0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 1
19 July 2021 to 23 July 2021
Time allowed
140 minutes
(30 mins per day)
Score:
/ 93
Questions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marks
4
9
7
6
7
10
5
6
6
8
6
8
5
6
Scores
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Week 1
2
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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12
On the axes below, sketch the graph of the curve y = 2x 2 - 5x - 3 , stating the coordinates of any points
where the curve meets the coordinate axes.
y
10
–1
0
4
x
–10
[4]
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24
The polynomial p (x) = 2x 3 + ax 2 + bx - 49 ,
x + 3 there is a remainder of -24 .
(i)
where a and b are constants.
Show that 6a - b = 78 .
It is given that 2x - 1
When pl (x) is divided by
[2]
is a factor of p (x).
(ii)
Find the value of a and of b.
[4]
(iii)
Write p (x) in the form (2x - 1) Q (x) , where Q (x) is a quadratic factor.
[2]
(iv)
Hence factorise p (x) completely.
[1]
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35
It is given that log 4 x = p. Giving your answer in its simplest form, find, in terms of p,
(i)
log 4 (16x) ,
[2]
(ii)
log 4 f
[2]
x7
p.
256
Using your answers to parts (i) and (ii),
(iii)
solve log 4 (16x) - log 4 f
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x7
p = 5, giving your answer correct to 2 decimal places.
256
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47
Do not use a calculator in this question.
All lengths in this question are in centimetres.
A
2+3 5
B
10 - 2 5
D
6+3 5
C
The diagram shows the trapezium ABCD, where AB = 2 + 3 5 , DC = 6 + 3 5 , AD = 10 - 2 5 and
angle ADC = 90°.
(i)
Find the area of ABCD, giving your answer in the form a + b 5 , where a and b are integers.
(ii)
Find cot BCD, giving your answer in the form c + d 5 , where c and d are fractions in their simplest
form.
[3]
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7
59
The area of a sector of a circle of radius r cm is 36 cm2.
72
.
r
(i)
Show that the perimeter, P cm, of the sector is such that P = 2r +
(ii)
Hence, given that r can vary, find the stationary value of P and determine its nature.
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8
116 (a) Solve
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sin x cos x =
1
tan x for 0° G x G 180° .
2
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9
(b) (i)
(ii)
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Show that sec i -
Hence solve
sin i
= cos i .
cot i
sec 3i -
sin 3i 1
=
cot 3i 2
[3]
for -
2r
2r
, where i is in radians.
GiG
3
3
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10
73
The function f is defined, for 0c G x G 360c, by f (x) = a + b sin cx , where a, b and c are constants with
b 2 0 and c 2 0 . The graph of y = f (x) meets the y-axis at the point (0, -1), has a period of 120° and an
amplitude of 5.
(i)
Sketch the graph of y = f (x) on the axes below.
[3]
y
6
0
0°
60°
120°
180°
240°
300°
360°
x
–6
(ii)
Write down the value of each of the constants a, b and c.
a = .....................................
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[2]
b = ..................................... c = .....................................
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11
84
(a) Find the values of x for which `2x + 1j G 3x + 4 .
2
(b) Show that, whatever the value of k, the equation
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[3]
x2
+ kx + k 2 + 1 = 0 has no real roots.
4
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12
95
Solutions to this question by accurate drawing will not be accepted.
The points A(3, 2), B(7, -4), C(2, -3) and D(k, 3) are such that CD is perpendicular to AB. Find the
equation of the perpendicular bisector of CD.
[6]
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13
106
The relationship between experimental values of two variables, x and y, is given by y = Ab x , where A and
b are constants.
(i)
Transform the relationship y = Ab x into straight line form.
[2]
The diagram shows ln y plotted against x for ten different pairs of values of x and y. The line of best fit has
been drawn.
ln y
8
7
6
5
4
3
2
1
0
1
2
3
4
x
(ii)
Find the equation of the line of best fit and the value, correct to 1 significant figure, of A and of b. [4]
(iii)
Find the value, correct to 1 significant figure, of y when x = 2.7.
