Cambridge IGCSE™
* 8 8 8 7 5 1 2 3 8 9 *
ADDITIONAL MATHEMATICS
0606/11
May/June 2024
Paper 1
2 hours
You must answer on the question paper.
No additional materials are needed.
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 calculator where appropriate.
●
You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
●
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.
INFORMATION
●
The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages.
DC (DE/JG) 332317/3
© UCLES 2024
[Turn over
2
Mathematical 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)
1-r
a
S3 =
( r 1 1)
1-r
Sn =
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
© UCLES 2024
0606/11/M/J/24
3
1
(a) On the axes, sketch the graph of
axes.
1
y =- (x + 2) (2x - 1) (x + 5) , stating the intercepts with the
5
[3]
y
O
1
(b) Hence solve the inequality - (x + 2) (2x - 1) (x + 5) H 0.
5
© UCLES 2024
0606/11/M/J/24
x
[2]
[Turn over
4
2
DO NOT USE A CALCULATOR IN THIS QUESTION.
The polynomial p is such that p (x) = 6x 3 - 35x 2 + 34x + 45.
(a) Find p (x) in the form (2x - 5) q (x) + r, where q (x) is a polynomial and r is a constant.
[3]
(b) Hence write the expression p (x) - 5 as a product of linear factors.
[2]
(c) Hence write down the solutions of the equation p (x) = 5.
[1]
© UCLES 2024
0606/11/M/J/24
5
3
(a) Write 1 + lg (x 2 - 1) - 2 lg (x - 1), where x 2 1, as a single logarithm to base 10. Give your
answer in its simplest form.
[4]
(b) Solve the equation 4 log 5 (x + 1) = 9 log (x + 1) 5, giving your answers in the form a + b c, where
a, b and c are constants.
[5]
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0606/11/M/J/24
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6
4
(a) The first three terms, in ascending powers of x, in the expansion of (3 + px) n are
243 + 810x + qx 2, where n, p and q are constants. Find the values of n, p and q.
(b) Find the term independent of y in the expansion of e2y -
© UCLES 2024
0606/11/M/J/24
6
[5]
1
o . Give your answer in exact form.
3y 2
[2]
7
5
(a) The diagram shows the graph of y = a cos bx + c, for - 360° G x G 360°, where a, b and c are
constants. Find the values of a, b and c.
[3]
y
2
1
– 360° – 300° – 240° – 180° – 120° – 60°
0
–1
60°
120°
180°
240°
300°
360°
x
–2
–3
–4
–5
–6
–7
–8
–9
– 10
(b) The line y = p is a tangent to the curve
© UCLES 2024
y = 3 - 2 sin 6i. Write down the possible values of p.
[2]
0606/11/M/J/24
[Turn over
8
6
c4 2
3
Find d e
o dx, giving your answer in exact form.
2
x
3
(3x - 5) 2
e2
[4]
7
Given that 2 + cot i = 3x and sin i = y, find y in terms of x.
[3]
© UCLES 2024
0606/11/M/J/24
9
8
Solve the equation 4 sin 2 b2a - l = 1 for - G a G . Give your answers in terms of r.
2
2
3
© UCLES 2024
r
r
0606/11/M/J/24
r
[5]
[Turn over
10
9
(a) Solve the following simultaneous equations.
e x + y # e 3x - 2y = 1
x 2 y = 256
© UCLES 2024
0606/11/M/J/24
[5]
11
(b) Solve the equation 10e (2x - 1) - 11 = 6e (1 - 2x), giving your answer in exact form.
© UCLES 2024
0606/11/M/J/24
[4]
[Turn over
12
10 In this question, all distances are in metres and time, t, is in seconds.
A particle P is at a fixed point O at time t = 0.
The velocity, v, of P is given by v = 3 sin 2t for t H 0.
(a) Find the exact value of t for which the velocity is zero for the first time after P leaves O.
[2]
(b) Find an expression, in terms of t, for the displacement of P from O at time t.
[4]
© UCLES 2024
0606/11/M/J/24
13
(c) Find the distance travelled by P for 0 G t G r.
© UCLES 2024
0606/11/M/J/24
[3]
[Turn over
14
11
1
The tangent to the curve y = (3x - 1) 3 at the point where x = 3 meets the coordinate axes at the
points A and B. The point with coordinates (a, a) lies on the perpendicular bisector of the line AB. Find
the exact value of a.
[10]
© UCLES 2024
0606/11/M/J/24
15
Continuation of working space for Question 11.
Question 12 is printed on the next page.
