Centre Number
71
Candidate Number
General Certificate of Secondary Education
2006
Mathematics
(Non-calculator)
Higher Tier
GMM41
Module M4 Paper 1
[GMM41]
MONDAY 5 JUNE
1.30 pm – 2.30 pm
TIME
1 hour.
INSTRUCTIONS TO CANDIDATES
Write your Centre Number and Candidate Number in the spaces
provided at the top of this page.
Write your answers in the spaces provided in this question paper.
Answer all ten questions.
Any working should be clearly shown in the spaces provided since
marks may be awarded for partially correct solutions.
You must not use a calculator for this paper
For Examiner’s
use only
Question
Number
1
2
3
4
INFORMATION FOR CANDIDATES
11 –– 6.1.06BP
6.1.06BP
The total mark for this paper is 49.
Figures in brackets printed down the right-hand side of pages indicate
the marks awarded to each question or part question.
You should have a ruler, compasses, set-square and protractor.
The Formula Sheet is on page 2.
5
6
7
8
9
10
Total
Marks
GMM41S6
2344
Marks
Formula Sheet
Area of trapezium = 1–2 (a + b)h
a
h
b
Volume of prism = area of cross section × length
Cross
section
length
In any triangle ABC
A
Area of triangle = 1–2 ab sin C
Sine rule :
c
b
a
b
c
=
=
sin A sin B sin C
B
a
Cosine rule: a2 = b2 + c2 – 2bc cos A
C
r
Volume of sphere =
4
– πr 3
3
Surface area of sphere = 4πr 2
Volume of cone = 1–3 πr 2h
l
h
Curved surface area of cone = πrl
r
Quadratic equation:
1 – 6.1.06BP
The solutions of ax2 + bx + c = 0, where a ≠ 0, are given by
– b ± b 2 – 4 ac
x=
2a
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B
1
Examiner Only
Marks
Remark
y
C
O
Diagram not
drawn accurately
z
100°
D
x
A
E
O is the centre of the circle. Find the sizes of the angles
(a) x
Answer _____________° [1]
(b) y
Answer _____________° [1]
(c) z
1 – 6.1.06BP
Answer _____________° [1]
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2
In a closing down sale, all prices were reduced by 70%.
The sale price of a camera was £48.
What was the original price before the sale?
Examiner Only
Marks
Remark
Answer £ _____________ [3]
3
Evaluate
(a) 10–3
Answer _____________ [1]
(b) 60
1 – 6.1.06BP
Answer _____________ [1]
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(a) Solve x2 + 14x – 15 = 0
4
Examiner Only
Marks
Remark
Answer x = _____________ [3]
(b) On the grid below, illustrate the inequalities
y ⬍ 8, y ⬍ 3x + 4,
y ⬎ x + 1,
x⬎0
Mark with R the region satisfying all four inequalities.
[4]
y
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
x
1 – 6.1.06BP
0
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5
Evaluate
Examiner Only
Marks
(a)
Remark
1–
36 2
Answer _________ [1]
(b) 320.2
Answer _________ [1]
3–
(c) 164
Answer _________ [2]
– 2–
3
(d) 125
1 – 6.1.06BP
Answer _________ [2]
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6
The two triangles shown are similar.
Examiner Only
Marks
Remark
Diagram not
drawn accurately
7.5 cm
xcm
9cm
12cm
(a) Calculate the length of the side marked x.
Answer _____________cm [2]
(b) Given that the area of the small triangle is 22.5 cm2, calculate the
shaded area between the two triangles.
1 – 6.1.06BP
Answer _____________cm2 [3]
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7
The table gives information about the heights of plants in a crop test.
Examiner Only
Marks
Height (h cm)
Frequency
Remark
0 ⬍ h ⭐ 6 6 ⬍ h ⭐ 8 8 ⬍ h ⭐ 10 10 ⬍ h ⭐ 15 15 ⬍ h ⭐ 20
120
320
260
50
[3]
1 – 6.1.06BP
(a) Show this information on a histogram.
150
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(b) The plants are arranged in order of height and every tenth one is
selected to form a sample.
Explain why this is not a suitable way to select a sample.
Examiner Only
Marks
Remark
[2]
(c) In a stratified sample of all plants over 9 cm in height, 20 up to 10 cm
were selected and 8 over 15 cm were selected.
How many plants in the original crop were over 9 cm in height?
1 – 6.1.06BP
Answer _____________ [3]
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8
Factorise fully
Examiner Only
Marks
Remark
(a) 6cd – 9c2
Answer _________________ [2]
(b) 3a2 – 27d 2
Answer _________________ [3]
9
Solve the equation
4
9
——— + ——— = 3
(x – 3) (2x – 1)
1 – 6.1.06BP
Show your working.
Answer _____________ [7]
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10 Find the values of c and d for which
Examiner Only
Marks
Remark
y2 – 12y + c ≡ (y + d)2
1 – 6.1.06BP
Answer c = ___________ d = ___________ [3]
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1 – 6.1.06BP
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Centre Number
71
Candidate Number
General Certificate of Secondary Education
2006
Mathematics
(With calculator)
Higher Tier
GMM42
Module M4 Paper 2
[GMM42]
MONDAY 5 JUNE
2.45 pm – 3.45 pm
TIME
1 hour.