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14
11
7
dy
ax 2 + b
=
, where a, b and p are positive constants. [4]
dx `x 2 + 1j p
(i)
Given that y = x x 2 + 1 , show that
(ii)
Explain why the graph of y = x x 2 + 1 has no stationary points.
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15
129
(a) It is given that g (x) = 6x 4 + 5 for all real x.
(i)
Explain why g is a function but does not have an inverse.
[2]
(ii)
Find g 2 (x) and state its domain.
[2]
It is given that h (x) = 6x 4 + 5 for x G k .
(iii)
State the greatest value of k such that h-1 exists.
[1]
(iv)
For this value of k, find h-1(x).
[3]
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16
132
(i)
Write down the amplitude of 4 sin 3x - 1.
[1]
(ii)
Write down the period of 4 sin 3x - 1.
[1]
(iii)
On the axes below, sketch the graph of y = 4 sin 3x - 1 for - 90° G x° G 90° .
y
6
4
2
–90
–60
–30
0
30
60
90
x
–2
–4
–6
[3]
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17
143
The polynomial p (x) = (2x - 1) (x + k) - 12 , where k is a constant.
(i)
Write down the value of p (- k) .
[1]
When p(x) is divided by x + 3 the remainder is 23.
(ii)
Find the value of k.
[2]
(iii)
Using your value of k, show that the equation p (x) =- 25 has no real solutions.
[3]
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0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 2
26 July 2021 to 30 July 2021
Time allowed
140 minutes
(30 mins per day)
Score:
/ 90
Questions
1
2
3
4
5
6
7
8
9
10
11
12
Marks
8
9
8
9
9
6
4
9
8
9
5
6
13
14
Scores
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Week 2
2
19
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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20
17
(a) Solve log 3 x + log 9 x = 12 .
[3]
1
(b) Solve log 4 (3y 2 - 10) = 2 log 4 ( y - 1) + .
2
[5]
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21
28
It is given that f (x) = 5e x - 1 for x d R .
(i)
Write down the range of f.
[1]
(ii)
Find f -1 and state its domain.
[3]
It is given also that g (x) = x 2 + 4 for x d R .
(iii)
Find the value of fg(1).
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22
(iv)
Find the exact solutions of g 2 (x) = 40 .
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23
39
In this question all lengths are in centimetres.
A closed cylinder has base radius r, height h and volume V. It is given that the total surface area of the
cylinder is 600r and that V, r and h can vary.
(i)
Show that V = 300rr - rr 3 .
[3]
(ii)
Find the stationary value of V and determine its nature.
[5]
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24
104 When lg y is plotted against x2 a straight line graph is obtained which passes through the points (2, 4) and
(6, 16).
2
(i)
Show that y = 10 A + Bx , where A and B are constants.
(ii)
Find y when x =
(iii)
Find the positive value of x when y = 2.
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.
3
[4]
[2]
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5
11
It is given that y = `x 2 + 1j`2x - 3j2 .
1
(i)
Show that
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dy Px 2 + Qx + 1
, where P and Q are integers.
=
1
dx
`2x - 3j2
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(ii)
Hence find the equation of the normal to the curve y = `x 2 + 1j`2x - 3j2 at the point where x = 2,
giving your answer in the form ax + by + c = 0, where a, b and c are integers.
[4]
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36
(i)
Sketch the graph of y = 5x - 3
graph meets the coordinate axes.
on the axes below, showing the coordinates of the points where the
y
O
x
[3]
(ii)
Solve the equation
5x - 3 = 2 - x .
[3]
` 5 - 3j
2
47
Without using a calculator, express
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5 +1
in the form p 5 + q , where p and q are integers.
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8
B
8 cm
E
A
2r
rad
9
D
C
The diagram shows a right-angled triangle ABC with AB = 8 cm and angle ABC =
r
radians. The points D
2
and E lie on AC and BC respectively. BAD and ECD are sectors of the circles with centres A and C
2r
respectively. Angle BAD =
radians.
9
(i)
Find the area of the shaded region.
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(ii)
Find the perimeter of the shaded region.
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30
10
9
Solutions to this question by accurate drawing will not be accepted.
The points A and B have coordinates ( p, 3) and (1, 4) respectively and the line L has equation 3x + y = 2 .
