M17/5/FURMA/HP1/ENG/TZ0/XX
Further mathematics
Higher level
Paper 1
Wednesday 10 May 2017 (afternoon)
2 hours 30 minutes
Instructions to candidates
yy Do not open this examination paper until instructed to do so.
yy Answer all questions.
yy Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
yy A graphic display calculator is required for this paper.
yy A clean copy of the mathematics HL and further mathematics HL formula booklet is
required for this paper.
yy The maximum mark for this examination paper is [150 marks].
9 pages
2217 – 7101
© International Baccalaureate Organization 2017
–2–
M17/5/FURMA/HP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a solution,
you should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised to
show all working.
1.
[Maximum mark: 10]
The mean weight of a certain breed of bird is claimed to be 5.5 kg . In order to test this claim,
a random sample of 10 birds of the breed was obtained and weighed, with the following
results in kg .
5.41 5.22 5.54 5.58 5.20 5.57 5.23 5.32 5.46 5.37
You may assume that the weights of this breed of bird are normally distributed.
(a)
State suitable hypotheses for testing the above claim using a two-tailed test.
[1]
(b)
Calculate unbiased estimates of the mean and the variance of the weights of this breed
of bird.
[4]
(c)
(i)
Determine the p-value of the above data.
(ii)
State whether or not the claim is supported by the data, using a significance level
of 5 %.
2.
[5]
[Maximum mark: 7]
(a)
Consider the linear congruence ax ≡ b(mod p) where a , b ∈ + , p is a prime
and gcd(a , p) = 1 . Using Fermat’s little theorem, show that x ≡ a p - 2b(mod p) .
[3]
(b)
Hence find the smallest value of x greater than 100 satisfying the linear congruence
3x ≡ 13(mod 19) .
[4]
–3–
3.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 10]
Consider the system of equations
1
2

5

3
4.
2
1
1
3
1
3
8
4
3   x1   2 
1   x2   3 
=
.
0   x3   λ 
   
4   x4   µ 
(a)
Determine the value of λ and the value of µ for which the equations are consistent.
[5]
(b)
For these values of λ and µ , solve the equations.
[3]
(c)
State the rank of the matrix of coefficients, justifying your answer.
[2]
[Maximum mark: 11]
The weights of male students in a college are modelled by a normal distribution with mean
80 kg and standard deviation 7 kg.
The weights of female students in the college are modelled by a normal distribution with
mean 54 kg and standard deviation 5 kg.
(a)
Find the probability that the weight of a randomly chosen male student is more than
twice the weight of a randomly chosen female student.
[6]
The college has a lift installed with a recommended maximum load of 550 kg. One morning,
the lift contains 3 male students and 6 female students. You may assume that the 9 students
are randomly chosen.
(b)
Determine the probability that their combined weight exceeds the recommended
maximum.
[5]
Turn over
–4–
5.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 6]
(a)
Given that the series
∞
∑u
n =1
n
is convergent, where un > 0 , show that the series
∞
∑u
n =1
2
n
is
also convergent.
(b)
6.
[4]
(i)
State the converse proposition.
(ii)
By giving a suitable example, show that it is false.
[2]
[Maximum mark: 8]
The permutation P is given by
1 2 3 4 5 6
P=
.
3
4
5
6
2
1


7.
(a)
Determine the order of P , justifying your answer.
[2]
(b)
Find P 2 .
[2]
(c)
The permutation group G is generated by P . Determine the element of G that is of
order 2, giving your answer in cycle notation.
[4]
[Maximum mark: 13]
The function f is defined by
f ( x) =
(a)
(b)
e x + e − x + 2 cos x
, x∈ .
4
(i)
Show that f (4)(x) = f (x) ;
(ii)
By considering derivatives of f , determine the first three non-zero terms of the
Maclaurin series for f (x) .
[8]
The random variable X has a Poisson distribution with mean µ .
(i)
Write down a series in terms of µ for the probability p = P[X ≡ 0(mod 4)] .
(ii)
Show that p = e-µ f (µ) .
(iii)
Determine the numerical value of p when µ = 3 .
[5]
–5–
8.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
The normal at the point T(at 2 , 2at) , t ≠ 0 , on the parabola y2 = 4ax meets the parabola again
at the point S(as2 , 2as) .
9.
(a)
Show that t 2 + st + 2 = 0 .
[7]
(b)
 T is a right-angle, where O is the origin, determine the possible values
Given that SO
of t .
[5]
[Maximum mark: 13]
(a)
Using l’Hôpital’s rule, show that
lim x n e − x = 0 where n ∈  .
[4]
x →∞
(b)
Let
∞
I n = ∫ x n e − x dx where n ∈  .
1
(i)
Show that, for n ∈ + ,
I n = α e −1 + β nI n −1
where α , β are constants to be determined.
10.
(ii)
Determine the value of I3 , giving your answer as a multiple of e-1 .
[9]
[Maximum mark: 9]
Let G denote the set of 2 × 2 matrices whose elements belong to  and whose determinant
is equal to 1. Let ∗ denote matrix multiplication which may be assumed to be associative.
(a)
Show that {G , ∗} is a group.
[5]
Let H denote the set of 2 × 2 matrices whose elements belong to  and whose determinant
is equal to 1.
(b)
Determine whether or not {H , ∗} is a subgroup of {G , ∗} .
[4]
Turn over
–6–
11.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
The simple connected planar graph J has the following adjacency table.
A
B
C
D
E
F
G
H
(a)
A
–
0
1
1
1
0
0
0
B
0
–
1
1
1
1
1
1
C
1
1
–
1
1
0
1
1
D
1
1
1
–
1
0
0
0
E
1
1
1
1
–
1
1
0
F
0
1
0
0
1
–
1
0
G
0
1
1
0
1
1
–
1
H
0
1
1
0
0
0
1
–
Without attempting to draw J ,
(i)
verify that J satisfies the handshaking lemma;
(ii)
determine the number of faces in J .
(b)
The vertices D and G are joined by a single edge to form the graph K . Show that K
is not planar.
(c)
(i)
Explain why a graph containing a cycle of length three cannot be bipartite.
(ii)
Hence by finding a cycle of length three, show that the complement of K is not
bipartite.
[4]
[3]
[5]
–7–
12.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
The vertices A , B , C of an acute angled triangle have position vectors a , b , c with respect
to an origin O .
(a)
The mid-point of [BC] is denoted by D . The point E lies on [AD] such that
AE = 2DE .
(i)
Show that the position vector of E is
1
(a + b + c ) .
3
(ii)
Explain briefly why this result shows that the three medians of a triangle are
concurrent.
(b)
[5]
The perpendiculars from B to [AC] and C to [AB] meet at the point F .
(i)
Show that the position vector f of F satisfies the equations
(ii)
Show, by expanding these equations, that
(iii)
Explain briefly why this result shows that the three altitudes of a triangle are
concurrent.
(b - f )�(c - a) = 0
(c - f )�(a - b) = 0.
(a - f )�(c - b) = 0.
[7]
Turn over
–8–
13.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 7]
(
)
A random sample X1 , X2 , … , Xn is taken from the normal distribution N µ , σ 2 , where the
2
value of µ is unknown but the value of σ is known. The mean of the sample is denoted
−
by X .
(a)
X −µ
σ .
n
(i)
State the distribution of
(ii)
Hence show that, with probability 0.95,
X − 1.96
(b)
σ
σ
.
≤ µ ≤ X + 1.96
n
n
[5]
A mathematics teacher, wishing to apply the above result, generates some artificial
data, assumes a value for the variance and calculates the following 95 % confidence
interval for µ ,
[3.12 , 3.25].
The teacher asks Alun to interpret this result. Alun makes the following statement.
“The value of µ lies in the interval [3.12 , 3.25] with probability 0.95.”
(i)
Explain briefly why this is an incorrect statement.
(ii)
Give a correct interpretation.
14.
[2]
[Maximum mark: 9]
(a)
By writing 10 = 11 - 1 , use the binomial theorem to show that
10n ≡ (-1)n (mod 11) for n ∈  .
[3]
A number is called palindromic if it reads the same backwards as forwards, for example
524425.
(b)
(i)
Show that all palindromic numbers to base 10 with an even number of digits are
divisible by 11.
(ii)
By finding a suitable example, show that this is not necessarily true for
palindromic numbers to base 10 with an odd number of digits.
[6]
–9–
15.
M17/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 11]
(a)
The non-zero vectors v1 , v2 , v3 form an orthogonal set of vectors in 3 .
(i)
By considering α1v1 + α2v2 + α3v3 = 0 , show that v1 , v2 , v3 are linearly
independent.
(ii)
Explain briefly why v1 , v2 , v3 form a basis for vectors in 3 .
(i)
Show that the vectors
(b)
[6]
1   −1  1 
0  ;  1  ;  2 
     