INSTRUCTIONS TO CANDIDAT ES
Write your Centre Number and Candidate Number in the spaces
provided at the top of this page.
Write your answers in the spaces provided in this question paper.
Answer all eleven questions.
Any working should be clearly shown in the spaces provided since
marks may be awarded for partially correct solutions.
For Examiner’s
use only
Question
Number
1
2
3
4
5
INFORMATION FOR CANDIDATES
The total mark for this paper is 49.
Figures in brackets printed down the right-hand side of pages indicate
the marks awarded to each question or part question.
You should have a calculator, ruler, compasses, set-square and
protractor.
The Formula Sheet is on page 2.
6
7
8
9
10
11
Total
Marks
GMM42S6
2345
Marks
Formula Sheet
Area of trapezium = 1–2 (a + b)h
a
h
b
Volume of prism = area of cross section × length
Cross
section
length
In any triangle ABC
A
Area of triangle = 1–2 ab sin C
Sine rule :
c
b
a
b
c
=
=
sin A sin B sin C
B
a
Cosine rule: a2 = b2 + c2 – 2bc cos A
C
r
Volume of sphere =
4
– πr 3
3
Surface area of sphere = 4πr 2
Volume of cone = 1–3 πr 2h
l
h
Curved surface area of cone = πrl
r
Quadratic equation:
The solutions of ax2 + bx + c = 0, where a ≠ 0, are given by
x=
– b ± b 2 – 4 ac
2a
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1
The cumulative frequency graph gives information about the percentage
marks obtained by 300 candidates in an examination.
Examiner Only
Marks
Remark
Use the graph to estimate
(a) the median percentage mark,
Answer _____________ [1]
(b) the interquartile range,
Answer _____________ [2]
(c) the percentage mark separating the top 20 candidates from the others.
Answer _____________ [2]
300
250
Cumulative frequency
200
150
100
50
0
10
20
30
40
50
60
70
80
% marks
(less than)
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90
2
(a) A is (0, 4). B is (3, 13)
Find the equation of the line AB.
Examiner Only
Marks
Remark
Answer _________________ [3]
(b) Find the equation of the line parallel to the line AB and passing
through (0, 8).
Answer _________________ [2]
(c) Find the equation of the line perpendicular to the line AB and passing
through (9, 0).
Answer _________________ [2]
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Q
3
Examiner Only
Marks
Diagram not
drawn accurately
8cm
P
Remark
R
11cm
In the triangle PQR, angle PQR is 90°, PR = 11 cm and PQ = 8 cm.
Calculate the size of the angle QPR.
Answer _________________° [3]
4
Calculate the volume of a sphere of radius 12 cm.
Answer _____________ [3]
5
Solve the simultaneous equations
6x + 7y = 1
2x – y = 7
Show your working.
A solution by trial and improvement will not be accepted.
Answer x = ___________ y = ___________ [3]
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5
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D
6
Examiner Only
G
Marks
Remark
7 cm
E
F
C
B
8 cm
O
10 cm
A
(a) Calculate the length of the space-diagonal OG in the cuboid
OABCDEFG.
Answer _____________ cm [2]
(b) Calculate the size of the angle GOA between the lines OA and OG.
Answer _____________° [2]
7
Prove that opposite angles of a cyclic quadrilateral add up to 180°.
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6
[3]
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8
The area of a rectangle is 6 cm2 and the perimeter is 20 cm.
Examiner Only
Marks
Remark
(a) If x is the length of one side of the rectangle, show that
x2 – 10x + 6 = 0
[3]
(b) Find the lengths of the sides of the rectangle. You may give your
answers correct to two decimal places.
Answer ______________________ cm [3]
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9
y
1
0
90
180
270
360
x
–1
y = cosx
Use the graph to find, for 0° ⬍ x ⬍ 360°, the solutions of
Examiner Only
Marks
Remark
(a) cos x = – 0.45
Answer __________________________ [2]
(b) 8 cos x = 5
Answer __________________________ [2]
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Start
10
5km
Bill
3 km
Examiner Only
Marks
Remark
Diagram not
drawn accurately
3k
m
Ben
Bill and Ben are hiking.
They hike together 5 km due East.
They then split up. Bill continues 3 km due East but Ben travels 3 km in a
straight line in a different direction.
When they have both stopped Ben is 1 km nearer the starting point than Bill.
Calculate Ben’s bearing from the starting point.
A solution by scale drawing will not be accepted.
Answer _____________ [4]
11 Solve simultaneously the equations
x2 + 4y2 = 10
2y + 3x = 10
A solution by trial and improvement will not be accepted.
Show your working.
Answer x = ____________ y = __________ [7]
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THIS IS THE END OF THE QUESTION PAPER
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S – 1/06 – 3200 – 302507(134)
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