Cambridge IGCSE™
* 0 9 3 1 6 8 4 5 7 9 *
CAMBRIDGE INTERNATIONAL MATHEMATICS
Paper 4 (Extended)
0607/42
February/March 2022
2 hours 15 minutes
You must answer on the question paper.
You will need:
Geometrical instruments
INSTRUCTIONS
●
Answer all questions.
●
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
●
Write your name, centre number and candidate number in the boxes at the top of the page.
●
Write your answer to each question in the space provided.
●
Do not use an erasable pen or correction fluid.
●
Do not write on any bar codes.
●
You should use a graphic display calculator where appropriate.
●
You may use tracing paper.
●
You must show all necessary working clearly and you will be given marks for correct methods, including
sketches, even if your answer is incorrect.
●
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
●
For r, use your calculator value.
INFORMATION
●
The total mark for this paper is 120.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 20 pages. Any blank pages are indicated.
DC (CE/CT) 301839/2
© UCLES 2022
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2
Formula List
ax 2 + bx + c = 0
For the equation
x=
- b ! b 2 - 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2rrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = rrl
Curved surface area, A, of sphere of radius r.
A = 4r r 2
Volume, V, of pyramid, base area A, height h.
1
V = Ah
3
Volume, V, of cylinder of radius r, height h.
V = rr 2 h
Volume, V, of cone of radius r, height h.
1
V = rr 2 h
3
Volume, V, of sphere of radius r.
4
V = rr 3
3
A
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
b
c
1
Area = bc sin A
2
B
© UCLES 2022
a
C
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Answer all the questions.
1
(a) Find the gradient and y-intercept of the line with equation 3x + 4y = 24 .
Gradient = ................................................
y-intercept = ................................................ [3]
(b)
y
L
NOT TO
SCALE
(8, 5)
(4, 3)
x
0
The diagram shows line L and the coordinates of two points on the line.
(i) Show that the equation of line L is 2y - x = 2 .
[3]
(ii) Find the equation of the line parallel to L that passes through the point (0, 7).
Give your answer in the form y = mx + c .
y = ................................................ [2]
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2
(a) Find 12 kg as a percentage of 80 kg.
.............................................. % [1]
(b) Find 19% of $250.
$ ................................................ [2]
(c) Xavier invests $500 at a rate of 1.5% per year simple interest.
At the end of y years, the value of Xavier’s investment is $612.50 .
Find the value of y.
y = ................................................ [3]
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(d) Each year the value of a car decreases by 12% of its value at the beginning of that year.
The original value of the car is $20 000.
(i) Calculate the value of the car at the end of 3 years.
Give your answer correct to the nearest dollar.
$ ................................................ [3]
(ii) Find the number of complete years for the value of $20 000 to decrease until it is first below
$1000.
................................................. [4]
(e) Each year the value of another car decreases by r % of its value at the beginning of that year.
At the end of 10 years, the value has decreased from $12 000 to $4673.
Find the value of r.
r = ................................................ [3]
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3
(a) The table shows the coursework grades for 20 students.
Grade
3
4
5
6
7
Frequency
1
3
6
2
8
Find
(i) the mode,
................................................. [1]
(ii) the range,
................................................. [1]
(iii) the median,
................................................. [1]
(iv) the lower quartile.
................................................. [1]
(b) The table shows some information about the heights, h cm, of 100 bushes.
Height (h cm)
100 1 h G 110
110 1 h G 115
115 1 h G 130
18
37
45
Frequency
Calculate an estimate of the mean height.
............................................ cm [2]
(c) The table shows some information about the times, t minutes, taken by some students to read a
magazine.
Time (t minutes)
Frequency
0 1 t G 10
10 1 t G 20
20 1 t G 30
30 1 t G 40
3
11
n
19
When using mid-interval values, an estimate of the mean value of t is 25.4 .
Find the value of n.
n = ................................................ [4]
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4
(a)
a°
b°
65°
NOT TO
SCALE
30°
c°
The diagram shows two straight lines crossing two parallel lines.