1   1   −1
form an orthogonal basis.
(ii)
Express the vector
2
8 
 
 0 
as a linear combination of these vectors.
[5]
M18/5/FURMA/HP1/ENG/TZ0/XX
Further mathematics
Higher level
Paper 1
Thursday 17 May 2018 (afternoon)
2 hours 30 minutes
Instructions to candidates
yyDo not open this examination paper until instructed to do so.
yyAnswer all questions.
yyUnless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
yyA graphic display calculator is required for this paper.
yyA clean copy of the mathematics HL and further mathematics HL formula booklet is
required for this paper.
yyThe maximum mark for this examination paper is [150 marks].
7 pages
2218 – 7101
© International Baccalaureate Organization 2018
–2–
M18/5/FURMA/HP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a solution,
you should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised to
show all working.
1.
2.
[Maximum mark: 9]
(a)
Use the Euclidean algorithm to find the greatest common divisor of 74 and 383.
[4]
(b)
Hence find integers s and t such that 74s + 383t = 1 .
[5]
[Maximum mark: 6]
Let A2 = 2A + I where A is a 2 × 2 matrix.
(a)
Show that A4 = 12A + 5I .
[3]
4 2 
.
1 −3
Let B = 
k 0 
 , find the value of k .
0 k 
[3]
(a)
A number written in base 5 is 4303. Find this as a number written in base 10.
[2]
(b)
1000 is a number written in base 10. Find this as a number written in base 7.
[5]
(b)
3.
Given that B 2 – B – 4I = 
[Maximum mark: 7]
–3–
4.
M18/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
The transformations T1 , T2 , T3 , T4 , in the plane are defined as follows:
T1 : A rotation of 360 about the origin
T2 : An anticlockwise rotation of 270 about the origin
T3 : A rotation of 180 about the origin
T4 : An anticlockwise rotation of 90 about the origin.
(a)
(b)
Copy and complete the following Cayley table for the transformations of T1 , T2 , T3 , T4 ,
under the operation of composition of transformations.
T1
T2
T3
T4
T1
T1
T2
T3
T4
T2
T2
T3
T3
T4
T4
[2]
(i)
Show that T1 , T2 , T3 , T4 under the operation of composition of transformations
form a group. Associativity may be assumed.
(ii)
Show that this group is cyclic.
[4]
The transformation T5 is defined as a reflection in the x-axis.
(c)
Write down the 2 × 2 matrices representing T3 , T4 and T5 .
(d)
The transformation T is defined as the composition of T3 followed by T5 followed
by T4 .
(i)
Find the 2 × 2 matrix representing T .
(ii)
Give a geometric description of the transformation T .
5.
[3]
[3]
[Maximum mark: 7]
Use the integral test to determine whether or not
∞
1
n=2
( ln n )
∑n
2
converges.
[7]
Turn over
–4–
6.
M18/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 9]
(a) Consider the integers between 1 and 20 inclusive.
Let A = {multiples of 2} , B = {multiples of 3} , C = {multiples of 4} .
Find the elements in each of the following sets,
(i)
A ∩ (B ∪ C) ;
(ii)
A \ (B \ C) .
(b)
7.
Let M = {x : x is an integer multiple of 10} and let N = {x : x is an integer multiple of 5}
Prove that M is a proper subset of N .
[5]
[4]
[Maximum mark: 9]
A sample of size 100 is taken from a normal population with unknown mean µ and known
variance 36.
(a)
An investigator wishes to test the hypotheses H0 : µ = 65 , H1 : µ > 65 .
He decides on the following acceptance criteria:
Accept H0 if the sample mean x ≤ 66.5
Accept H1 if x > 66.5
Find the probability of a Type I error.
(b)
[3]
Another investigator decides to use the same data to test the hypotheses
H0 : µ = 65 , H1 : µ = 67.9 .
(i)
(ii)
She decides to use the same acceptance criteria as the previous investigator.
Find the probability of a Type II error.
Find the critical value for x if she wants the probabilities of a Type I error and a
Type II error to be equal.
[6]
–5–
8.
M18/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 13]
Consider the simultaneous linear equations
where a and b are constants.
9.
10.
x+z=-1
3x + y + 2z = 1
2x + ay - z = b
(a)
Using row reduction, find the solutions in terms of a and b when a ≠ 3 .
[8]
(b)
Explain why the equations have no unique solution when a = 3 .
[1]
(c)
Find all the solutions to the equations when a = 3 , b = 10 in the form r = s + λt .
[4]
[Maximum mark: 13]
(a)
Given that A is the interval {x : 0 ≤ x ≤ 3} and B is the interval {y : 0 ≤ y ≤ 4} then
describe A × B in geometric form.
(b)
Let f :  ×  →  ×  be defined by f (x , y) = (x + 3y , 2x - y) .
(i)
Show that the function f is a bijection.
(ii)
Hence find the inverse function f -1 .
[3]
[10]
[Maximum mark: 12]
(a)
By considering the images of the points (1, 0) and (0, 1),
(i)
(ii)
determine the 2 × 2 matrix P which represents a reflection in the line
y = (tan θ)x ;
determine the 2 × 2 matrix Q which represents an anticlockwise rotation of θ
about the origin.
[5]
(b)
Describe the transformation represented by the matrix PQ .
[5]
(c)
A matrix M is said to be orthogonal if M TM = I where I is the identity. Show that Q is
orthogonal.
[2]
Turn over
–6–
11.
M18/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
y−x
Given that y is a function of x , the function z is given by z =
, where x ∈  , x ≠ 3 ,
y+x
y + x ≠ 0.
dz
2
 dy

=
x − y.
2 
dx ( y + x )  d x

(a)
Show that
(b)
Show that the differential equation f ( x)  x
f ( x)
(c)
[3]
 dy