1
, find the value of p.
3
[2]
(i)
Given that the gradient of AB is
(ii)
Show that L is the perpendicular bisector of AB.
[3]
(iii)
Given that C `q, - 10j lies on L, find the value of q.
[1]
(iv)
Find the area of triangle ABC.
[2]
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31
10
11
(a) (i)
(ii)
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Show that
Hence solve
cosec i - cot i
1
.
=
1 + cos i
sin i
cosec i - cot i 5
=
2
sin i
for 180° 1 i 1 360° .
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32
(b) Solve tan `3z - 4j =-
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r
1
for 0 G z G
radians.
2
2
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33
2
11
Do not use a calculator in this question.
Find the coordinates of the points of intersection of the curve y = (2x + 3) 2 (x - 1) and the line
y = 3 (2x + 3) .
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34
` pr 2j2 qr
3
4
12
(a) Given that
b and c.
(b) Solve
2`
q pr
2j-1
can be written in the form paqbr c, find the value of each of the constants a,
[3]
1
1
3x 2 - y - 2 = 4,
1
1
4x 2 + 3y - 2 = 14.
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0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 3
2 August 2021 to 6 August 2021
Time allowed
140 minutes
(30 mins per day)
Score:
/ 88
Questions
1
2
3
4
5
6
7
8
9
10
11
12
Marks
6
9
9
7
6
5
8
9
4
7
10
8
13
14
Scores
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Week 3
2
36
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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37
15
A
12 cm
9.6 cm
O
i rad
C
B
r
radians
2
and angle AOB = i radians. AC is an arc of the circle, centre O, radius 12 cm and AC has length 9.6 cm.
The diagram shows the right-angled triangle OAB. The point C lies on the line OB. Angle OAB =
(i)
Find the value of i.
[2]
(ii)
Find the area of the shaded region.
[4]
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38
28
When e y is plotted against
1
, a straight line graph passing through the points (2, 20) and (4, 8) is obtained.
x
(i)
Find y in terms of x.
[5]
(ii)
Hence find the positive values of x for which y is defined.
[1]
(iii)
Find the exact value of y when x = 3.
[1]
(iv)
Find the exact value of x when y = 2.
[2]
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3
10
h cm
4x cm
x cm
The diagram shows an open container in the shape of a cuboid of width x cm, length 4x cm and height h cm.
The volume of the container is 800 cm3.
(i)
Show that the external surface area, S cm2, of the open container is such that S = 4x 2 +
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40
(ii)
Given that x can vary, find the stationary value of S and determine its nature.
[5]
Question 11 is printed on the next page.
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41
114
The normal to the curve y = `x - 2j`3x + 1j3
Find the exact coordinates of the point P.
2
7
at the point where x = , meets the y-axis at the point P.
3
[7]
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35
(i)
Given that x - 2 is a factor of ax 3 - 12x 2 + 5x + 6 , use the factor theorem to show that a = 4 . [2]
(ii)
Showing all your working, factorise 4x 3 - 12x 2 + 5x + 6 and hence solve 4x 3 - 12x 2 + 5x + 6 = 0 .
[4]
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56
(i)
Express
5x 2 - 15x + 1 in the form p `x + qj + r , where p, q and r are constants.
2
(ii) Hence state the least value of x 2 - 3x + 0.2 and the value of x at which this occurs.
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44
97
(a) Solve
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6 sin 2 x - 13 cos x = 1 for 0° G x G 360° .
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(b) (i)
(ii) Hence solve
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r
r
Show that, for - 1 y 1 ,
2
2
integer.
4 tan y
2
1 + tan y
4 tan y
1 + tan 2 y
+ 3 = 0 for -
r
2
can be written in the form a sin y, where a is an
[3]
1y1
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r
2
radians.
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8
12
(a) The functions f and g are defined by
f (x) = 5x - 2 for x 2 1,
g (x) = 4x 2 - 9 for x 2 0 .
(i)
State the range of g.
[1]
(ii)
Find the domain of gf.
[1]
(iii)
Showing all your working, find the exact solutions of gf (x) = 4 .