© UCLES 2024
0606/11/M/J/24
[Turn over
16
ln 3x
for x 2 0.
x2
dy
A + B ln 3x
, where A and B are integers.
Find . Give your answer in the form
dx
x3
12 (a) It is given that y =
(b) Hence find <
ln 3x
dx.
x3
[4]
[4]
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 2024
0606/11/M/J/24
* 0019655405201 *
,
,
Cambridge IGCSE™
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* 7 0 5 8 4 4 3 8 6 1 *
ADDITIONAL MATHEMATICS
0606/12
Paper 1
May/June 2024
2 hours
You must answer on the question paper.
No additional materials are needed.
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 calculator where appropriate.
●
You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
●
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.
INFORMATION
●
The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages.
DC (DE/CGW) 332388/3
© UCLES 2024
[Turn over
2
,
,
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
DO NOT WRITE IN THIS MARGIN
* 0019655405202 *
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
DO NOT WRITE IN THIS MARGIN
- b ! b 2 - 4ac
2a
DO NOT WRITE IN THIS MARGIN
x=
DO NOT WRITE IN THIS MARGIN
For the equation ax 2 + bx + c = 0 ,
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
© UCLES 2024
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0606/12/M/J/24
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Formulae for ∆ABC
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* 0019655405203 *
3
,
,
1
y
2
– 360°
– 180°
0
180°
360°
x
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–2
–4
–6
–8
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y = a sin bx + c for - 360c G x G 360c, where a, b and c are
The diagram shows the graph of
constants. Find the values of a, b and c.
[3]
Given that log 3 r + 2 log9 s = 8 , find the value of rs.
[3]
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2
© UCLES 2024
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0606/12/M/J/24
[Turn over
4
,
dy
x
r
Given that y = tan , find the exact value of
when x = .
3
2
dx
[4]
© UCLES 2024
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0606/12/M/J/24
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3
,
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* 0019655405204 *
5
,
4
,
A team of 8 people is to be formed from 6 teachers, 5 doctors and 4 police officers.
(a) Find the number of teams that can be formed.
[1]
(b) Find the number of teams that can be formed without any teachers.
[1]
(c) Find the number of teams that can be formed with the same number of doctors as teachers.
[4]
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* 0019655405205 *
© UCLES 2024
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Ĭ׶ìÁĨ°¶Û÷Āßī÷ûíĂ
ĥĥµÕõÕÅÕÕĕÅĥõĥąĕĥÕ
0606/12/M/J/24
[Turn over
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* 0019655405206 *
6
,
5
,
DO NOT USE A CALCULATOR IN THIS QUESTION.
In this question, all lengths are in centimetres.
A
8 7 -7
B
D
9 7 -9
C
The diagram shows the trapezium ABCD. The lengths of AB, BC and CD are 8 7 - 7 ,
9 7 - 9 respectively. The line BC is perpendicular to the lines AB and CD.
[3]
© UCLES 2024
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0606/12/M/J/24
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(b) Find the area of the trapezium, giving your answer in the form p 7 + q , where p and q are rational
numbers.
[3]
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(a) Find the perimeter of the trapezium, giving your answer in its simplest form.
7 + 2 and
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7 +2
7
,
,
(c) Find cot DBC , giving your answer in the form r 7 + s , where r and s are simplified rational
numbers.
[3]
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* 0019655405207 *
© UCLES 2024
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0606/12/M/J/24
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* 0019655405208 *
8
,
6
,
In this question, all lengths are in metres and all angles are in radians.
A
B
D
C
The diagram shows a circle with centre O and radius 5. The points A, B, C and D lie on the circumference
of the circle. Angle DOC = i . Angle AOD = angle COB = 0.5. The length of the minor arc DC is 3.75.
[1]
(b) Find the perimeter of the shaded region.
[5]
© UCLES 2024
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Ĭ׶êÆĢêÉÑČċĜíÛØăąĂ
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0606/12/M/J/24
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(a) Show that i = 0.75.
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i
5
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O
9
,
,
(c) Find the area of the shaded region.
[3]
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* 0019655405309 *
© UCLES 2024
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Ĭ׶ìÉĝÙßâĉ÷ßÆěسíĂ
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0606/12/M/J/24
[Turn over
10
,
(a) The line y = 3x - 2 intersects the curve 2x 2 - xy + y 2 = 2 at the points A and B. The point C
7
with coordinates bk, l lies on the perpendicular bisector of the line AB. Find the exact value of k.
8
[9]
© UCLES 2024
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0606/12/M/J/24
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7
,
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* 0019655405310 *
11
,
,
(b) The point D lies on the perpendicular bisector of AB such that D is a reflection of C in the line AB.