− y  = y 2 − x 2 can be written as
 dx

dz
= 2z .
dx
[2]
 dy

− y  = y2 − x2
 dx

Hence show that the solution to the differential equation ( x − 3)  x
y − x  x −3
=
given that x = 4 when y = 5 is
 .
y+x  3 
2
12.
[7]
[Maximum mark: 15]
(a)
Solve the recurrence relation un = 4un-1 - 4un-2 given that u0 = u1 = 1 .
[6]
Consider vn which satisfies the recurrence relation 2vn = 7vn-1 - 3vn-2 subject to the initial
conditions v0 = v1 = 1 .
n
(b)
13.
41 1
Prove by using strong induction that vn =   + (3) n for n ∈  .
52 5
[9]
[Maximum mark: 9]
 2 −4 
.
 −1 −1
Consider the matrix M = 
(a)
Show that the linear transformation represented by M transforms any point on the line
y = x to a point on the same line.
[2]
(b)
Explain what happens to points on the line 4y + x = 0 when they are transformed
by M .
[3]
(c)
State the two eigenvalues of M .
[2]
(d)
State two eigenvectors of M which correspond to the two eigenvalues.
[2]
–7–
14.
M18/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 8]
At an early stage in analysing the marks scored by candidates in an examination paper,
the examining board takes a random sample of 250 candidates and finds that the marks, x ,
x = 10985 and
x 2 = 598736 .
of these candidates give
∑
15.
∑
(a)
Calculate a 90% confidence interval for the population mean mark µ for this paper.
[4]
(b)
The null hypothesis µ = 46.5 is tested against the alternative hypothesis µ < 46.5
at the λ% significance level. Determine the set of values of λ for which the null
hypothesis is rejected in favour of the alternative hypothesis.
[4]
[Maximum mark: 9]
Given that the tangents at the points P and Q on the parabola y2 = 4ax are perpendicular,
find the locus of the midpoint of PQ.
[9]
No part of this product may be reproduced in any form or by any electronic
or mechanical means, including information storage and retrieval systems,
without written permission from the IB.
Additionally, the license tied with this product prohibits commercial use of
any selected files or extracts from this product. Use by third parties,
including but not limited to publishers, private teachers, tutoring or study
services, preparatory schools, vendors operating curriculum mapping
services or teacher resource digital platforms and app developers, is not
permitted and is subject to the IB’s prior written consent via a license. More
information on how to request a license can be obtained from http://
www.ibo.org/contact-the-ib/media-inquiries/for-publishers/guidance-forthird-party-publishers-and-providers/how-to-apply-for-a-license.
Aucune partie de ce produit ne peut être reproduite sous quelque forme ni
par quelque moyen que ce soit, électronique ou mécanique, y compris des
systèmes de stockage et de récupération d’informations, sans l’autorisation
écrite de l’IB.
De plus, la licence associée à ce produit interdit toute utilisation
commerciale de tout fichier ou extrait sélectionné dans ce produit.
L’utilisation par des tiers, y compris, sans toutefois s’y limiter, des éditeurs,
des professeurs particuliers, des services de tutorat ou d’aide aux études,
des établissements de préparation à l’enseignement supérieur, des
fournisseurs de services de planification des programmes d’études, des
gestionnaires de plateformes pédagogiques en ligne, et des développeurs
d’applications, n’est pas autorisée et est soumise au consentement écrit
préalable de l’IB par l’intermédiaire d’une licence. Pour plus d’informations
sur la procédure à suivre pour demander une licence, rendez-vous à
l’adresse http://www.ibo.org/fr/contact-the-ib/media-inquiries/for-publishers/
guidance-for-third-party-publishers-and-providers/how-to-apply-for-alicense.
No se podrá reproducir ninguna parte de este producto de ninguna forma ni
por ningún medio electrónico o mecánico, incluidos los sistemas de
almacenamiento y recuperación de información, sin que medie la
autorización escrita del IB.
Además, la licencia vinculada a este producto prohíbe el uso con fines
comerciales de todo archivo o fragmento seleccionado de este producto. El
uso por parte de terceros —lo que incluye, a título enunciativo, editoriales,
profesores particulares, servicios de apoyo académico o ayuda para el
estudio, colegios preparatorios, desarrolladores de aplicaciones y
entidades que presten servicios de planificación curricular u ofrezcan
recursos para docentes mediante plataformas digitales— no está permitido
y estará sujeto al otorgamiento previo de una licencia escrita por parte del
IB. En este enlace encontrará más información sobre cómo solicitar una
licencia: http://www.ibo.org/es/contact-the-ib/media-inquiries/for-publishers/
guidance-for-third-party-publishers-and-providers/how-to-apply-for-alicense.
M19/5/FURMA/HP1/ENG/TZ0/XX
Further mathematics
Higher level
Paper 1
Thursday 23 May 2019 (afternoon)
2 hours 30 minutes
Instructions to candidates
yyDo not open this examination paper until instructed to do so.
yyAnswer all questions.
yyUnless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
yyA graphic display calculator is required for this paper.
yyA clean copy of the mathematics HL and further mathematics HL formula booklet is
required for this paper.
yyThe maximum mark for this examination paper is [150 marks].
8 pages
2219 – 7101
© International Baccalaureate Organization 2019
–2–
M19/5/FURMA/HP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a solution,
you should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised to
show all working.
1.
[Maximum mark: 12]
The graph G with vertices A , B , C , D and E has the weights shown in the following table.
A
B
C
D
E
2.
A
–
8
11
17
12
B
8
–
14
9
13
C
11
14
–
16
10
D
17
9
16
–
15
E
12
13
10
15
–
(a)
Justifying your answer, explain whether or not G contains an Eulerian circuit.
[2]
(b)
Prove that G cannot be drawn as a planar graph.
[3]
(c)
Starting at A , use the nearest-neighbour algorithm to find an upper bound for the
travelling salesman problem for G .[3]
(d)
By deleting vertex A , use the deleted vertex algorithm to find a lower bound for this
travelling salesman problem.
[4]
[Maximum mark: 9]
The function f is defined for x ≥ 0 by f (x) = ln(2ex - 1) .
(a)
Determine the Maclaurin series for f (x) as far as the term in x3 .[7]
(b)
Hence determine the value of
lim
x 0
f ( x) 2 x
.[2]
x2
–3–
3.
[Maximum mark: 12]
(a)
(b)
4.
M19/5/FURMA/HP1/ENG/TZ0/XX
a b .
c d The matrix A is given by A (i)
Show that the eigenvalues of A are real if (a - d )2 + 4bc ≥ 0 .
(ii)
Deduce that the eigenvalues are real if A is symmetric.
[6]
3 2
.
2 3
The matrix B is given by B (i)
Determine the eigenvalues of B .
(ii)
Determine the corresponding eigenvectors.
[6]
[Maximum mark: 8]
The positive integer N is given by 1321 when expressed in base b and 521 when expressed
in base b + 2 .
5.
(a)
Determine the value of b .[4]
(b)
Express N
(i)
in base 10 ;
(ii)
in base 16 .
[4]
[Maximum mark: 9]
Consider the differential equation
�
dy
2 y tan x sin x where 0 ≤ x <
2
dx
Given that y = 2 when x = 0 , solve the differential equation giving your answer in the
form y = f (x) .
[9]
Turn over
–4–
6.
M19/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 8]
The group {G , *} has the following Cayley table.
*
0
1
2
3
4
5
7.
0
4
5
0
1
2
3
1
5
2
1
4
3
0
2
0
1
2
3
4
5
3
1
4
3
0
5
2
4
2
3
4
5
0
1
5
3
0
5
2
1
4
(a)
Determine the order of each of the elements of {G , *} .
(b)
Hence find the subgroup S2 of order 2 and the subgroup S3 of order 3.[2]
(c)
Write down the coset with respect to S2 of each element of {G , *} not included in S2 .[2]
[4]
[Maximum mark: 12]
(2) n x n
.[5]
n
n 1
(a)
Determine the radius of convergence of the power series
(b)
(i)
Use l’Hôpital’s rule to determine the value of lim
(ii)
Use the limit comparison test together with an appropriate series to determine
ln 1 x .
x 0
x
whether the series
1
ln 1 n is convergent or divergent.
[7]
n 1
8.
[Maximum mark: 7]
The line AD is a median of the acute-angled triangle ABC and E is the midpoint of AD.
The line BE meets AC at the point F.
(a)
Draw a diagram to illustrate this situation.
(b)
Determine the value of the ratio
(c)
[1]
CF
.[4]
AF
The line CE meets AB at the point G. Giving a reason, write down the value of the
BG
ratio
.[2]
AG
–5–
9.
M19/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
Consider the system of equations
1 2 5 x 1 2 3 8 y 3 4 1 6 z 1 2 5 where the matrix 2 3 8 is singular and µ is a constant.
4 1 6 10.
(a)
Determine the value of µ for which the equations are consistent.
(b)
For this value of µ
[4]
(i)
solve the system of equations;
(ii)
find the values of x , y and z which minimize x2 + y2 + z2 and interpret your result
geometrically.[8]
[Maximum mark: 10]
The continuous random variable X has cumulative distribution function F given by
0, x < 0