[3]
Question 12(b) is printed on the next page.
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47
(b) The function h is defined by h (x) =
(i)
x 2 - 1 for x G- 1.
State the geometrical relationship between the graphs of y = h (x) and y = h -1 (x) .
(ii) Find an expression for h -1 (x) .
[1]
[3]
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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
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48
39
Show that the line y = mx + 4 will touch or intersect the curve y = x 2 + 3x + m for all values of m.
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49
10
5
(i)
On the axes below, sketch the graph of y = 3x 2 - 14x - 5 , showing the coordinates of the points
where the graph meets the coordinate axes.
y
–1
0
6
x
[4]
(ii)
Find the exact value of k such that
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50
6
11
(a) (i)
(ii)
Show that sec i -
Solve sec 2i -
tan i
= cos i .
cosec i
3
tan 2i
=
for 0c G i G 180c.
cosec 2i
2
(b) Solve 2 sin 2 bz + l = 1 for 0 1 z 1 2r radians.
3
r
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12
7
Do not use a calculator in this question.
In this question, all lengths are in centimetres.
A
2 5 -1
B
2 +2 5
C
The diagram shows the triangle ABC such that AB = 2 5 - 1, BC = 2 + 5 and angle ABC = 90c.
(i)
Find the exact length of AC.
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52
(ii)
Find tan ACB, giving your answer in the form p + q r , where p, q and r are integers.
[3]
(iii)
Hence find sec 2 ACB , giving your answer in the form s + t u where s, t and u are integers.
Congratulation, although you will not be graded, you have completed half of this
revision booklet. Please send an email to alvintan@cempaka.edu.my with the subject,
“(your name) has complete 50%”. Thank you.
[2]
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0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 4
9 August 2021 to 13 August 2021
Time allowed
140 minutes
(30 mins per day)
Score:
/ 94
Questions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marks
10
5
5
8
10
5
7
5
7
6
10
5
3
8
Scores
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Week 4
2
54
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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55
f | x 7 e 3x for x d R
g | x 7 2x 2 + 1 for x H 0
81
(i)
Write down the range of g.
[1]
(ii)
Show that f - 1 g ( 62) = ln 5 .
[3]
(iii)
Solve f l (x) = 6gll (x) , giving your answer in the form ln a, where a is an integer.
[3]
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56
(iv)
On the axes below, sketch the graph of y = g and the graph of y = g -1 , showing the points where the
graphs meet the coordinate axes.
y
O
x
[3]
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57
23
The points A, B and C have coordinates (4, 7), (-3, 9) and (6, 4) respectively.
(i)
Find the equation of the line, L, that is parallel to the line AB and passes through C. Give your answer
in the form ax + by = c, where a, b and c are integers.
[3]
(ii)
The line L meets the x-axis at the point D and the y-axis at the point E. Find the length of DE.
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58
34
The function f is defined, for 0° G x G 360°, by f (x) = 4 + 3 sin 2x .
(i)
Sketch the graph of y = f(x) on the axes below.
y
8
0
0°
60°
120°
180°
240°
300°
360°
x
–8
[3]
(ii)
State the period of f.
[1]
(iii)
State the amplitude of f.
[1]
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59
64
A curve has equation
y = `3x - 5j - 2x .
3
dy
d2y
and 2 .
dx
dx
(i)
Find
(ii)
Find the exact value of the x-coordinate of each of the stationary points of the curve.
[2]
(iii)
Use the second derivative test to determine the nature of each of the stationary points.
[2]
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60
5
12
Do not use a calculator in this question.
The line y = 4x - 6 intersects the curve y = 10x3 - 19x2 - x at the points A, B, and C. Given that C is
the point (2, 2), find the coordinates of the midpoint of AB.
[10]
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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
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61
63
2y
Given that 7 x # 49 y = 1 and 5 5x # 125 3 =
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, calculate the value of x and of y.
25
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62
57
f (x) = 3e 2x + 1
for x d R
g (x) = x + 1
for x d R
(i)
Write down the range of f and of g.
[2]
(ii)
Evaluate fg 2 (0) .