Find the coordinates of D.
[2]
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* 0019655405311 *
© UCLES 2024
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0606/12/M/J/24
[Turn over
12
,
,
1
Ax 2 + Bx + C
2
(3x 2 - 5) 3 (x + 4) 2
, where A, B and C are integers.
[5]
© UCLES 2024
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0606/12/M/J/24
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(b) Hence find the x-coordinates of the stationary points on the curve. Give your answers in their
simplest exact form.
[3]
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8
(3x 2 - 5) 3
A curve has equation y =
.
x+4
dy
can be written in the form
(a) Show that
dx
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* 0019655405312 *
13
,
9
,
In this question, all distances are in metres and time, t, is in seconds.
- 20
o.
A particle P moves with a speed of 14.5 parallel to the vector e
21
(a) Find the velocity vector of P.
[2]
3
Initially, P has position vector e o.
5
(b) Write down the position vector of P at time t.
[2]
-1
-5
o t at time t.
A second particle Q has position vector e o + e
3
7.5
(c) Find, in terms of t, the distance between P and Q at time t. Simplify your answer.
[4]
(d) Hence show that P and Q never collide.
[2]
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* 0019655405313 *
© UCLES 2024
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0606/12/M/J/24
[Turn over
14
,
,
10 (a) The first 3 terms of an arithmetic progression are 3 sin 2x , 5 sin 2x , 7 sin 2x .
2r
, find the exact sum of the first 20 terms.
3
[2]
© UCLES 2024
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0606/12/M/J/24
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(ii) Given that x =
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(i) Show that the sum to n terms of this arithmetic progression can be written in the form
n (n + a) sin 2x , where a is a constant.
[3]
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* 0019655405314 *
15
,
,
(b) The first 3 terms of a geometric progression are ln 2y , ln 4y 2 , ln 16y 4 .
(i) Find the nth term of this geometric progression.
[2]
(ii) Find the sum to n terms of this geometric progression, giving your answer in its simplest
form.
[2]
12
13
1
(c) The first 3 terms of a different geometric progression are b2w - l, b2w - l , b2w - l .
4
4
4
Find the values of w for which this geometric progression has a sum to infinity.
[3]
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© UCLES 2024
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Question 11 is printed on the next page.
0606/12/M/J/24
[Turn over
11
,
(a) Given that y = x 2 ln x , find
[2]
y x ln x dx .
[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 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 2024
ĬÍĉ¯Ġ³íÅõéāÝĪºę¶Ĕ×
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0606/12/M/J/24
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(b) Hence find
dy
.
dx
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16
,
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* 0019655405316 *
Cambridge IGCSE™
* 9 1 7 9 2 4 2 7 6 4 *
ADDITIONAL MATHEMATICS
0606/13
May/June 2024
Paper 1
2 hours
You must answer on the question paper.
No additional materials are needed.
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 calculator where appropriate.
●
You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
●
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.
INFORMATION
●
The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages. Any blank pages are indicated.
DC (PQ/SW) 332389/2
© UCLES 2024
[Turn over
3
1
y
6
4
2
– 320°
– 240°
– 160°
– 80°
0
80°
160°
240°
320°
x
–2
The diagram shows the graph of y = a sin bx + c , for - 320° G x G 320° , where a, b and c are
constants. Find the values of a, b and c.
[3]
2
Solve the equation 3 `2 2x + 1j - 11 `2 xj + 3 = 0 , giving your answers correct to 2 decimal places.
© UCLES 2024
0606/13/M/J/24
[4]
[Turn over
4
3
(a) Find the coordinates of the stationary points on the curve y = (2x + 1) 2 (x - 3) .
[4]
(b) On the axes, sketch the graph of y = (2x + 1) 2 (x - 3) , stating the intercepts with the axes.
[3]
y
O
© UCLES 2024
0606/13/M/J/24
x
5
(c) Write down the values of k for which the equation (2x + 1) 2 (x - 3) = k has exactly one solution.
[2]
4
Find
© UCLES 2024
y
2
0
2
`1 + e 2xj dx , giving your answer in exact form.
0606/13/M/J/24
[5]
[Turn over
6
5
When e 2y is plotted against x 3 , a straight line graph that passes through the points (2, 5) and (6.4, 7.2)
is obtained.
(a) Find y in terms of x.
[4]
(b) Find the values of x for which y exists.