Find the values of a, b and c.
a = ................................................
b = ................................................
c = ................................................ [3]
(b)
L
20°
E
K
D
y° 70°
w°
30°
u°
C
NOT TO
SCALE
v°
x°
z°
A
B
A, B, C, D and E are points on the circle.
KL is a tangent to the circle at E.
AC = AD.
Find the values of u, v, w, x, y and z.
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u = ................................................
x = ................................................
v = ................................................
y = ................................................
w = ................................................
z = ................................................ [6]
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5
(a) (i) Expand and simplify `2x + 3j .
2
.................................................. [2]
(ii) The equation 4x 2 + 12x + 5 = 0 can be written as
`2x + 3j = k .
2
Find the value of k.
k = ................................................ [1]
(iii) Use your answer to part(ii) to solve the equation 4x 2 + 12x + 5 = 0 .
x = ................... or x = .................. [2]
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(b) x varies inversely as the square root of (w – 1).
When w = 10, x = 2.
(i) Find x in terms of w.
x = ................................................ [2]
(ii) Find x when w = 3.25 .
x = ................................................ [1]
(iii) Find w in terms of x.
w = ................................................ [3]
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6
In this question all lengths are in centimetres.
NOT TO
SCALE
2x – 1
5x + 1
7
13 – x
The area of the larger rectangle is 84 cm 2 greater than the area of the smaller rectangle.
(a) Show that 5x 2 + 2x - 88 = 0 .
[4]
(b) Factorise 5x 2 + 2x - 88 .
................................................. [2]
(c) Find the area of the smaller rectangle.
......................................... cm 2 [2]
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7
y
12
0
–4
4
x
– 12
f (x) = 4 - x 2 for - 4 G x G 4
(a) On the diagram, sketch the graph of y = f (x) .
[2]
(b) Write down the zeros of f(x).
................................................. [2]
(c) Write down the coordinates of the local maximum.
(...................... , ......................) [1]
(d) The equation
4 - x 2 = k has 4 solutions and k is an integer.
Write down a possible value of k.
k = ................................................ [1]
(e) (i) On the diagram, sketch the graph of y = 2x .
(ii) Solve the equation
[1]
4 - x 2 = 2x .
............................................................. [2]
(iii) On the diagram, shade the regions where y H 0, y G 2x and y G 4 - x 2 .
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8
f (x) = 2x + 1
g (x) = 3 - 2x
h (x) = log (x + 1)
(a) Find the value of
(i) f(12),
................................................. [1]
(ii) g(f(12)).
................................................. [1]
(b) Find the value of x when f (x) = g (x) .
x = ................................................ [2]
(c) Find f(g(x)), giving your answer in its simplest form.
................................................. [2]
(d) Find g -1 (x) .
g -1 (x) = ................................................ [2]
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(e) Find x when h (x) = f (0.5) .
x = ................................................ [2]
(f) Find h -1 (x) .
h -1 (x) = ................................................ [2]
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9
(a)
x cm
NOT TO
SCALE
4 cm
40°
Calculate the value of x.
x = ................................................ [3]
(b)
C
8 cm
A
NOT TO
SCALE
9 cm
B
10 cm
(i) Calculate angle ABC.
Angle ABC = ................................................ [3]
(ii) T is the point on AB that is the shortest distance from C.
Calculate BT.
BT = ........................................... cm [3]
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(c) Another triangle PQR has QR = 12 cm, PR = 7 cm and angle PQR = 35°.
Calculate the difference between the two possible values of angle QPR.
................................................. [5]
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16
10 When Zena wears a sweatshirt, the probability that she goes for a walk is
7
.
10
When Zena does not wear a sweatshirt, the probability that she goes for a walk is
On any day, the probability that she wears a sweatshirt is
1
.
5
9
.
10
(a) Complete the tree diagram.
Wears a sweatshirt
Goes for a walk
..........
1
5
Yes
..........
..........
..........
