subject.
3
[3 marks]

Answer

7 (a) a is a value greater than 1

4 2m
Work out the value of m for which (am) = (a5)
[2 marks]

m=

7 (b) w3x2y5 = w13x7

Write y in terms of w and x.

Give your answer in its simplest form.
[2 marks]

y=

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7
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8 A function f is given by
f(x) = 4x x<0
= x2 – 8x 0⩽x⩽8
= 16 – 2x x>8

A sketch of y = f(x) is shown.

Work out all the values of x for which f(x) = –12

[4 marks]

Answer
11

Turn over ►

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8
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1 1 box
9 (a) Circle the expression that is equivalent to +
a b
[1 mark]

2 ab 2 b+a
a+b b+a ab ab

6c 4 – c 3
9 (b) Simplify fully
36c 2 – 1
[3 marks]

Answer

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9
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3k box
10 The radius of a sphere, in cm, is
2
The volume of the sphere, in cm3, is 972π

4 3
Volume of a sphere = πr where r is the radius
3

Work out the value of k.

[3 marks]

Answer

11 Expand and simplify fully (5x + 3y2)(4x – y2)

[3 marks]

Answer

10

Turn over ►

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10
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12 A and B are points on the line y = 3x + 2
B, C and D (5, 0) are points on the line L.
OA : AC = 1 : 4

Work out the x-coordinate of B.

[5 marks]

Answer

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11
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13 P is the point on the curve y = ax3 + 10x2 where x=2
1
The gradient of the normal to the curve at P is –
4

Work out the value of a.

[4 marks]

Answer

Turn over for the next question

Turn over ►

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12
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1 0  box
14 (a) A=  
 0 –1

Describe geometrically the single transformation represented by A.

[1 mark]

Answer

 0 1
14 (b) B=  
 –1 0 

Describe geometrically the single transformation represented by B2

[2 marks]

Answer

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13
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15 A, B and C are points on a circle, centre O.
ACD is a straight line.
Angle BCD = w

Prove that w = x + 90°

[5 marks]

Turn over ►

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14
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16 The coefficient of x4 in the expansion of (a + 2x)6 is 1500

Work out the two possible values of a.

[3 marks]

Answer and

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15
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17 ABCDEFGH is a cube with side length 32 cm
M and N are points on DH and CG respectively.

Work out the size of the angle that the line BM makes with the plane ABCD.
[5 marks]

Answer degrees 8

Turn over ►

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16
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18 y = 12x +
x

Show that y has a minimum value when x = 0.5

[5 marks]

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17
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19 (a) f(x) = (x + 2)3
g is a function such that gf(x) = (x + 2)12

Work out an expression for g(x)

[1 mark]

Answer

19 (b) h(x) = x2 + 5
k is a function such that hk(x) = 4x2 + 5

Work out an expression for kh(x)

[2 marks]

Answer

Turn over for the next question

Turn over ►

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18
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2sin x + cos x 1 box
20 Show that – can be written in the form acos x + bsin x
tan x sin x
where a and b are integers.
[4 marks]

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19

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21 3x2 + 2bx + 8a can be written in the form 3(x + a)2 + b + 2

Work out the two possible pairs of values of a and b.

[6 marks]

a= b=

a= b=

END OF QUESTIONS

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DO NOT WRITE ON THIS PAGE

ANSWER IN THE SPACES PROVIDED

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Question Additional page, if required.
number Write the question numbers in the left-hand margin.

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Question Additional page, if required.
number Write the question numbers in the left-hand margin.

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Question Additional page, if required.
number Write the question numbers in the left-hand margin.

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24
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There are no questions printed on this page box

DO NOT WRITE ON THIS PAGE

ANSWER IN THE SPACES PROVIDED

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