[2]
(iii)
On the axes below, sketch the graphs of y = f (x) and y = f - 1 (x) , stating the coordinates of the points
where the graphs meet the coordinate axes.
[3]
y
O
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63
68
1
Find the equation of the normal to the curve y = 8x + 5 at the point where x = , giving your answer in
2
[5]
the form ax + by + c = 0, where a, b and c are integers.
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64
97
When lg y is plotted against x, a straight line graph passing through the points (2.2, 3.6) and (3.4, 6) is
obtained.
(i)
Given that y = Ab x, find the value of each of the constants A and b.
[5]
(ii)
Find x when y = 900.
[2]
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65
10
8
Do not use a calculator in this question.
In this question, all lengths are in centimetres.
A
4+3 5
B
7+ 5
D
11 + 2 5
C
The diagram shows the trapezium ABCD in which angle ADC is 90° and AB is parallel to DC. It is given
that AB = 4 + 3 5, DC = 11 + 2 5 and AD = 7 + 5 .
(i)
Find the perimeter of the trapezium, giving your answer in simplest surd form.
[3]
(ii)
Find the area of the trapezium, giving your answer in simplest surd form.
[3]
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66
119
15 cm
A
B
10 cm
O
D
10 cm
C
The diagram shows a circle with centre O and radius 10 cm. The points A, B, C and D lie on the circle such
that the chord AB = 15 cm and the chord CD = 10 cm. The chord AB is parallel to the chord DC.
(i)
Show that the angle AOB is 1.70 radians correct to 2 decimal places.
[2]
(ii)
Find the perimeter of the shaded region.
[4]
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(iii)
Find the area of the shaded region.
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68
121
On the axes below, draw the graph of y = 2x - 3 .
(i)
y
8
6
4
2
0
-2
4 x
2
-2
-4
[2]
Solve the equation 7 - 2x - 3 = 0 .
(ii)
[3]
132
y
6
4
2
0
2r
r
x
-2
The figure shows part of the graph of y = p + q cos rx . Find the value of each of the integers p, q and r.
p=
q=
r=
[3]
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69
143
(a) Solve e 2x + 1 = 3e 4 - 3x .
[3]
(b) Solve lg (y - 6) + lg (y + 15) = 2 .
[5]
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0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 5
16 August 2021 to 20 August 2021
Time allowed
150 minutes
(30 mins per day)
Score:
/ 98
Questions
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Marks
5
7
10
5
5
7
7
7
8
11
5
5
7
9
Scores
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Week 5
2
71
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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72
14
Do not use a calculator in this question.
Solve the following simultaneous equations, giving your answers for both x and y in the form a + b 2 ,
where a and b are integers.
2x + y = 5
3x - 2 y = 7
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73
26
Do not use a calculator in this question.
The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B. The mid-point of AB is the
[7]
point C. Show that the point C lies on the line x + y = 11.
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74
37
(a) (i)
(ii)
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Use the factor theorem to show that 2x - 1 is a factor of p (x) , where p (x) = 4x 3 + 9x - 5 .
Write p (x) as a product of linear and quadratic factors.
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75
(b) (i)
(ii)
Show that 13 tan x sec x - 4 sin x - 5 sec 2 x = 0 can be written as 4 sin 3 x + 9 sin x - 5 = 0 .
Using your answers to part (a)(ii) and part (b)(i) solve the equation
13 tan x sec x - 4 sin x - 5 sec 2 x = 0 for 0 1 x 1 2r radians.
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76
41
(i)
On the axes below, sketch the graph of y = 2 cos 3x - 1 for - 90° G x G 90° .
y
4
3
2
1
-90°
-60°
0
-30°
30°
60°
90° x
-1
-2
-3
-4
[3]
(ii) Write down the amplitude of 2 cos 3x - 1.
(iii)
Write down the period of 2 cos 3x - 1.
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77
52
When lg y 2 is plotted against x, a straight line is obtained passing through the points (5, 12) and (3, 20).
[5]
Find y in terms of x, giving your answer in the form y = 10 ax + b , where a and b are integers.