Yes
No
Yes
No
..........
No
[3]
(b) (i) Find the probability that on one day Zena does not wear a sweatshirt and she goes for a walk.
................................................. [2]
(ii) Find the probability that on one day Zena goes for a walk.
................................................. [2]
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(c) In the tree diagram below, the value of J is the answer to part (b)(i) and the value of K is the
answer to part (b)(ii).
Goes for a walk
Wears a sweatshirt
..........
No
J
Yes
K
..........
..........
..........
Yes
No
No
..........
Yes
(i) Find the probability that Zena does not wear a sweatshirt when she goes for a walk.
................................................. [2]
(ii) Complete the tree diagram above.
[3]
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11
(a)
2r
NOT TO
SCALE
r
y°
The diagram shows a sector of a circle with radius r and angle y°.
The length of the arc of the sector is 2r.
Calculate the value of y.
y = ................................................ [3]
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(b)
NOT TO
SCALE
8 cm
x°
The diagram shows a sector of a circle with radius 8 cm and angle x°.
The area of the shaded segment is A cm 2 .
(i) Show that A =
8x
r - 32 sin x .
45
[2]
(ii) Find the value of A when x = 90.
................................................. [1]
(iii) By sketching the graph of A =
8x
r - 32 sin x , find the value of x when A = 5.5 .
45
A
20
0
90
x
x = ................................................ [3]
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BLANK PAGE
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge
Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.
Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.
© UCLES 2022
0607/42/F/M/22
Cambridge IGCSE™
* 6 4 8 3 8 9 3 0 8 1 *
CAMBRIDGE INTERNATIONAL MATHEMATICS
Paper 4 (Extended)
0607/41
May/June 2022
2 hours 15 minutes
You must answer on the question paper.
You will need:
Geometrical instruments
INSTRUCTIONS
●
Answer all questions.
●
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
●
Write your name, centre number and candidate number in the boxes at the top of the page.
●
Write your answer to each question in the space provided.
●
Do not use an erasable pen or correction fluid.
●
Do not write on any bar codes.
●
You should use a graphic display calculator where appropriate.
●
You may use tracing paper.
●
You must show all necessary working clearly and you will be given marks for correct methods, including
sketches, even if your answer is incorrect.
●
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
●
For r, use your calculator value.
INFORMATION
●
The total mark for this paper is 120.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 20 pages. Any blank pages are indicated.
DC (CJ/CB) 215444/1
© UCLES 2022
[Turn over
2
Formula List
ax 2 + bx + c = 0
For the equation
x=
- b ! b 2 - 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2rrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = rrl
Curved surface area, A, of sphere of radius r.
A = 4rr 2
Volume, V, of pyramid, base area A, height h.
1
V = Ah
3
Volume, V, of cylinder of radius r, height h.
V = rr 2 h
Volume, V, of cone of radius r, height h.
1
V = rr 2 h
3
Volume, V, of sphere of radius r.
4
V = rr 3
3
A
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
b
c
1
Area = bc sin A
2
B
© UCLES 2022
a
C
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3
Answer all the questions.
1
y
3
2
1
–4
–3
–2
0
–1
T
1
2
3
4
x
–1
–2
P
–3
-2
(a) Translate triangle T by the vector e o.
2
[2]
(b) Reflect triangle T in the line y = 0.5 .
[2]
(c) Describe fully the single transformation that maps triangle P onto triangle T.
.....................................................................................................................................................
..................................................................................................................................................... [3]
(d) Enlarge triangle P with scale factor - 2 , centre (3, - 1) .
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[2]
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2
(a) The cumulative frequency curve shows the marks for 300 students in a history test.
300
250
200
Cumulative
150
frequency
100
50
0
0
10
20
30
40
50
History mark
(i) Find an estimate for the median.
................................................. [1]
(ii) Estimate the number of students with a mark of more than 20.
................................................. [2]
(iii) 70% of the students pass the test.
Find the pass mark.
................................................. [2]
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(b) The table shows the marks for 100 students in a geography test.