[2]
© UCLES 2024
0606/13/M/J/24
7
6
It is given that y =
(a) Find
ln `2x 2 + 1j
, x ! -2 .
x+2
dy
.
dx
[3]
(b) Given that x increases from 1 to 1 + h , where h is small, find the approximate corresponding
change in y.
[2]
(c) When x = 1, the rate of change in y is 3 units per second. Find the corresponding rate of change
in x.
[2]
© UCLES 2024
0606/13/M/J/24
[Turn over
8
7
(a) A 6-digit number is to be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The 6-digit number
cannot start with 0. Each digit can be used at most once in any 6-digit number. Find how many of
these 6-digit numbers are divisible by 5.
[3]
(b) The number of combinations of (n + 1) objects taken 13 at a time is equal to 16 times the number
of combinations of n objects taken 12 at a time. Find the value of n.
[3]
© UCLES 2024
0606/13/M/J/24
9
8
1
The line L is the normal to the curve y = 3 (5x + 6) 2 at the point where x = 2 . The point (- 2, k) ,
where k is a constant, lies on L. Find the exact value of k.
[7]
© UCLES 2024
0606/13/M/J/24
[Turn over
10
9
In this question, all lengths are in metres, and time, t, is in seconds.
A particle P moves in a straight line such that, t seconds after leaving a fixed point O, its displacement,
s, is given by s = 4t - 4 cos 2t + 4 .
(a) Find the velocity, v, of P at time t.
[2]
(b) On the axes, sketch the velocity–time graph for P for 0 G t G r , stating the intercepts with the
axes in exact form.
[5]
v
O
© UCLES 2024
r
0606/13/M/J/24
t
11
(c) Find the acceleration of P at time t.
[1]
(d) Find the times when the acceleration of P is zero for 0 G t G r . Give your answers in terms of r .
[2]
© UCLES 2024
0606/13/M/J/24
[Turn over
12
10 (a) In an arithmetic progression, the first term is a and the common difference is d. The sum of the
first three terms of this arithmetic progression is 42. The product of the first three terms of this
arithmetic progression is -6720 .
(i) Show that a (a + 2d ) = -480 .
[3]
(ii) Hence, given that a is positive, find the values of a and d.
[4]
© UCLES 2024
0606/13/M/J/24
13
e 4x
e 11x
(b) In a geometric progression, the 3rd term is
and the 10th term is
. Find the first term and
4
512
the common ratio.
[5]
© UCLES 2024
0606/13/M/J/24
[Turn over
14
11
Solve the following simultaneous equations, giving your answers in exact form.
8 log 3 x + 12 log 81 y = 5
4 log 9 x + 3 log3 y = 2
© UCLES 2024
0606/13/M/J/24
[6]
15
J
N
r
r
r
12 Solve the equation sec KK3i - OO = 2 for - G i G . Give your answers in exact form.
2
2
2
L
P
© UCLES 2024
0606/13/M/J/24
[5]
Cambridge IGCSE™
* 1 8 4 6 3 1 0 2 4 0 *
ADDITIONAL MATHEMATICS
0606/21
May/June 2024
Paper 2
2 hours
You must answer on the question paper.
No additional materials are needed.
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 calculator where appropriate.
●
You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
●
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.
INFORMATION
●
The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages. Any blank pages are indicated.
DC (DE/SW) 332410/2
© UCLES 2024
[Turn over
2
Mathematical 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)
1-r
a
S3 =
( r 1 1)
1-r
Sn =
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
© UCLES 2024
0606/21/M/J/24
3
1
(a) On the axes, sketch the graph of
axes.
y = 4x - 6 , showing the points where the graph meets the
[2]
y
O
(b) Solve the equation
© UCLES 2024
4x - 6 = 2x .
x
[3]
0606/21/M/J/24
[Turn over
4
2
(a) Write
3 + 4x - 2x 2
in the form a + b (x + c) 2 , where a, b and c are integers.
(b) Hence write down the range of the function
3
f (x) = 3 + 4x - 2x 2 ,
where x ! R .
Use algebra to show that the equation 5x (x - 3) = 5x - 26 has no real solutions.
© UCLES 2024
0606/21/M/J/24
[3]
[1]
[3]
5
4
DO NOT USE A CALCULATOR IN THIS QUESTION.
(a) Find the exact distance between the two points where the curve 9 (x - 1) 2 + 4 (y - 3) 2 = 36
cuts the y-axis.
[4]
(b) Find the coordinates of the points where the curve with equation 2x 2 + 83xy = x 3 y - 20x
1
intersects the curve with equation y = . Give each of your answers in the form a + b c , where
x
a and b are rational and c is the smallest integer possible.