F ( x) =  2
.
0
≤
<
∞
arctan
x
,
x
 �
(a)
(b)
(i)
Sketch the graph of F(x) for x ≥ 0 .
(ii)
Explain how it can be deduced from the graph of F(x) that the mode of X is zero.
(iii)
Determine the median of X .[5]
It is often stated that for certain probability distributions, the following approximation is true:
Median – Mode ≈ 2(Mean – Median) .
Explain why this approximation is not valid for the probability distribution defined above.
[5]
Turn over
–6–
11.
M19/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 10]
(a)
(i)
Show that the set S1 of three-dimensional vectors given by
1 3 2 S1 5 , 5 , 8 2 1 4 is a basis for three-dimensional vectors.
(b)
(ii)
9
Express the vector 17 in terms of S1 .
3 (i)
Show that the set S2 of three-dimensional vectors given by
[5]
1 3 2 S 2 5 , 5 , 8 2 1 3 is not a basis for three-dimensional vectors.
12.
(ii)
State the dimension of the subspace spanned by S2 .
(iii)
2
Determine whether or not the vector 7 belongs to this subspace.
2 [5]
[Maximum mark: 9]
The relation R is defined on + such that xRy if and only if x2 - y2 ≡ 0 (mod N ) where N ≥ 3
is a positive integer.
(a)
Show that R is an equivalence relation for all values of N .[6]
(b)
Show that N - 1 and N + 1 are in the same equivalence class as 1.[3]
–7–
13.
M19/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 11]
The function f : M → M where M is the set of 2 × 2 matrices, is given by f (X ) = AX
where A is a 2 × 2 matrix.
(a)
Given that A is non-singular, prove that f is a bijection.
[7]
It is now given that A is singular.
(b)
14.
By considering appropriate determinants, prove that f is not a bijection.
[4]
[Maximum mark: 12]
The Poisson random variable X with mean m has probability function
P( X x) (a)
e m m x
, x∈.
x!
Show that the probability generating function of X is given by
Gx (t ) e m (t 1) .[3]
(b)
A random sample X 1 , X 2 , X 3 is taken from the distribution of X . The random
variable Y is defined by Y = X 1 + 2X 2 + 3X 3 .
(i)
3 m m ( t t
Show that the probability generating function of Y is given by G y (t ) e e
(ii)
3 m m ( t t
G y (t ) eof
e
By considering the series expansion
terms of m for P(Y = 4) .
2
t3 )
, determine an expression in
2
t3 )
.
[9]
Turn over
–8–
15.
M19/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 9]
An ellipse E has equation x2 + 2y2 = 2 . The point P has coordinates (x1 , y1) and is external
to the ellipse.
(a)
Write down the equation of the line L with gradient m passing through the point P .[1]
(b)
Show that the x coordinates of the points of intersection of the line L and the ellipse E
are given by the roots of the quadratic equation
x2 (1 + 2m2) + 4mx ( y1 - mx1) + 2y12 + 2m2x12 - 4mx1 y1 - 2 = 0 .[3]
(c)
Show that the condition for the line L to be a tangent to E is given by
m2 (x12 - 2) - 2mx1 y1 + y12 - 1 = 0 .[3]
(d)
Hence show that the equation of the locus of points from which the two tangents to E
are perpendicular is x2 + y2 = 3 .
[2]
M16/5/FURMA/HP1/ENG/TZ0/XX
Further mathematics
Higher level
Paper 1
Thursday 19 May 2016 (afternoon)
2 hours 30 minutes
Instructions to candidates
yyDo not open this examination paper until instructed to do so.
yyAnswer all questions.
yyUnless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
yyA graphic display calculator is required for this paper.
yyA clean copy of the mathematics HL and further mathematics HL formula booklet is
required for this paper.
yyThe maximum mark for this examination paper is [150 marks].
8 pages
2216 – 7101
© International Baccalaureate Organization 2016
–2–
M16/5/FURMA/HP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a solution,
you should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised to
show all working.
1.
[Maximum mark: 12]
The set P contains all prime numbers less than 2500.
The set Q is the set of all subsets of P .
(a)
Explain why only one of the following statements is true
(i)
17 ⊂ P ;
(ii)
{7 , 17 , 37 , 47 , 57} ∈ Q ;
(iii)
φ ⊂ Q and φ ∈ Q, where φ is the empty set.
[4]
The set S contains all positive integers less than 2500.
The function f : S → Q is defined by f (s) as the set of primes exactly dividing s , for s ∈ S .
For example f (4) = {2} , f (45) = {3 , 5} .
(b)
(i)
(ii)
(c)
State the value of f (1) , giving a reason for your answer.
Find n( f (2310)) .
[4]
Determine whether or not f is
(i)
injective;
(ii)
surjective.
[4]
–3–
2.
M16/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 11]
The lifetime, in years, of a randomly chosen basic vacuum cleaner is assumed to be
modelled by the normal distribution B ∼ N(14 , 32) .


1

Var ( B)  .
2

(a)
Find P  B > E ( B ) +
(b)
Find the probability that the total lifetime of 7 randomly chosen basic vacuum cleaners
is less than 100 years.
[2]
[4]
The lifetime, in years, of a randomly chosen robust vacuum cleaner is assumed to be
modelled by the normal distribution R ∼ N(20 , 42) .
(c)
3.
Find the probability that the total lifetime of 5 randomly chosen robust vacuum cleaners
is greater than the total lifetime of 7 randomly chosen basic vacuum cleaners.
[5]
[Maximum mark: 8]
Consider the Diophantine equation 7x - 5y = 1 , x , y ∈  .
(a)
Find the general solution to this equation.
[3]
(b)
Hence find the solution with minimum positive value of xy .
[2]
(c)
Find the solution satisfying xy = 2014 .
[3]
Turn over
–4–
4.
M16/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 10]
All members of a large athletics club take part in an annual shotput competition.
The following data give the distances achieved, in metres, by a random selection of
10 members of the club in the 2016 competition
11.8 , 14.3 , 13.8 , 10.3 , 14.9 , 14.7 , 12.4 , 13.9 , 14.0 , 11.7
The president of the club wishes to test whether these data provide evidence that distances
achieved have increased since the 2015 competition, when the mean result for the club
was 12.4 m. You may assume that the distances achieved follow a normal distribution with
mean µ , variance σ 2 , and that the membership of the club has not changed from 2015
to 2016.
(a)
State suitable hypotheses.
(b)
(i)
Give a reason why a t test is appropriate and write down its degrees of freedom.
(ii)
Find the critical region for testing at each of the 5 % and 10 % significance levels.
(i)
Find unbiased estimates of µ and σ 2 .
(ii)
Find the value of the test statistic.
(c)
(d)
5.
State the conclusions that the president of the club should reach from this test,
giving reasons for your answer.
[Maximum mark: 8]
Consider the curve C given by y = x3 .
The tangent at a point P on C meets the curve again at Q . The tangent at Q meets the
curve again at R . Denote the x-coordinates of P , Q and R , by x1 , x2 and x3 respectively
where x1 ≠ 0 . Show that, x1 , x2 , x3 form the first three elements of a divergent geometric
sequence.
[1]
[4]
[3]
[2]
–5–
6.
M16/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 8]
Consider the recurrence relation Hn+1 = 2Hn + 1 , n ∈ + where H1 = 1 .
7.
(a)
Find H2 , H3 and H4 .
[2]
(b)
Conjecture a formula for Hn in terms of n , for n ∈ + .
[1]
(c)
Prove by mathematical induction that your formula is valid for all n ∈ + .
[5]
[Maximum mark: 9]
 -3 x + 1 for x < 0

The function f :  →  is defined by f : x → 
for x = 0 .
1
2 x 2 - 3 x + 1 for x > 0

By considering limits prove that f is
8.
(a)
continuous at x = 0 ;
[4]
(b)
differentiable at x = 0 .
[5]
[Maximum mark: 14]
The points A , B have coordinates (-3 , 0) , (5 , 0) respectively. Consider the Apollonius
circle C which is the locus of point P where
AP
= k for k ≠ 1 .
BP
Given that the centre of C has coordinates (13 , 0) , find
(a)
(i)
the value of k ;
(ii)
the radius of C ;
(iii)
(b)
the x-intercepts of C .
Let M be any point on C and N be the x-intercept of C between A and B .
 N = NM
 B.
Prove that AM
[11]
[3]
Turn over
–6–
9.
M16/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 11]
(a)
Use the Euclidean algorithm to find gcd(162 , 5982) .
(b)
The relation R is defined on + by nRm if and only if gcd(n , m) = 2 .
(i)
By finding counterexamples show that R is neither reflexive nor transitive.
(ii)
Write down the set of solutions of nR6 .
10.
[7]
[Maximum mark: 10]
(a)
Show that 2n ≡ (-1)n (mod 3) , where n ∈  .
[3]
(b)
Hence show that an integer is divisible by 3 if and only if the difference between the
sum of its binary (base 2) digits in even-numbered positions and the sum of its binary
digits in odd-numbered positions is divisible by 3.
[3]
Express the hexadecimal (base 16) number ABBA16 in binary.
[4]
(c)
11.
[4]
[Maximum mark: 8]
The points P , Q and R , lie on the sides [AB] , [AC] and [BC] , respectively, of the
triangle ABC . The lines (AR) , (BQ) and (CP) are concurrent.
Use Ceva’s theorem to prove that [PQ] is parallel to [BC] if and only if R is the midpoint
of [BC] .
M16/5/FURMA/HP1/ENG/TZ0/XX
–7–
12.
[Maximum mark: 14]
In this question, x , y and z denote the coordinates of a point in three-dimensional Euclidean
space with respect to fixed rectangular axes with origin O . The vector space of position
vectors relative to O is denoted by 3 .
(a)
Explain why the set of position vectors of points whose coordinates satisfy x - y - z = 1
does not form a vector subspace of 3 .
(b)
(i)
Show that the set of position vectors of points whose coordinates satisfy
x - y - z = 0 forms a vector subspace, V , of 3 .
(ii)
1
 
Determine an orthogonal basis for V of which one member is  2  .
 -1
 
(iii)
Augment this basis with an orthogonal vector to form a basis for 3 .
(iv)
Express the position vector of the point with coordinates (4 , 0 , -2) as a linear
combination of these basis vectors.
13.
[1]
[13]
[Maximum mark: 11]
The discrete random variables Xn , n ∈ + have probability generating functions given
by Gn (t) =
(a)
t  tn - 1