10 1 m G 20 20 1 m G 30 30 1 m G 40 40 1 m G 50
Mark m
Frequency
2
28
57
13
Calculate an estimate of the mean.
................................................. [2]
(c) The table shows the marks for 9 students in chemistry and in physics.
Chemistry
mark (x)
33
28
39
40
22
25
38
43
36
Physics
mark (y)
45
32
26
49
18
36
29
40
35
(i) Find the equation of the regression line for y in terms of x.
y = ................................................ [2]
(ii) What type of correlation is seen in this data?
................................................. [1]
(iii) Use your answer to part (c)(i) to estimate the physics mark for a student with a mark of 30 in
chemistry.
................................................. [1]
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3
y
5
0
–3
1
x
–5
1
x2
(a) On the diagram, sketch the graph of y = f (x) for values of x between - 3 and 1.
f (x) = 2x + 4 -
[3]
(b) Write down the equation of the asymptote of the graph.
................................................. [1]
(c) Find the coordinates of the local maximum.
(....................... , .......................) [1]
(d) g (x) = x 3 - 5x for - 3 G x G 1.
Solve f (x) G g (x) .
................................................. [4]
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4
(a) $216 is shared in the ratio 5 : 1.
Work out the larger share.
$ ................................................ [2]
(b) Luis shares some money between Ali, Betty and Clare in the ratio 3 : 4 : 6.
Ali receives $171.
Find the total amount of money Luis shared.
$ ................................................ [2]
(c) Farima invests $1400 in a savings account paying simple interest at a rate of 2.5% per year.
Calculate the total amount in the account at the end of 3 years.
$ ................................................ [3]
(d) Emir invests $3000 at a rate of 2% per year compound interest.
(i) Calculate the value of Emir’s investment at the end of 4 years.
$ ................................................. [2]
(ii) Find the number of complete years until Emir’s investment is first worth more than $4000.
................................................. [4]
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5
A sequence of patterns is made using grey tiles and white tiles.
Pattern 1
Pattern 2
Pattern 3
(a) Complete the table.
Pattern number
1
2
Number of grey tiles
6
10
Number of white tiles
0
2
3
4
n
[6]
(b) Find and simplify an expression for the total number of tiles in Pattern n.
................................................. [1]
(c) Pattern k has a total of 600 tiles.
Find the number of grey tiles in Pattern k.
................................................. [4]
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(d) The tiles in a pattern are put in a bag.
The probability of taking a grey tile from the bag at random is
5
.
12
A tile is taken from the bag at random and replaced.
This is repeated 3 times.
Find the probability that all 3 tiles are white.
................................................. [2]
(e) All the grey tiles from Pattern 4 are put in a bag.
Two tiles are taken from the bag at random without replacement.
Find the probability that one tile came from a corner of the pattern and the other did not.
................................................. [3]
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6
(a)
NOT TO
SCALE
C
5 cm
30°
B
O
A
The diagram shows a circle, centre O, with radius 5 cm.
BA and BC are tangents to the circle at A and C.
Angle ABC = 30° .
Calculate the area of the shaded minor segment.
......................................... cm 2 [4]
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(b)
O
height
O
NOT TO
SCALE
40°
12 cm
DE
D
E
The circle, centre O, has radius 12 cm.
Angle DOE = 40° .
The minor sector DOE is removed.
The major sector is formed into a cone by joining OD to OE.
Calculate the height of the cone.
............................................ cm [5]
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7
Abbi makes wooden boards in three sizes, small, medium and large.
They are all cuboids.
The medium board has height 2 cm, width 23 cm and length 50 cm.
(a) Calculate the volume of the medium board.
......................................... cm 3 [2]
(b) The small board is mathematically similar to the large board.
The small board has a volume of 287.5 cm 3 and a height of 1.15 cm.
The large board has a volume of 18400 cm 3 .
(i) Find the height of the large board.
............................................ cm [3]
(ii) Is the medium board mathematically similar to the large board?
