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General Certificate of Secondary Education
Higher Tier
November 2011
Mark
3
4 –5
6 –7
Mathematics (Modular)
(Specification B)
Module 5
43055/1H
8 –9
10 –11
H
Paper 1 Non-calculator
12 –13
14 –15
16 –17
Wednesday 9 November 2011
1.30 pm to 2.45 pm
TOTAL
For this paper you must have:
l
mathematical instruments.
You must not use a calculator.
Time allowed
l 1 hour 15 minutes
Instructions
l Use black ink or black ball-point pen. Draw diagrams in pencil.
l Fill in the boxes at the top of this page.
l Answer all questions.
l You must answer the questions in the spaces provided. Do not write
outside the box around each page or on blank pages.
l Do all rough work in this book. Cross through any work you do not want
to be marked.
Information
l The marks for questions are shown in brackets.
l The maximum mark for this paper is 70.
l You may ask for more answer paper, graph paper and tracing paper.
These must be tagged securely to this answer booklet.
Advice
l In all calculations, show clearly how you work out your answer.
(NOV11430551H01)
WMP/Nov11/43055/1H
43055/1H
2
Formulae Sheet: Higher Tier
a
1
Area of trapezium = –
(a + b)h
h
2
b
Volume of prism = area of cross-section × length
crosssection
h
lengt
r
4
Volume of sphere = –
πr3
3
Surface area of sphere = 4 π r 2
1
Volume of cone = –
πr2 h
3
l
h
Curved surface area of cone = π rl
r
In any triangle ABC
C
1
Area of triangle = 2 ab sin C
Sine rule
a
sin A
=
b
sin B
=
b
a
c
sin C
A
c
B
Cosine rule a 2 = b 2 + c 2 – 2bc cos A
The Quadratic Equation
The solutions of ax 2 + bx + c = 0, where a ≠ 0, are given by
x=
(02)
– b ± √ (b2 – 4ac)
2a
WMP/Nov11/43055/1H
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outside the
box
3
Answer all questions in the spaces provided.
1
Draw the graph of
y = 3x – 5 for values of x from –2 to 2.
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y
2
1
–2
–1
O
1
2
x
–1
–2
–3
–4
–5
–6
–7
–8
–9
– 10
– 11
– 12
(3 marks)
3
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outside the
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4
2
Here is a conversion graph.
23
22
21
20
19
18
17
16
15
14
13
Miles per hour
(mph)
12
11
10
9
8
7
6
5
4
3
2
1
0
0
(04)
1
2
3
4
6
8
5
7
Metres per second (m/s)
9
10
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5
An athlete runs 200 metres in 20 seconds.
Work out his average speed in miles per hour.
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Answer .............................................................. mph
3
Here is a sequence.
1
3 (a)
(3 marks)
5
9
13
17
........
Work out an expression for the nth term of the sequence.
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Answer ......................................................................
3 (b)
(2 marks)
Explain why 99 is not a term in this sequence.
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(1 mark)
Turn over for the next question
6
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6
An aeroplane flies due North from A to B.
The distance from A to B on the motorway is 23 miles.
4
N
Scale: 1cm represents 2 miles
B
Motorway
A
4 (a)
How much further is it from A to B on the motorway than by aeroplane?
............................................................................................................................................
............................................................................................................................................
Answer ........................................................... miles
4 (b)
C is 12 miles North-East of A.
4 (b) (i)
Write down the three-figure bearing of C from A.
Answer .................................................................. º
(3 marks)
(1 mark)
4 (b) (ii) Mark with a cross the point C on the diagram.
(2 marks)
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7
5 (a)
Use π = 3.14 to work out the area of a circle of radius 10 cm.
State the units of your answer.
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Answer .....................................................................
5 (b)
(3 marks)
The diagram shows a shape made from two identical trapezia.
Not drawn
accurately
5 cm
8 cm
9 cm
Work out the area of the shape.
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Answer .....................................................................
(3 marks)
12
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8
6 (a)
Solve
Do not write
outside the
box
4(3w – 7) = 32
............................................................................................................................................
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Answer w = ................................................................
6 (b)
Solve
(3 marks)
26 – y
––––– = 4
5
............................................................................................................................................
............................................................................................................................................
............................................................................................................................................
Answer y = .................................................................
6 (c)
Multiply out
(2 marks)
x2(x – 4)
............................................................................................................................................
Answer ......................................................................