.
n t -1
Use the formula for the sum of a finite geometric series to show that
1

P( Xn = k) =  n
 0
(b)
for 1 ≤ k ≤ n
.
[4]
otherwise
Find E(Xn) .
[3]
Let Xn-1 and Xn+1 be independent.
(c)
Find the set of values of n for which E(Xn-1 × Xn+1) < 2n .
[4]
Turn over
–8–
14.
M16/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 16]
A matrix M is called idempotent if M2 = M .
(a)
(i)
Explain why M is a square matrix.
(ii)
Find the set of possible values of det(M) .
[4]
The idempotent matrix N has the form
 a -2a 
N =

 a -2a 
where a ≠ 0 .
(i)
Find the value of a .
(ii)
Find the eigenvalues of N .
(iii)
Find corresponding eigenvectors.
(b)
[12]
m15/5/furma/HP1/eng/TZ0/XX
Further mathematics
Higher level
Paper 1
Wednesday 20 May 2015 (afternoon)
2 hours 30 minutes
Instructions to candidates
yyDo not open this examination paper until instructed to do so.
yyAnswer all questions.
yyUnless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
yyA graphic display calculator is required for this paper.
yyA clean copy of the mathematics HL and further mathematics HL formula booklet is
required for this paper.
yyThe maximum mark for this examination paper is [150 marks].
8 pages
2215 – 7101
© International Baccalaureate Organization 2015
–2–
m15/5/furma/HP1/eng/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be
supported by working and/or explanations. In particular, solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a solution,
you should sketch these as part of your answer. Where an answer is incorrect, some marks may be
given for a correct method, provided this is shown by written working. You are therefore advised to
show all working.
1.
[Maximum mark: 5]
Use l’Hôpital’s rule to find lim (csc x − cot x) .
x →0
2.
3.
[Maximum mark: 7]
(a)
Find the general solution to the Diophantine equation 3x + 5y = 7 .
[5]
(b)
Find the values of x and y satisfying the equation for which x has the smallest positive
integer value greater than 50 .
[2]
[Maximum mark: 11]
Consider the set S = {0 , 1 , 2 , 3 , 4 , 5} under the operation of addition modulo 6 ,
denoted by +6 .
(a)
Construct the Cayley table for {S , + 6 } .
[2]
(b)
Show that {S , + 6 } forms an Abelian group.
[5]
(c)
State the order of each element.
[2]
(d)
Explain whether or not the group is cyclic.
[2]
–3–
4.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 10]
A simple graph G is represented by the following adjacency table.
A
B
C
D
E
F
A
−
1
−
−
1
1
B
1
−
1
−
1
−
C
−
1
−
1
−
−
D
−
−
1
−
1
1
E
1
1
−
1
−
−
F
1
−
−
1
−
−
(a)
Draw the simple graph G .
[1]
(b)
Explain why G does not contain an Eulerian circuit.
[1]
(c)
Show that G has a Hamiltonian cycle.
[2]
(d)
State whether or not G is planar, giving a reason for your answer.
[2]
(e)
State whether or not the simple graph G is bipartite, giving a reason for your answer.
[2]
(f)
Draw the complement G' of G .
[2]
Turn over
–4–
5.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 9]
Jim is investigating the relationship between height and foot length in teenage boys.
A sample of 13 boys is taken and the height and foot length of each boy are measured.
The results are shown in the table.
Height
x cm
Foot
length
y cm
129
135
156
146
155
152
139
166
148
179
157
152
160
25.8 25.9 29.7 28.6 29.0 29.1 25.3 29.9 26.1 30.0 27.6 27.2 28.0
You may assume that this is a random sample from a bivariate normal distribution.
Jim wishes to determine whether or not there is a positive association between height and
foot length.
6.
(a)
Calculate the product moment correlation coefficient.
[2]
(b)
Find the p-value.
[2]
(c)
Interpret the p-value in the context of the question.
[1]
(d)
Find the equation of the regression line of y on x .
[2]
(e)
Estimate the foot length of a boy of height 170 cm.
[2]
[Maximum mark: 9]
Find the interval of convergence of the series
( x − 3) k
.
∑
k2
k =1
∞
–5–
7.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 12]
(a)
Sami is undertaking market research on packets of soap powder. He considers the
brand “Gleam”. The weight of the contents of a randomly chosen packet of “Gleam”
follows a normal distribution with mean 750 grams and standard deviation 20 grams.
The weight of the packaging follows a different normal distribution with mean
40 grams and standard deviation 5 grams.
Find:
(b)
8.
the probability that a randomly chosen packet of “Gleam” has a total weight
exceeding 780 grams.
(ii)
the probability that the total weight of the contents of five randomly chosen
packets of “Gleam” exceeds 3800 grams.
[8]
Sami now considers the brand “Bright”. The weight of the contents of a randomly
chosen packet of “Bright” follow a normal distribution with mean 650 grams and
standard deviation 16 grams. Find the probability that the contents of six randomly
chosen packets of “Bright” weigh more than the contents of five randomly chosen
packets of “Gleam”.
[4]
[Maximum mark: 10]
(a)
(b)
9.
(i)
Differentiate the expression x2 tan y with respect to x , where y is a function of x .
[3]
dy
+ x sin 2 y = x3 cos 2 y given that y = 0
dx
when x = 1 . Give your answer in the form y = f (x) .
[7]
Hence solve the differential equation x 2
[Maximum mark: 10]
An integer N given in base r , can be expressed in base s in the form
N = a0 + a1s + a2s2 + a3s3 + … where a0 , a1 , a2 , … ∈ {0 , 1 , 2 , … , s − 1} .
(a)
In base 5 an integer is written 1031 . Express this integer in base 10 .
[2]
(b)
Given that N = 365, r = 10 and s = 7 find the values of a0 , a1 , a2 , …
[2]
(c)
(i)
Given that N = 899 , r = 10 and s = 12 find the values of a0 , a1 , a2 , …
(ii)
Hence write down the integer in base 12 , which is equivalent to 899 in base 10 .
(d)
Show that 121 is always a square number in any base greater than 2 .
[3]
[3]
Turn over
–6–
10.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 12]
A wheel of radius r rolls, without slipping, along a straight path with the plane of the wheel
remaining vertical. A point A on the circumference of the wheel is initially at O . When the
wheel is rolled, the radius rotates through an angle of θ and the point of contact is now at B ,
where the length of the arc AB is equal to the distance OB . This is shown in the following
diagram.
A
O
r
x
B
(a)
Find the coordinates of A in terms of r and θ .
(b)
As the wheel rolls, the point A traces out a curve. Show that the gradient of this curve
[3]
1 
2 
is cot  θ  .
(c)
11.
Find the equation of the tangent to the curve when θ =
[6]
π
.
3
[Maximum mark: 7]
Prove that the function f :  ×  →  ×  defined by f (x , y) = (2x + y , x + y) is a bijection.
[3]
–7–
12.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 12]
A transformation T is a linear mapping from 3 to 4 , represented by the matrix.
 1 2 1