Explain how you decide.
........................... because ...................................................................................................
............................................................................................................................................. [3]
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8
(a) A is the point (- 11, 7) and B is the point (8, - 13) .
Find the length of AB.
................................................. [3]
(b) P is the point (2, - 5) and Q is the point (6, 11) .
Line L is perpendicular to PQ and crosses PQ at point R.
The ratio PR : RQ = 3 : 1.
Find the equation of line L.
................................................. [6]
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9
f (x) = 2x + 3
(a)
g (x) = 2 - 4x
h (x) = 3 x
(i) Find f(5).
................................................. [1]
(ii) Find and simplify g(f(x)).
................................................. [2]
(iii) Find g -1 (x) .
g -1 (x) = ................................................ [2]
(iv) Solve h (x) = 48 .
................................................. [2]
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(b) (i) The diagram shows a sketch of the graph of y = j (x) .
y
10
j(x)
5
–5
– 10
0
10 x
5
–5
– 10
On the same diagram, sketch the graph of y = j (x + 2) .
[1]
(ii) The diagram shows the graphs of y = k (x) and y = m (x) .
y
10
k(x)
m(x)
5
–5
0
3
x
–5
– 10
Write k (x) in terms of m (x) .
k (x) = ................................................ [1]
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10 (a) Simplify fully.
4x 2 y
x
'
3
12y
................................................. [2]
(b) Write as a single fraction in its simplest form.
1
x-3
x-3
2
................................................. [3]
(c) The nth term of a sequence is an 2 + bn - 5.
The second term of this sequence is - 3 and the third term is 4.
Find the value of a and the value of b.
You must show all your working.
a = ................................................
b = ................................................ [6]
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11
NOT TO
SCALE
2.1 m
110°
110°
0.9 m
The diagram shows the symmetrical cross-section of a ditch containing water.
The angle between the base and each side of the ditch is 110°.
The width of the base is 0.9 m and the depth of the water is 2.1 m.
The ditch is 100 m long.
(a) Calculate the volume of water in the ditch.
.............................................m3 [4]
(b) On a different day, the ditch contains 300 m 3 of water.
Water is pumped out of the ditch at a rate of 4.2 litres per second.
Calculate the time taken to empty the ditch completely.
Give your answer in hours and minutes, correct to the nearest minute.
................... h ................... min [4]
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12
B
A
80°
9m
NOT TO
SCALE
16 m
D
22 m
115°
C
(a) Calculate the area of triangle BCD.
............................................ m2 [2]
(b) Calculate angle ADB.
Angle ADB = ................................................ [6]
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BLANK PAGE
© UCLES 2022
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BLANK PAGE
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge
Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.
Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.
© UCLES 2022
0607/41/M/J/22
Cambridge IGCSE™
* 1 1 5 2 7 3 1 1 8 0 *
CAMBRIDGE INTERNATIONAL MATHEMATICS
Paper 4 (Extended)
0607/43
May/June 2022
2 hours 15 minutes
You must answer on the question paper.
You will need:
Geometrical instruments
INSTRUCTIONS
●
Answer all questions.
●
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
●
Write your name, centre number and candidate number in the boxes at the top of the page.
●
Write your answer to each question in the space provided.
●
Do not use an erasable pen or correction fluid.
●
Do not write on any bar codes.
●
You should use a graphic display calculator where appropriate.
●
You may use tracing paper.
●
You must show all necessary working clearly and you will be given marks for correct methods, including
sketches, even if your answer is incorrect.
●
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
●
For r, use your calculator value.
INFORMATION
●
The total mark for this paper is 120.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 20 pages. Any blank pages are indicated.
DC (RW/JG) 303207/1
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2
Formula List
ax 2 + bx + c = 0
For the equation
x=
- b ! b 2 - 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2rrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = rrl
Curved surface area, A, of sphere of radius r.
A = 4rr 2
Volume, V, of pyramid, base area A, height h.
1
V = Ah
3
Volume, V, of cylinder of radius r, height h.