[6]
© UCLES 2024
0606/21/M/J/24
[Turn over
6
5
There are 3 women, 2 men and 4 children in a choir.
(a) The choir stands in a single straight line.
(i) Find the number of possible arrangements if the first person and last person are both women.
[2]
(ii) Find the number of possible arrangements if all the children stand next to each other.
[2]
(b) Four of the choir are selected to sing in a group.
(i) Find the number of different selections if no man is chosen.
[2]
(ii) Find the number of different selections if at least 2 women are chosen.
[2]
© UCLES 2024
0606/21/M/J/24
7
6
7
Variables x and y are such that y = cos x sin 2 x . Use differentiation to find the approximate change in
y as x increases from 3 to 3 + h , where h is small.
[5]
x
It is given that
y = mx 2 + + n , where m and n are non-zero constants. It is also given that
2
2
d2y
dy
e
o
3f 2 p =
- y for all values of x. Find the values of m and n.
[4]
dx
dx
© UCLES 2024
0606/21/M/J/24
[Turn over
8
8
(a) In an arithmetic progression, the sum of the first 30 terms is -1065.
The sum of the next 20 terms is - 2210 .
Find the first term and the common difference.
© UCLES 2024
0606/21/M/J/24
[5]
9
(b) A geometric progression is such that the first term is 4 and the sum of the first three terms is 7.
Find the two possible values of the common ratio and find the sum to infinity for the convergent
progression.
[5]
© UCLES 2024
0606/21/M/J/24
[Turn over
10
9
The functions f and g are defined by
3x 2
for x 1 0
4x - 1
1
g (x) = 2
for x 1 0 .
x
(a) Explain why the function fg does not exist.
[1]
(b) Given that the function gf does exist, find and simplify an expression for gf (x) .
[2]
f (x) =
(c) Show that f -1 (x) can be written as
© UCLES 2024
px - x (qx + r)
where p, q and r are integers.
3
0606/21/M/J/24
[4]
11
10 (a) Show that
(tan x + sec x) 2
can be written as
1 + sin x
.
1 - sin x
(b) Hence solve the equation (tan 3i + sec 3i) 2 = 6 for 0° G i G 180° .
© UCLES 2024
0606/21/M/J/24
[4]
[4]
[Turn over
12
11
In this question all lengths are in centimetres.
A
B
x
C
D
The diagram shows a rectangle ABCD with BC = x .
The area of the rectangle is 400 cm 2 .
x
Two identical quarter-circles of radius , with centres A and C, are removed from the rectangle to
2
make the shaded shape.
Given that x can vary, find the value of x that gives the minimum value of the perimeter of the shaded
shape and hence find this minimum value.
[7]
© UCLES 2024
0606/21/M/J/24
13
Continuation of working space for Question 11.
© UCLES 2024
0606/21/M/J/24
[Turn over
14
12
O
A
D
P
B
C
The diagram shows a triangle OBC.
OA : OB = 4 : 7 and OD : OC = 4 : 7.
OB = b and OC = c
The point P is the point of intersection of AC and BD such that AP = m AC and BP = n BD where m
and n are scalars.
(a) Find two expressions for OP , each in terms of b, c and a scalar, and hence show that P divides
both AC and DB in the ratio 4 : 7.
[7]
© UCLES 2024
0606/21/M/J/24
15
2
2
(b) The point Q is such that OQ = b + c .
7
7
Use a vector method to show that O, Q and P are collinear. Justify your answer.
© UCLES 2024
0606/21/M/J/24
[2]
* 0019655485301 *
,
,
Cambridge IGCSE™
¬OŠ. 3mEuW©M~S5 W
¬W8dK¡LPirwz/uVdm‚
¥• •5•Eu Uue55e•uU
* 6 3 8 4 3 5 3 9 0 0 *
ADDITIONAL MATHEMATICS
Paper 2
0606/22
May/June 2024
2 hours
You must answer on the question paper.
No additional materials are needed.
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 calculator where appropriate.
●
You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
●
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.
INFORMATION
●
The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 16 pages. Any blank pages are indicated.
DC (DE/SG) 332420/2
© UCLES 2024
[Turn over
2
,
,
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
DO NOT WRITE IN THIS MARGIN
* 0019655485302 *
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)
1-r
a
S3 =
( r 1 1)
1-r
Sn =
2. TRIGONOMETRY
Identities
sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A
DO NOT WRITE IN THIS MARGIN
- b ! b 2 - 4ac
2a
DO NOT WRITE IN THIS MARGIN
x=
DO NOT WRITE IN THIS MARGIN
For the equation ax 2 + bx + c = 0 ,
a
b
c
=
=
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2
© UCLES 2024
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ĥÅĕĕõõąĕµĕåĥµõÅÕÕÕ
0606/22/M/J/24
DO NOT WRITE IN THIS MARGIN
Formulae for ∆ABC
3
,
1
,
(a) On the axes, sketch the graph of
coordinate axes.