2 7 5
M =
 −3 1 4 


 1 5 4
(a)
(b)
13.
14.
(i)
Find the row rank of M .
(ii)
Hence or otherwise find the kernel of T .
(i)
State the column rank of M .
(ii)
Find the basis for the range of this transformation.
[8]
[4]
[Maximum mark: 9]
(a)
Two line segments [AB] and [CD] meet internally at the point Y . Given that
YA × YB = YC × YD show that A , B , C and D all lie on the circumference of a circle.
[6]
(b)
Explain why the result also holds if the line segments meet externally at Y .
[3]
[Maximum mark: 9]
Sarah is the quality control manager for the Stronger Steel Corporation which makes steel
sheets. The steel sheets should have a mean tensile strength of 430 MegaPascals (MPa) .
If the mean tensile strength drops to 400 MPa, then Sarah must recommend a change in
composition. The tensile strength of these steel sheets follows a normal distribution with a
standard deviation of 35 MPa. Sarah defines the following hypotheses
H0 : μ = 430
H1 : μ = 400
where μ denotes the mean tensile strength in MPa. She takes a random sample of n steel
k
sheets and defines the critical region as x ≤ k , where x ≤denotes
the mean tensile strength
of the sample in MPa and k is a constant.
Given that the P (Type I Error) = 0.0851 and P (Type II Error) = 0.115 , both correct to three
significant figures, find the value of k and the value of n .
Turn over
–8–
15.
m15/5/furma/HP1/eng/TZ0/XX
[Maximum mark: 13]
The relations ρ1 and ρ2 are defined on the Cartesian plane as follows
( x1 , y1 ) ρ1 ( x2 , y2 ) ⇔ x12 − x2 2 = y12 − y2 2
( x1 , y1 ) ρ2 ( x2 , y2 ) ⇔ x12 + x2 2 ≤ y12 +
16.
y2 2 .
(a)
For ρ1 and ρ2 determine whether or not each is reflexive, symmetric and transitive.
(b)
For each of ρ1 and ρ2 which is an equivalence relation, describe the equivalence
classes.
[Maximum mark: 5]
A circle x2 + y2 + dx + ey + c = 0 and a straight line lx + my + n = 0 intersect. Find the general
equation of a circle which passes through the points of intersection, justifying your answer.
[11]
[2]
M14/5/FURMA/HP1/ENG/TZ0/XX
22147101
FURTHER MATHEMATICS
HIGHER LEVEL
PAPER 1
Wednesday 21 May 2014 (afternoon)
2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
 Do not open this examination paper until instructed to do so.
 Answer all questions.
 Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
 A graphic display calculator is required for this paper.
 A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required
for this paper.
 The maximum mark for this examination paper is [150 marks].
2214-7101
7 pages
© International Baccalaureate Organization 2014
M14/5/FURMA/HP1/ENG/TZ0/XX
–2–
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should
be supported by suitable working. For example, if graphs are used to find a solution, you should sketch
these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method,
provided this is shown by written working. You are therefore advised to show all working.
1.
[Maximum mark: 6]
Find the positive square root of the base 7 number (551662)7 , giving your answer as a
base 7 number.
2.
[Maximum mark: 7]
dy
= y 3 − x3 for which y = 1 when x = 0 . Use Euler’s
dx
method with a step length of 0.1 to find an approximation for the value of y when x = 0.4 .
Consider the differential equation
3.
[Maximum mark: 6]
The following table shows the probability distribution of the discrete random variable X .
(a)
x
1
2
3
P ( X = x)
1
4
1
2
1
4
Show that the probability generating function of X is given by
G (t ) =
(b)
t (1 + t ) 2
.
4
Given that Y = X 1 + X 2 + X 3 + X 4 , where X 1 , X 2 , X 3 , X 4 is a random sample from the
distribution of X ,
(i)
state the probability generating function of Y ;
(ii)
hence find the value of P (Y = 8) .
2214-7101
[2]
[4]
–3–
4.
M14/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
a b
The matrix M is defined by M = 
.
c d 
The eigenvalues of M are denoted by λ1 , λ2 .
5.
(a)
Show that λ1 + λ2 =a + d and λ1λ2 = det ( M ) .
[3]
(b)
Given that a + b = c + d = 1 , show that 1 is an eigenvalue of M .
[2]
(c)
 2 −1 
Find eigenvectors for the matrix 
.
 3 −2 
[7]
[Maximum mark: 7]
(a)
(b)
6.
Assuming the Maclaurin series for e x , determine the first three non-zero terms in the
e x − e− x
Maclaurin expansion of
.
2
The random variable X has a Poisson distribution with mean µ . Show that
P ( X ≡ 1(mod 2) ) =
a + becµ where a , b and c are constants whose values are to
be found.
[3]
[4]
[Maximum mark: 9]
The parabola P has equation y 2 = 4ax . The distinct points U ( au 2 , 2au ) and V ( av 2 , 2av ) lie
ˆ is a right angle, where O denotes the origin,
on P , where u , v ≠ 0 . Given that UOV
(a)
4
show that v = − ;
u
[3]
(b)
find expressions for the coordinates of W, the midpoint of [UV], in terms of a and u ;
[2]
(c)
y 2 2ax − 8a 2 ;
show that the locus of W, as u varies, is the parabola P′ with equation =
[2]
(d)
determine the coordinates of the vertex of P′ .
[2]
2214-7101
Turn over
M14/5/FURMA/HP1/ENG/TZ0/XX
–4–
7.
[Maximum mark: 11]
The weights, in grams, of 10 apples were measured with the following results:
212.2 216.9
209.0
215.5
215.9
213.5
208.9
213.8
216.4
209.9
You may assume that this is a random sample from a normal distribution with mean µ and
variance σ 2 .
(a)
Giving all your answers correct to four significant figures,
(i)
determine unbiased estimates for µ and σ 2 ;
(ii)
find a 95 % confidence interval for µ .
[5]
Another confidence interval for µ , [211.5, 214.9], was calculated using the above data.
(b)
8.
Find the confidence level of this interval.
[6]
[Maximum mark: 12]
The group {G , *} has a subgroup { H , *} . The relation R is defined, for x , y ∈ G , by xRy if
and only if x −1 * y ∈ H .
9.
(a)
Show that R is an equivalence relation.
[8]
(b)
Given that G = {0, ± 1, ± 2, …} , H = {0, ± 4, ± 8, …} and * denotes addition, find the
equivalence class containing the number 3.
[4]
[Maximum mark: 5]
ABCDEF is a hexagon. A circle lies inside the hexagon and touches each of the six sides.
Show that AB + CD + EF = BC + DE + FA .
2214-7101
M14/5/FURMA/HP1/ENG/TZ0/XX
–5–
10.
[Maximum mark: 12]
1 2 1
The matrix A is given by A =  1 1 2  .
2 3 1


(a)
Given that A3 can be expressed in the form A3 = aA2 + bA + cI , determine the values of
the constants a , b , c .
(b)
(i)
Hence express A−1 in the form A−1 = dA2 + eA + fI where d , e , f ∈  .
(ii)
Use this result to determine A−1 .
11.
[7]
[5]
[Maximum mark: 9]
The random variables X , Y follow a bivariate normal distribution with product moment
correlation coefficient ρ . The following table gives a random sample from this distribution.
x
5.1
3.8
3.7
2.5
4.0
3.7
1.6
2.8
3.3
2.9
y
4.6
4.9
4.1
5.9
4.2
1.6
5.1
2.1
6.4
4.7
(a)
Determine the value of r , the product moment correlation coefficient of this sample.
(b)
(i)
Write down hypotheses in terms of ρ which would enable you to test whether or
not X and Y are independent.
(ii)
Determine the p-value of the above sample and state your conclusion at the
5 % significance level. Justify your answer.
(i)
Determine the equation of the regression line of y on x .
(ii)
State whether or not this equation can be used to obtain an accurate prediction of
the value of y for a given value of x . Give a reason for your answer.
(c)
2214-7101
[2]
[5]
[2]
Turn over
–6–
12.
M14/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 11]
xn
.
Consider the infinite series S = ∑ 2n
2
n =1 2 ( 2n −1 )
∞
13.
(a)
Determine the radius of convergence.
[4]
(b)
Determine the interval of convergence.
[7]
[Maximum mark: 9]