V = rr 2 h
Volume, V, of cone of radius r, height h.
1
V = rr 2 h
3
Volume, V, of sphere of radius r.
4
V = rr 3
3
A
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
b
c
1
Area = bc sin A
2
B
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C
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Answer all the questions.
1
(a) Anneka invests $2500 in an account paying compound interest at a rate of 1.6% per year.
Find the amount in the account at the end of 3 years.
$ ................................................. [2]
(b) Bashir invests $2500 in an account paying simple interest at a rate of r% per year.
At the end of 5 years the amount in the account is $2718.75 .
Calculate the value of r.
r = ................................................. [3]
(c) Chanda invests $2500 in an account paying compound interest at a rate of 1.55% per year.
Find the number of complete years until Chanda’s investment is first worth more than $4000.
................................................. [4]
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2
The heights, h cm, of 100 seedlings are shown in the table.
h cm
Frequency
4.5 1 h G 5.5
9
5.5 1 h G 6.5
18
6.5 1 h G 7.5
27
7.5 1 h G 8.5
19
8.5 1 h G 9.5
16
9.5 1 h G 10.5
11
Total
100
(a) Calculate an estimate for the mean.
............................................ cm [2]
(b) Write down the modal group.
.................. 1 h G .................. [1]
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(c) (i) Draw a cumulative frequency curve for the heights of the seedlings.
100
90
80
70
Cumulative
frequency
60
50
40
30
20
10
0
4
5
6
8
7
Height (cm)
9
11 h
10
[4]
(ii) Use your curve to estimate the median.
............................................ cm [1]
(iii) Use your curve to estimate the interquartile range.
............................................ cm [2]
(iv) Find an estimate of the percentage of the seedlings that were more than 8 cm in height.
.............................................. % [2]
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3
y
10
9
8
D
7
6
5
B
4
3
2
A
1
– 6 –5 – 4 –3 –2 –1 0
–1
1C2
3
4
5
6
7
8
9 10 x
–2
–3
–4
The diagram shows triangles A, B, C and D and the line with equation x + y = 9 .
(a) Enlarge triangle A with centre (4, 3) and scale factor 3.
[2]
(b) Describe fully the single transformation that maps triangle A onto
(i) triangle B,
.............................................................................................................................................
............................................................................................................................................. [2]
(ii) triangle C.
.............................................................................................................................................
............................................................................................................................................. [3]
(c) Triangle A can be mapped onto triangle D by
a rotation of 90° clockwise about a point on the line x + y = 9 followed by a reflection.
Find one possible centre of rotation and the equation of the corresponding mirror line.
Centre ( ....................... , ....................... )
Equation of mirror line ................................................. [2]
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4
(a) Solve 4x - 3 = 7 .
x = ................................................. [2]
y=
(b)
3x + 1
z
Find the value of y when x = 4.3 and z =- 2 .
y = ................................................. [2]
(c) Solve the simultaneous equations.
You must show all your working.
4x - 3y = 14
3x + 5y = 25
x = .................................................
y = ................................................. [4]
(d) Simplify
2x 2 + 4x x 2 - 4
'
.
10y
5y 2
................................................. [4]
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5
y
20
0
–3
3
x
– 10
f (x) = x 3 - 5x + 3 for - 3 G x G 3
(a) On the diagram, sketch the graph of y = f (x) .
[2]
(b) Find the coordinates of the local maximum.
( ...................... , ...................... ) [2]
(c) Describe fully the symmetry of the graph of y = f (x) .
.....................................................................................................................................................
..................................................................................................................................................... [3]
(d) Find the zeros of the graph of y = f (x) .
............................................................................................ [3]
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(e) g (x) = x 2 - 2x + 2 for - 3 G x G 3
(i) On the same diagram, sketch the graph of y = g (x) .
[2]
(ii) Use your graphs to solve x 3 - x 2 - 3x + 1 = 0 .