DO NOT WRITE IN THIS MARGIN
y = (2x - 5) (x + 3) (1 - x) ,
x
O
(b) Hence
(i) solve the inequality (2x - 5) (x + 3) (1 - x) G 0
(ii) on the axes below, sketch the graph of
[2]
y = (2x - 5) (x + 3) (1 - x) .
[1]
DO NOT WRITE IN THIS MARGIN
y
O
DO NOT WRITE IN THIS MARGIN
stating the intercepts with the
[3]
y
DO NOT WRITE IN THIS MARGIN
DO NOT WRITE IN THIS MARGIN
* 0019655485303 *
© UCLES 2024
ĬÏĊ®Ġ³íÅõ×ĩÍþÒď·Ğ×
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ĥÅĥÕµĕĥõåĥõĥµĕåĕÅÕ
0606/22/M/J/24
x
[Turn over
4
,
(a) Evaluate
,
r
2
x
cos dx . You must show all your working.
r
4
3
y
[4]
y e4x1- 3 + x1 o dx .
[3]
3
© UCLES 2024
ĬÍĊ®Ġ³íÅõ×ĩÍþÔč·Ġ×
Ĭ×µáÂĩ²¿Ô÷÷ÛñÛĔČíĂ
ĥõµÕõĕĥÕÅÅąĥõĕąĕÕÕ
0606/22/M/J/24
DO NOT WRITE IN THIS MARGIN
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(b) Find
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2
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* 0019655485304 *
5
,
3
,
(a) Determine whether the equation
roots or no real roots.
(4x + 1) (3x + 2)
= x + 1 has two distinct real roots, two equal
5x - 3
[4]
(b) Solve the equation
12 3
- x = 4.
x
[4]
3
DO NOT WRITE IN THIS MARGIN
DO NOT WRITE IN THIS MARGIN
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* 0019655485305 *
© UCLES 2024
ĬÏĊ®Ġ³íÅõ×ĩÍþÔď·Ġ×
Ĭ׶âÊğ®¯åāĊĎåãÈČýĂ
ĥõÅĕµõąµÕµÕĥõõĥÕÅÕ
0606/22/M/J/24
[Turn over
The polynomial p is such that p (x) = 6x 3 + x 2 - 12x + 5.
(a) Find the remainder when p (x) is divided by x - 2 .
[1]
(b) (i) Show that 2x - 1 is a factor of p (x) .
[1]
(ii) Hence write p (x) as a product of linear factors.
[3]
(iii) Hence solve the equation 6 sin 3 i + sin 2 i - 12 sin i + 5 = 0 for 0° G i G 90° .
[2]
DO NOT WRITE IN THIS MARGIN
,
© UCLES 2024
ĬÑĊ®Ġ³íÅõ×ĩÍþÑď¶Ğ×
Ĭ×µâÅĥÖµÝĆûħÅÛåäíĂ
ĥåĥĕµÕąĕĕÕõĥµõÅĕĕÕ
0606/22/M/J/24
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4
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6
,
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* 0019655485306 *
7
,
5
,
A curve has equation y = 5e 2x - 1 + e . The tangent to the curve at the point where x = 1 cuts the x-axis
at the point P.
Find the equation of the tangent in the form y = mx + c , where m and c are exact values, and hence find
the x-coordinate of P.
[6]
DO NOT WRITE IN THIS MARGIN
DO NOT WRITE IN THIS MARGIN
DO NOT WRITE IN THIS MARGIN
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* 0019655485307 *
© UCLES 2024
ĬÓĊ®Ġ³íÅõ×ĩÍþÑč¶Ğ×
Ĭ׶á½ģÚÅÜôĆâđãñäýĂ
ĥåĕÕõµĥõąååĥµĕåÕąÕ
0606/22/M/J/24
[Turn over
8
,
(a) Show that
sin 3 x b
cosec xl
cot x
,
can be written as sin 2 x tan x .