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78
64
(i) On the axes below, sketch the graph of
where the graph meets the axes.
y = 2x 2 - 9x - 5
showing the coordinates of the points
[4]
y
x
O
(ii) Find the values of k for which
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2x 2 - 9x - 5 = k has exactly 2 solutions.
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79
75
(a) It is given that
f :x7 x
g : x 7 x+5
for x H 0 ,
for x H 0 .
Identify each of the following functions with one of
(i)
(ii)
f -1, g-1, fg, gf, f 2, g2.
[1]
x+5
[1]
x-5
(iii) x2
(iv)
[1]
x + 10
(b) It is given that
[1]
h (x) = a +
b
where a and b are constants.
x2
(i) Why is - 2 G x G 2 not a suitable domain for h(x)?
[1]
(ii) Given that h (1) = 4 and h l (1) = 16 , find the value of a and of b.
[2]
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80
86
qp 2
k
r
in the form p a q b r c , where a, b and c are constants.
-1 3
p
qr
pa
[3]
(b) Solve log 7 x + 2 log x 7 = 3.
[4]
(a) Write
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81
9
A solid circular cylinder has a base radius of r cm and a height of h cm. The cylinder has a volume of
1200r cm 3 and a total surface area of S cm2.
r
(i) Show that S = 2rr 2 + 2400
.
r
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(ii) Given that h and r can vary, find the stationary value of S and determine its nature.
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83
10
A
18 cm
B
10 cm
O
18 cm
C
The diagram shows a circle centre O, radius 10 cm. The points A, B and C lie on the circumference of the
circle such that AB = BC = 18 cm.
(i) Show that angle AOB = 2.24 radians correct to 2 decimal places.
[3]
(ii) Find the perimeter of the shaded region.
[5]
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84
Continuation of working space for Question 10(ii).
(iii) Find the area of the shaded region.
[3]
Question 11 is printed on the next page.
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4
85
112
Given that y = 2 sin 3x + cos 3x ,
determined.
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d 2 y dy
+ + 3y = k sin 3x ,
dx 2 dx
0606/22/O/N/19
where k is a constant to be
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6
86
12
4
Find the values of k for which the line y = kx + 3 does not meet the curve y = x 2 + 5x + 12 .
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87
13
5
At the point where x = 1 on the curve y =
(i) Find the value of the constant k.
k
1
2 , the normal has a gradient of 3 .
^x + 1h
[4]
(ii) Using your value of k, find the equation of the tangent to the curve at x = 2 .
Amazing, you truly enjoyed with Additional Mathematics homework. The solutions will
be posted to the schoology after the welcome Test, which will be held on the 1st and 2nd
day of the school reopening. The test questions are 100% similar to this revision
booklet. I wish you all the best!
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88
14
6
(i) Show that
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tan x
1 + sec x
2
.
+
=
tan x
1 + sec x
sin x
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89
(ii) Hence solve the equation
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tan x
1 + sec x
+
= 1 + 3 sin x for 0c G x G 180c.
tan x
1 + sec x
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0606 Additional Mathematics
Revision Booklet 1
Holiday Homework
IGCSE
0606 Additional Mathematics
Week 6
23 August 2021 to 27 August 2021
Time allowed
145 minutes
(30 mins per day)
Score:
/ 96
Questions
1
2
3
4
5
6
7
8
9
10
11
12
13
Marks
8
11
6
6
8
4
3
5
8
10
6
10
11
14
Scores
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Week 6
2
91
Mathematical
MathematicalFormulae
Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax 2 + bx + c = 0 ,
x=
- b ! b 2 - 4ac
2a
Binomial Theorem
n
n
n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1
2
r
n
n!
where n is a positive integer and e o =
(n - r) !r!
r
Arithmetic series
un = a + (n - 1) d
1
1
Sn = n (a + l ) = n {2a + (n - 1) d}
2
2
Geometric series
un = ar n - 1
a (1 - r n )
( r ! 1)
Sn =
1-r
a
S3 =
( r 1 1)
1-r
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
Formulae for ∆ABC
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
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10
92
17
(a) The cubic equation x 3 + ax 2 + bx - 40 = 0 has three positive integer roots. Two of the roots are 2
and 4. Find the other root and the value of each of the integers a and b.