x
The function f :  + ×  + →  + ×  + is defined by f ( x , y ) =  xy ,  .
y

Prove that f is a bijection.
14.
[Maximum mark: 12]
(a)
The function g is defined by g ( x , y ) = x 2 + y 2 + dx + ey + f and the circle C1 has
equation g ( x , y ) = 0 .
(i)
e
 d
Show that the centre of C1 has coordinates  − , −  and the radius of C1
2
 2
is
(ii)
(b)
d 2 e2
+ −f .
4 4
The point P (a , b) lies outside C1 . Show that the length of the tangents from P
to C1 is equal to g (a , b) .
[6]
0
The circle C2 has equation x 2 + y 2 − 6 x − 2 y + 6 =.
The line y = mx meets C2 at the points R and S.
2214-7101
(i)
Determine the quadratic equation whose roots are the x-coordinates of R and S.
(ii)
Hence, given that L denotes the length of the tangents from the origin O to C2 ,
L2 .
show that OR × OS =
[6]
–7–
15.
M14/5/FURMA/HP1/ENG/TZ0/XX
[Maximum mark: 12]
(a)
Show that the solution to the linear congruence ax ≡ b (mod p ) , where a , x , b , p ∈  + ,
p is prime and a , p are relatively prime, is given by x ≡ a p−2 b (mod p ) .
(b)
Consider the congruences
[4]
7 x ≡ 13(mod 19)
2 x ≡ 1(mod 7) .
(i)
Use the result in (a) to solve the first congruence, giving your answer in the form
x ≡ k (mod 19) where 1 ≤ k ≤ 18 .
(ii)
Find the set of integers which satisfy both congruences simultaneously.
16.
[8]
[Maximum mark: 10]
{G , *} is a group of order N and {H , *}
is a proper subgroup of {G , *} of order n .
(a)
Define the right coset of { H , *} containing the element a ∈ G .
[1]
(b)
Show that each right coset of { H , *} contains n elements.
[2]
(c)
Show that the union of the right cosets of { H , *} is equal to G .
[2]
(d)
Show that any two right cosets of { H , *} are either equal or disjoint.
[4]
(e)
Give a reason why the above results can be used to prove that N is a multiple of n .
[1]
2214-7101
INTERNATIONAL BACCALAUREATE
BACCALAURÉAT INTERNATIONAL
BACHILLERATO INTERNACIONAL
M01/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Monday 14 May 2001 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
•
•
•
Do not open this examination paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or
to three significant figures as appropriate.
•
Write the make and model of your calculator on the front cover of your answer
booklets e.g. Casio fx-9750G, Sharp EL-9400, Texas Instruments TI-85.
221–254
3 pages
–2–
M01/540/S(1)
A correct answer with no indication of the method used will usually receive no marks. You are therefore
advised to show your working. In particular, where graphs from a graphic display calculator are being
used to find solutions, you should sketch these graphs as part of your answer.
1.
(a) Explain when the Yates continuity correction needs to be used, giving a reason.
(b) In 200 tosses of a coin, 108 tails and 92 heads were observed. Test the hypothesis that it
is a fair coin, at a significance level of 1% .
2.
Let (G , ∞) be a group with identity element e . Given that x ∞ x = e for all x G , prove that
(G , ∞) is an Abelian group.
3.
The profit of an internet company at the end of a given year is 8000 dollars more than twice
the profit for the previous year. If the profit at the end of the first year is $30 000, find an
expression for profit at the end of the nth year, for n = 1 , 2 , . . . .
4.
Let (R , +) be the group of real numbers under addition, and (R+ , ¥) be the group of positive
real numbers under multiplication. Prove that the two groups are isomorphic.
5.
The points T , C and D lie on a circle with centre S . A tangent [OT] and a secant [OCD] are
drawn from a point O to this circle. Prove that OT2 = OC ¥ OD .
•
6.
(a) Prove that the series
(–1)n
 (n + 1)
7
converges, for n N .
n =0
(b) Approximate the sum of the series to an accuracy of six decimal places.
7.
Let I =
Ú
5
2
e – x dx . Find the number n , of intervals necessary to approximate correct to two
0
decimal places, the value of I by the trapezium rule.
221–254
–3–
M01/540/S(1)
x2
+
y2
= 1 if c2 = a2 m2 + b2 .
8.
Prove that a line y = mx + c is a tangent to an ellipse
9.
A manager of two coal mines wants to test the heat-producing capacity of coal from each mine.
The heat-producing capacity (in millions of calories per ton) of random samples of coal from
each mine is given in the following table.
a2
b2
Mine 1
8260
8130
8350
8070
8340
Mine 2
7950
7890
7900
8140
7920
7840
The manager knows that the two population variances are equal.
(a) Describe the test to be used with the choice of the test statistic, giving reasons for your
answers.
(b) At the 5% level of significance, test if the average heat-producing capacity of the coal from
the two mines is equal.
10.
Let G be a simple graph. Prove that G has a spanning tree if and only if G is connected.
A
221–254
INTERNATIONAL BACCALAUREATE
BACCALAURÉAT INTERNATIONAL
BACHILLERATO INTERNACIONAL
N01/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Monday 12 November 2001 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
•
•
•
•
Do not open this examination paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or
to three significant figures as appropriate.
Write the make and model of your calculator on the front cover of your answer
booklets e.g. Casio fx-9750G, Sharp EL-9600, Texas Instruments TI-85.
881–254
4 pages
–2–
N01/540/S(1)
A correct answer with no indication of the method used will usually receive no marks. You are therefore
advised to show your working. In particular, where graphs from a graphic display calculator are being
used to find solutions, you should sketch these graphs as part of your answer.
1.
Let S = {1 , 2 , 3 , 4} and let A = S S . Define the relation R on A by:
(a , b) R (x , y) if and only if a + b = x + y .
Show that R is an equivalence relation and find the partition it creates on A .
∞
2.
Determine whether the series
∑e
k
k
converges or diverges. Note the test you use.
k =1
3.
Find the order of a group G generated by two elements x and y , subject only to the
following relations x3 = y2 = (xy)2 = 1 . List all subgroups of G .
4.
Draw a graph given by the following adjacency matrix.
0
1

0

1

1
1
0
1
0
1
0
1
0
1
1
1
0
1
0
0
1
1

1

0

0
Determine how many graphs with the same number of edges are possible on this set of vertices.
881–254
–3–
5.
N01/540/S(1)
The following diagram shows an isosceles triangle ABC , and 2 circles. The circle whose centre
is I and radius is r is inscribed in ABC . The circle whose centre is E and radius is R is
the escribed circle, ie it is outside ABC , and the lines (BC) , (AB) and (AC) are tangents
to this circle.
A
r
I
B
C
E
R
(a) Show that angle IBE is a right angle.
(b) Find BC in terms of r and R .
6.
Find the solution to the recurrence relation
an = 7an – 1 – 6an – 2 , with a0 = –1 and a1 = 4 .
7.
Use a binary search tree to find 43 on the following list
10 , 15 , 20 , 28 , 37 , 39 , 43 , 58 , 67 , 77 , 81 , 99 .
Show all steps.
8.
A computer repair shop replaces corrupt hard disks at a rate of 4 per week. Assuming that
such repairs occur at random, find the probability that
(a) exactly 7 hard disks are replaced in one week;
(b) in a 3-week period, at least 7 disks are replaced in two of these weeks.
881–254
Turn over
–4–
9.
10.
N01/540/S(1)
In a triangle ABC , AB = 8 , AC = 10 , and the median to the side [BC] has length 8 . Find
the area of the triangle.
Estimate e0.2 correct to 3 decimal places, using the Taylor approximation
f ( a + x ) = f ( a) + x f ( a) + . . . +
881–254
x n (n )
x n +1
(n +1)
(c )
f ( a) +
f
( n + 1)!
n!
IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M02/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Wednesday 15 May 2002 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
•
•
•
•
Do not open this examination paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or
to three significant figures.
Write the make and model of your calculator on the front cover of your answer
booklets e.g. Casio fx-9750G, Sharp EL-9600, Texas Instruments TI-85.
222-254R
3 pages
–2–
M02/540/S(1)
You are advised to show all working, where possible. Solutions found from a graphic display calculator
should be supported by suitable working. For example, if graphs are used to find a solution, you should
sketch these as part of your answer. Incorrect answers with no working will normally receive no marks.
1.
Two independent random samples each of size of n observations are to be selected, one from
each of two populations. If you wish to estimate the difference between the two population
means correct to within 0.12 with probability equal to 90%, how large should n be? Assume
that both variances are equal to 0.25.
2.
The normal daily human potassium requirement is in the range of 2000 to 6000 milligrams.
The amount of potassium in bananas is normally distributed with mean 630 mg. and standard
deviation of 40 mg per banana. Anwar eats three bananas per day.
(a) Find the mean and standard deviation of Anwar’s daily potassium intake.
(b) Find the probability that Anwar’s daily intake exceeds the minimum requirement.
3.
Find all possible remainders when (2k + 1)151 is divided by 8, for k Z+ .
4.
A graph contains 22 vertices and 43 edges. Every vertex has a degree of 3 or 5. Find the
number of vertices of degree 3.
5.
Given that the order of a group is a prime number, prove that the group is cyclic.
6.