[3]
[5]
© UCLES 2024
ĬÑĊ®Ġ³íÅõ×ĩÍþÓď¶Ġ×
Ĭ׶ä½ĩìÄßîýé³ÿÓ´õĂ
ĥĕÅÕµµĥÕĥąÕĥõĕąÕĕÕ
0606/22/M/J/24
DO NOT WRITE IN THIS MARGIN
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1
(b) Solve the equation cos 2 x tan x - tan x = 0 for - r 1 x 1 r .
2
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6
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* 0019655485308 *
9
,
7
,
Find the number of different ways the 9 letters of the word POLYMATHS can be arranged when
(a) the O and A are not next to each other
[2]
(b) the letters MATHS are together in this order.
[2]
DO NOT WRITE IN THIS MARGIN
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* 0019655485409 *
© UCLES 2024
ĬÓĉ¯Ġ³íÅõ×ĩÍþÔď¶Ġ×
Ĭ׶âÂĦÛæÐïñĎČ¿ÓĄýĂ
ĥåĥĕµĕŵąąõĥµÕąĕåÕ
0606/22/M/J/24
[Turn over
,
An experiment was carried out and values of y for certain values of x were recorded. The table shows
the values recorded.
x
15
30
45
60
75
y
10
13
22
35
50
The relationship between y and x is modelled by y = Ae kx , where A and k are constants.
(a) Draw a straight line graph for ln y against x.
[2]
ln y
3
2
1
0
© UCLES 2024
15
ĬÑĉ¯Ġ³íÅõ×ĩÍþÑč¸Ğ×
Ĭ׸áÃĢÏÄçĂćñ´µģ¼ąĂ
ĥõõĕõõąÕµÅååµĕĥÕąÕ
30
45
0606/22/M/J/24
60
75 x
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4
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8
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10
,
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* 0019655485410 *
11
,
,
(b) Find the equation of the line in part (a) and hence find the values of A and k. Give each value
correct to 1 significant figure.
[5]
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* 0019655485411 *
(c) Find the value of x for which y = 17 .
DO NOT WRITE IN THIS MARGIN
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[2]
© UCLES 2024
ĬÓĉ¯Ġ³íÅõ×ĩÍþÑď¸Ğ×
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ĥõąÕµĕĥµåµõåµõąĕĕÕ
0606/22/M/J/24
[Turn over
9
,
y
B(5, 12)
y = 32x - 4x 2 - 48
D
O
A
C
x
© UCLES 2024
ĬÑĉ¯Ġ³íÅõ×ĩÍþÓč¸Ġ×
Ĭ×·ãËĞáµåúñ¯ÆġĕìýĂ
ĥÅÕÕõĕĥĕÅĕąåõõåĕąÕ
0606/22/M/J/24
DO NOT WRITE IN THIS MARGIN
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The diagram shows part of the curve y = 32x - 4x 2 - 48 and the line AB.
The curve and the line AB meet the x-axis at A and meet again at the point B(5, 12).
The line CD extended is parallel to the y-axis and passes through the maximum point of the curve.
Find the area of the shaded region.
[9]
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12
,
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* 0019655485412 *
13
,
,
Continuation of working space for Question 9.
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* 0019655485413 *
© UCLES 2024
ĬÓĉ¯Ġ³íÅõ×ĩÍþÓď¸Ġ×
Ĭ׸äÃĬÝÅÔĀĀúĒęÁìíĂ
ĥÅåĕµõąõÕĥÕåõĕÅÕĕÕ
0606/22/M/J/24
[Turn over
,
10 The functions f and fg are defined by
2
f (x) = e x + 3
fg (x) = e 2x
for x 1 0
3
for x 2 .
2
[1]
(b) Find an expression for f -1 (x) and state the domain and range of f -1 .
[5]
(c) Hence find and simplify an expression for g (x) .
[2]
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0606/22/M/J/24
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© UCLES 2024
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(a) Explain why f -1 exists.
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14
,
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* 0019655485414 *
15
,
11
,
xn
In the binomial expansion of b2 + l , the first three terms in increasing powers of x are
2
9
b + abx + abx 2 . Find the values of the constants n, a and b.
8
DO NOT WRITE IN THIS MARGIN
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* 0019655485415 *
© UCLES 2024
ĬÏĉ¯Ġ³íÅõ×ĩÍþÔϵĢ×
Ĭ׸áÊĤν×ðöĔĪ¼đĜąĂ
ĥĥÕĕµõåµåąąĥµµąÕÕÕ
0606/22/M/J/24
[8]
Cambridge IGCSE™
* 9 9 6 5 8 9 0 9 6 0 *
ADDITIONAL MATHEMATICS
0606/23
May/June 2024
Paper 2
2 hours
You must answer on the question paper.
No additional materials are needed.
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.