[4]
(b) Do not use a calculator in this question.
Solve the equation x 3 - 5x 2 - 46x - 40 = 0 given that it has three integer roots, only one of which is
positive.
[4]
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93
29
P (r,s)
A (-3,5)
M
B (5,-1)
The diagram shows the points A (–3, 5) and B (5, –1). The mid-point of AB is M and the line PM is
perpendicular to AB. The point P has coordinates (r, s).
(i) Find the equation of the line PM in the form y = mx + c , where m and c are exact constants.
[5]
(ii) Hence find an expression for s in terms of r.
[1]
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13
94
(iii) Given that the length of PM is 10 units, find the value of r and of s.
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95
113
Do not use a calculator in this question.
Solve the quadratic equation ^ 5 - 3h x 2 + 3x + ^ 5 + 3h = 0 , giving your answers in the form a + b 5 ,
where a and b are constants.
[6]
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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.
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96
24
(i)
On the axes below, sketch the graph of y = 5 cos 4x - 3 for - 90° G x G 90°.
y
6
4
2
-90°
-45°
0
45°
90° x
-2
-4
-6
-8
- 10
[4]
(ii)
Write down the amplitude of y.
[1]
(iii)
Write down the period of y.
[1]
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97
58
(a) Given that log a x = p and log a y = q, find, in terms of p and q,
(i)
log a axy 2,
[2]
(ii)
log a e
x3 o
,
ay
[2]
(iii)
log x a + log y a.
[1]
(b) Using the substitution m = 3 x, or otherwise, solve 3 x - 3 1 + 2x + 4 = 0.
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98
69
P
Q
r cm
S
R
i rad
2r cm
O
The diagram shows a sector OPQ of the circle centre O, radius 3r cm. The points S and R lie on OP and OQ
respectively such that ORS is a sector of the circle centre O, radius 2r cm. The angle POQ = i radians. The
perimeter of the shaded region PQRS is 100 cm.
(i)
Find i in terms of r.
[2]
(ii)
Hence show that the area, A cm2, of the shaded region PQRS is given by A = 50r - r 2 .
[2]
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99
3x + 2 = x + 4 .
17
Solve
28
(i) Show that cosecx - cot x = cosecx .
1 - cos x
(ii) Hence solve cosecx - cot x = 2
1 - cos x
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for 0° 1 x 1 180° .
0606/23/O/N/19
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100
94
(i) Given that y = 2x 2 - 4x - 7 , write y in the form a (x - b) 2 + c , where a, b and c are constants.
[3]
(ii) Hence write down the minimum value of y and the value of x at which it occurs.
[2]
(iii) Using your answer to part (i), solve the equation 2p - 4 p - 7 = 0 , giving your answer correct to
2 decimal places.
[3]
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101
10
5
(a) Solve 3 cot 2 b y - rl = 1 for 0 1 y 1 r radians.
4
[4]
(b) Solve 7 cot z + tan z = 7 cosec z for 0° G z G 360° .
[6]
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102
116
Do not use a calculator in this question.
C
3– 3
A
3+ 3
B
(i) Find tan ACB in the form r + s 3 , where r and s are integers.
[3]
(ii) Find AC in the form t u, where t and u are integers and t ! 1.
[3]
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103
12
8
The roots of the equation x 3 + ax 2 + bx + 24 = 0 are 2, 3 and p, where p is an integer.
(i) Find the value of p.
[1]
(ii) Show that a =-1 and find the value of b.
[4]
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104
Given that a curve has equation y = x 3 - x 2 + bx + 24 find, using your value of b,
(iii)
dy
,
dx
[1]
(iv) the integer value of x for which the gradient of the curve is 2 and the corresponding value of y.
[3]
The coordinates of the point P on the curve are given by the values of x and y found in part (iv).
(v) Find the equation of the tangent to the curve at P.
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105
13 The functions f and g are defined by
10
2
f (x) = ln (3x + 2) for x 2 - ,
3
2x
g (x) = e - 4
for x ! R .
(i) Solve gf (x) = 5 .
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106
(ii) Find f -1 (x) .
[2]
(iii) Solve f -1 (x) = g (x) .
[4]
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