6 (d)
Work out the smallest integer value of t that satisfies the inequality
(2 marks)
4t + 1 > 25
............................................................................................................................................
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............................................................................................................................................
Answer ......................................................................
(08)
(3 marks)
WMP/Nov11/43055/1H
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9
7
The diagram shows a solid triangular prism made of wood.
11 cm
6 cm
8 cm
The density of the wood is 0.5 grams per cm3.
Work out the mass of the prism.
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Answer ........................................................... grams
(4 marks)
Turn over for the next question
14
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10
8 (a)
Write down the value of w.
w
54°
Not drawn accurately
25°
(1 mark)
Answer ........................................................ degrees
8 (b)
O is the centre of the circle.
Work out the value of y.
y
Not drawn accurately
O
126°
............................................................................................................................................
Answer ........................................................ degrees
(10)
(1 mark)
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11
8 (c)
The diagram shows two circles with centres O and P.
The circles have common tangents, ABC and EDC.
Not drawn accurately
A
B
C
P
74°
O
x
D
E
Work out the size of angle OPD, marked x on the diagram.
You must show your working, which may be on the diagram.
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Answer ....................................................... degrees
(4 marks)
6
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12
9
9 (a)
Here are six equations of straight lines.
A
y = 3x + 2
B
y = 2x + 3
C
2x + 3y = 6
D
3x + 2y = 6
E
x = 3y + 2
F
x = 2y + 3
Which lines pass through the point (0, 3)?
Answer .....................................................................
9 (b)
Which line has gradient 3?
Answer .....................................................................
9 (c)
(2 marks)
(1 mark)
2
Which line has gradient – –
?
3
............................................................................................................................................
............................................................................................................................................
Answer .....................................................................
9 (d)
(1 mark)
Which lines pass through the point (5, 1)?
............................................................................................................................................
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............................................................................................................................................
Answer .....................................................................
(12)
(2 marks)
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13
10 (a)
Simplify fully
5s6t 2 × s3t 5
............................................................................................................................................
............................................................................................................................................
Answer ......................................................................
10 (b) (i)
Make y the subject of
(2 marks)
1 – 2y
x = –––––
y–6
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Answer ......................................................................
10 (b) (ii) Given that
(4 marks)
1 – 2y
x = –––––
y–6
work out the value of y when x = –1
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Answer ......................................................................
(2 marks)
14
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14
Prove that triangle ABC is congruent to triangle DEF.
11
Not drawn accurately
A
D
110°
110°
46°
24°
B
C
10 cm
F
E
10 cm
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............................................................................................................................................
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............................................................................................................................................
(3 marks)
12
(x + a)2 ≡ x2 – 12x + b
Work out the values of a and b.
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............................................................................................................................................
............................................................................................................................................
Answer a = ........................... , b = ...........................
(14)
(3 marks)
WMP/Nov11/43055/1H
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15
13
The diagram shows a shape made from a cuboid and a pyramid.
The cuboid and the pyramid have the same height.
The length of the cuboid is 8 cm.
The width of the cuboid is 3 cm.
The total volume of the shape is 160 cm3.
3 cm
8 cm
You are given
1
Volume of a pyramid = –
× area of base × perpendicular height
3
Work out the height of the shape.
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............................................................................................................................................
............................................................................................................................................
Answer ................................................................ cm
(5 marks)
11
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16
14
This is the graph of y = cos x for 0° ⭐ x ⭐ 360°
y
3
2
1
0
90
180
270
360 x
–1
–2
–3
14 (a)
In each case the graph of y = cos x is shown to help you.
14 (a) (i)
Draw the graph of y = 3cos x for 0° ⭐ x ⭐ 360°
y
3
2
1
0
90
180
270
360 x
–1
–2
–3
(1 mark)
(16)
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outside the
box
17
14 (a) (ii) Draw the graph of y = 2 + cos x for 0° ⭐ x ⭐ 360°
y
3
2
1
0
90
180
270
360 x
–1
–2
–3
(1 mark)
14 (b)
You are given that cos 60° = 0.5
Use the symmetries of the graph of y = cos x to
14 (b) (i)
write down the value of the obtuse angle x such that cos x = –0.5
Answer ........................................................ degrees
(1 mark)
14 (b) (ii) write down the value of the reflex angle x such that cos x = 0.5
Answer ........................................................ degrees
(1 mark)
END OF QUESTIONS
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