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Write your answer to each question in the space provided.
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Do not use an erasable pen or correction fluid.
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Do not write on any bar codes.
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You should use a calculator where appropriate.
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You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
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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.
INFORMATION
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The total mark for this paper is 80.
●
The number of marks for each question or part question is shown in brackets [ ].
This document has 12 pages.
DC (KN/SG) 332423/3
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2
Mathematical 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)
1-r
a
S3 =
( r 1 1)
1-r
Sn =
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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3
1
The point A has coordinates (1, 4) and the point B has coordinates (5, 6). The perpendicular bisector of
AB intersects the x-axis at the point C and the y-axis at the point D. Given that O is the origin, find the
area of triangle OCD.
[5]
2
Given that the equation kx 2 + (2k - 1) x + k + 1 = 0 has no real roots, find the set of possible values
of k.
[4]
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4
3
(a)
y
10
9
8
7
6
5
4
3
2
1
-2
-1
0
1
2
3
4
5
6
x
-1
-2
-3
Draw the graphs of y = 2x - 5
and y = 4 - x
(b) Use your graphs to solve the inequality
© UCLES 2024
for - 2 G x G 6 .
4 - x G 2x - 5 .
0606/23/M/J/24
[4]
[2]
5
4
(a) Find and simplify the term independent of x in the expansion of ex 2 -
10
1
o .
2x 3
[2]
(b) DO NOT USE A CALCULATOR IN THIS PART OF THE QUESTION.
(i) Use the binomial theorem to show that `1 + 2 2j - `1 - 2 2j = k 2 , where k is an integer
to be found.
[4]
4
`1 + 2 2j - `1 - 2 2j
4
(ii) Hence write
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1+ 2
4
4
in the form a + b 2 , where a and b are integers.
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[2]
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6
5
r
r
1 + 2 sin 2 x
for - 1 x 1 .
2
2
cos 2 x
2
(i) Show that f (x) can be written as a tan x + b , where a and b are integers.
[2]
(ii) Hence solve the equation f (x) = 4 .
[3]
(iii) Hence also find the gradient of the curve y = f (x) at each of the points where y = 4 .
[4]
(a) The function f is defined by f (x) =
© UCLES 2024
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7
(b) Solve the equation 50 cos 2 i = 5 sin i + 47 for 0° G i G 360° .
© UCLES 2024
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[5]
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8
6
DO NOT USE A CALCULATOR IN THIS QUESTION.
(a) Given that x - 3 and
2x 3 - 3x 2 - 8x - 3 = 0 .
x+1
are both factors of
2x 3 - 3x 2 - 8x - 3,
solve the equation
[2]
(b) The polynomial p (x) = x 3 + ax 2 + bx + c , where a, b and c are constants, has remainder - 5
4
when divided by x - 1. The curve y = p (x) has stationary points at x = and x = 2 .
3
(i) Find the values of a, b and c.
[7]
(ii) Hence use the second derivative test to show that the stationary point at x = 2 is a minimum.
[2]
© UCLES 2024
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9
7
C
D
O
i rad
A
5 cm
4 cm
B
In the diagram, AD and BC are arcs of circles with common centre O.
ODC and OAB are straight lines with OA = 5 cm and AB = 4 cm . Angle BOC = i radians .
The area of the shaded region ABCD is 4r cm 2 .
(a) Find i.
[3]
(b)
C
D
O
i rad
5 cm
A
4 cm
B
The straight line AC is added to the diagram and the region ACD is now shaded.
Find the perimeter of the shaded region ACD.
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[5]
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10
8
d2y
dy 3
3r r o
r
e
b
l
A curve is such that
2 = cos 4x - 4 . Given that dx = 4 at the point 16 , 4 on the curve, find
dx
the equation of the curve.
[7]
© UCLES 2024
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11
9
y
x=9
y = 4 + `3x - 1j
-1
A
O
B
x
The diagram shows a sketch of part of the curve y = 4 + (3x - 1) -1 and the line x = 9 .
The point A has x-coordinate 1. The tangent to the curve at A meets the x-axis at the point B.
Find the area of the shaded region.
[10]
Question 10 is printed on the next page.
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12
10
A
B
P
O
D
C
The diagram shows a parallelogram OABC. The point D divides the line OC in the ratio 2 : 3.
OA = a and OC = c
The point P lies on AD such that OP = m OB and AP = nAD , where m and n are scalars.
Find two expressions for OP , each in terms of a, c and a scalar, and hence show that P divides both
DA and OB in the ratio m : n, where m and n are integers to be found.
[7]
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