............................................................................................ [3]
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6
V
NOT TO
SCALE
12 cm
B
12 cm
A
O
D
C
VABC is a pyramid with a triangular base.
All the edges have length 12 cm.
O is vertically below V.
2
D is the mid-point of AC and BO = BD .
3
(a) Show that BO = 6.928 cm , correct to 3 decimal places.
[4]
(b) Calculate the volume of the pyramid.
.......................................... cm 3 [4]
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7
(a) Shade the region indicated below each of these Venn diagrams.
U
U
A
P
B
(A , B) l
Q
( P + Q l ) , ( P l + Q)
[2]
(b)
Bag A
Bag B
Bag A contains 4 white balls and 3 black balls.
Bag B contains 4 white balls and 5 black balls.
A ball is taken at random from bag A.
If the ball is white, it is replaced in Bag A.
If the ball is black, it is put in bag B.
A ball is then taken at random from bag B.
Find the probability that
(i) the ball taken from bag A is white,
................................................. [1]
(ii) both balls are black,
................................................. [2]
(iii) the balls are different colours.
................................................. [3]
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8
North
NOT TO
SCALE
B 120°
North
65 km
C
55°
A
The diagram shows the route of a ship between three ports, A, B and C.
The bearing of B from A is 055° and the bearing of C from B is 120°.
BC = 65 km .
The ship takes 7 hours to sail from A to B.
It sails at a speed of 20 km/h.
(a) Find the distance AB.
............................................ km [1]
(b) Show that angle ABC = 115° .
[1]
(c) (i) Calculate the distance CA.
............................................ km [3]
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(ii) Calculate the bearing of A from C.
................................................. [4]
(d) The ship takes 3.6 hours to sail from B to C.
It then sails from C to A at a speed of 21.5 km/h.
Find the average speed for the complete journey from A to B to C and back to A.
......................................... km/h [3]
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9
f (x) = 2 - 3x
g (x) = (x + 1) 2
h (x) = log x
(a) Find.
(i)
f ( - 4)
................................................. [1]
(ii)
f (g (3))
................................................. [2]
(iii)
f -1 (4)
................................................. [2]
(iv)
h -1 (2)
................................................. [2]
(b) Solve (f (x)) -1 = 5.
x = ................................................. [3]
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(c) Find g (f (x)) .
Write your answer in the form ax 2 + bx + c .
................................................. [3]
(d) y = h (f (x))
Find x in terms of y.
x = ................................................. [3]
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10 (a)
A
NOT TO
SCALE
E
54°
O
F
D
62°
B
C
G
A, B, C, D and E are points on the circle centre O.
FBG is a tangent to the circle at B.
Angle ABF = 62° and angle BED = 54° .
Find
(i) angle AEB,
Angle AEB = ................................................. [1]
(ii) angle BAD,
Angle BAD = ................................................. [1]
(iii) angle EAD,
Angle EAD = ................................................. [1]
(iv) angle BCD,
Angle BCD = ................................................. [1]
(v) angle FBD.
Angle FBD = ................................................. [1]
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(b)
A
NOT TO
SCALE
6 cm
P
O 120°
B
PA and PB are tangents to the circle centre O.
The radius of the circle is 6 cm and angle AOB = 120° .
The shaded area = (a 3 - br) cm 2 .
Find the value of a and the value of b.
a = .................................................
b = ................................................. [5]
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11
A tank has a capacity of 400 litres.
Water from Tap A flows at x litres per minute.
Water from Tap B flows at 2 litres per minute less than the water from tap A.
(a) Write down an expression in terms of x for the time, in minutes, for tap A to fill the tank.
................................................. [1]
(b) Tap B takes 10 minutes longer to fill the tank than tap A.
Write down an equation in terms of x and show that it simplifies to
x 2 - 2x - 80 = 0 .
[4]
(c) Solve x 2 - 2x - 80 = 0 and find the time it takes to fill the tank when both taps are turned on.
Give your answer in minutes and seconds, correct to the nearest second.
.......... minutes .......... seconds [4]
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