1
Let A =  5 , −3, , 2π, 6, 20  .
5


a
Q.
b
The relation R is defined on A by aRb if
(a) Prove that R is an equivalence relation.
(b) Find the partition of the set A.
∞
7.
Determine whether the series
∑
n =0
222–254R
n
 n 

 converges or diverges, giving clear reasons.
 n + 4
–3–
8.
M02/540/S(1)
The convergent infinite sequence of positive real numbers un is defined recursively by
un +1 = 5 − 2 un , n Z+ .
Find the exact value of the limit of the sequence.
9.
The following diagram shows ABC . [AM] is a median. D is the midpoint of [AM] .
A
N
D
B
M
C
Prove that the line (BD) trisects [AC].
10.
A hyperbola is defined by the parametric equations
1
1
x =t+ ; y=t− .
t
t
Find its foci.
222–254R
Turn over
–4–
222–254
M02/540/S(1)
c
IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
N02/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Tuesday 12 November 2002 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
!
!
!
!
Do not open this examination paper until instructed to do so.
Answer all ten questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or to three
significant figures.
Write the make and model of your calculator on the front cover of your answer booklets
e.g. Casio fx-9750G, Sharp EL-9600, Texas Instruments TI-85.
882-254
4 pages
–2–
N02/540/S(1)
You are advised to show all working, where possible. Solutions found from a graphic display
calculator should be supported by suitable working. For example, if graphs are used to find a
solution, you should sketch these as part of your answer. Incorrect answers with no working will
normally receive no marks.
1.
2.
3.
Consider the group ( Z12 , + ) .
(a)
Find the order of the elements 4 , 5 and 9 .
(b)
Show that this group is cyclic. Find all possible generators.
Consider κ n a complete graph with n vertices.
(a)
Draw κ 5 and find an Eulerian circuit in it.
(b)
Find the value of n such that κ n contains an Eulerian path but not an Eulerian circuit.
Justify your answer.
Determine whether the following series converges or diverges.
1 3
5
7
9
+ +
+ +
+….
2 2 2 2 4 4 2
4.
Find all the integers x that satisfy the equation 2 x 3 − 3x +1 ≡ 4 ( mod 6) .
5.
Eggs are packed in boxes of four. During one day 200 boxes were selected and the number of
broken eggs in each box was recorded.
Number of broken eggs
0
1
2
3
4
Number of boxes
73
80
31
14
2
Test at the 5 % level of significance whether this data follows a binomial distribution with
n = 4 and p = 0.24 .
882-254
–3–
6.
7.
N02/540/S(1)
1
The function f : R → R is defined by f ( x) = 3cos x + .
6
(a)
Determine whether the function is injective or surjective, giving your reasons.
(b)
If the domain of f is restricted to [ 0, π ] find its inverse function.
Consider the triangle ABC. The points M, N and P are on the sides [BC], [CA] and [AB]
respectively, such that the lines (AM), (BN) and (CP) are concurrent.
Given that
NA
CM
ΑP
+
.
= µ , where λ , µ ,∈ R , find
= λ , and
CN
CB
AB
π
.
4
8.
Find a cubic Taylor polynomial approximation for the function f (x) = tan x , about x =
9.
A school newspaper consists of three sections. The number of misprints in each section
has a Poisson distribution with parameters 0.9 , 1.1 and 1.5 respectively. Misprints occur
independently.
(a)
Find the probability that there will be no misprints in the newspaper.
(b)
The probability that there are more than n misprints in the newspaper is less than 0.5 .
Find the smallest value of n.
882-254
Turn over
–4–
10.
N02/540/S(1)
Consider the hyperbola H with equation b 2 x 2 − a 2 y 2 = a 2b 2 . The angle between the asymptotes
of H is π , as shown in the diagram below.
3
y
H
H
π
3
x
(a)
Calculate the eccentricity of H.
(b)
Find the equations of the directrices of H, giving your answers in terms of a.
882-254
c
IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M03/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Thursday 22 May 2003 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
y
y
y
y
Do not open this examination paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or to three
significant figures.
Write the make and model of your calculator in the appropriate box on your cover sheet
e.g. Casio fx-9750G, Sharp EL-9600, Texas Instruments TI-85.
223-254
4 pages
–2–
M03/540/S(1)
You are advised to show all working, where possible. Where an answer is wrong, some marks may
be given for correct method, provided this is shown by written working. Solutions found from a
graphic display calculator should be supported by suitable working e.g. if graphs are used to find a
solution, you should sketch these as part of your answer.
1.
Consider two sets S and T with a mapping α : S → T . If A and B are subsets of S, show that
α ( A ∪ B) = α ( A) ∪ α ( B) .
2.
Use the Euclidean algorithm to show that 7 is the greatest common divisor of 1001 and 357.
Hence find two integers m and n such that 7 = 1001n + 357 m .
3.
Let S3 be the set of permutations of {1, 2, 3} , under composition of permutations (i.e. S3 is
the symmetric group of degree 3).
4.
(a)
 1 2 3
Find a proper subgroup containing 
.
 2 3 1
(b)
Show that this subgroup is cyclic.
Use the graph below to answer the following.
(a)
Use the depth-first search to obtain a “depth-first” numbering of the vertices.
(b)
Use the numbering obtained to form a spanning tree of the graph.
A
E
B
C
D
H
J
F
223-254
I
G
–3–
5.
6.
M03/540/S(1)
A random variable X has a Poisson distribution with mean 9.
(a)
If X 1 and X 2 are two such independent variables, find P ( X 1 + X 2 = 10) .
(b)
The random variable X is the mean of a random sample of 64 values of X.
Find P ( X < 8.5) .
∞
xk
A series expansion of e is ∑ . A random variable X has a Poisson distribution.
k =0 k !
x
∞
Show that
∑ P ( X = x) = 1 .
x=0
7.
A circle with centre A and radius 3 cm, and a circle with centre B and radius 5 cm are given
with their centres 10 cm apart. A third circle is tangent to the two given circles simultaneously.
The following diagrams show two of the four possible cases.
(i)
I
5
B
3
A
(ii)
I
5
A
B
3
Find and describe the locus of the centres of the third circle in the two cases shown.
223-254
Turn over
–4–
8.
M03/540/S(1)
The diagram below shows the line (AD), where the points C and D are in harmonic ratio to
points A and B. The point M is outside line (AB). Line (PQ) is drawn through B parallel
to (AM), where Q lies on (MD) and P on (MC).
M
Q
A
C
D
B
P
Show that B is the midpoint of [PQ].
9.
The probability density function of a random variable X is given in the table below. Calculate
each of the following:
x
P ( X = x)
(a)
E(X )
(b)
E(X 2)
(c)
Var ( X )
(d)
Var (3 X − 2)
∞
10.
Determine whether
n=2
223-254
1
∑ ln n
4
0.35
6
0.51
8
0.14
is convergent or divergent, giving reasons for your answer.
c
IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M04/540/S(1)
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Monday 24 May 2004 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
y
y
y
y
Do not open this examination paper until instructed to do so.
Answer all the questions.
Unless otherwise stated in the question, all numerical answers must be given exactly or to three
significant figures.
Write the make and model of your calculator in the appropriate box on your cover sheet
e.g. Casio fx-9750G, Sharp EL-9600, Texas Instruments TI-85.
224-258
4 pages
–2–
M04/540/S(1)
You are advised to show all working, where possible. Where an answer is wrong, some marks may
be given for correct method, provided this is shown by written working. Solutions found from a
graphic display calculator should be supported by suitable working, e.g. if graphs are used to find
a solution, you should sketch these as part of your answer.
1.
The table below defines the operation ⊗ on the set {1, a, b, c, d}.
⊗
1
a
b
c
d
1
1
a
b
c
d
a
a
c
1
d
b
b
b
d
c
a
1
c
c
b
d
1
a
d
d
1
a
b
c
Give two reasons, based on different group properties, why the table does not define a group.
2.
Let X and Y be two points on the sides [AB] and [AC] of a triangle ABC such that (XY) is
parallel to (BC). Let Z be the point of intersection of (BY) and (CX). Show that Z lies on the
median from A.
3.
Define the operation ∆ on Z × Z by
(a , b) ∆ (c , d ) = (ac + bd , ad + bc) , where a , b , c , d ∈Z .
Find the identity element for this operation.
4.
Show that the series
∞
1
5.
6.
1+ r
∑1+ r
2
diverges.
Industrial accidents in a factory occur at random at a rate of three per month.
(a)
Calculate the probability that more than three accidents happen in a particular month.
(b)
Let X denote the number of accidents occurring in a particular month. Find the minimum
value of k such that P ( X > k ) < 0.05 .
Points A and B are chosen on the x-axis and y-axis respectively of a coordinate system such
AP 4
= . Give
that the length AB = 18 units. P is a point on the line segment [AB] such that
PB 5
a full geometric description of the locus of P as A and B move along the axes.
224-258
–3–
7.
8.
M04/540/S(1)
The amount of liquid dispensed into a bottle by an automatic filling machine is normally
distributed with preset mean µ cl and standard deviation 3 cl. The output from the machine is
monitored every hour and regulations require that the mean volume of liquid contained in a
randomly chosen sample of bottles from the output of the machine should lie within 0.9 cl of
the preset mean µ cl.
(a)
Find the probability that the regulations are satisfied when the sample contains nine
bottles.
(b)
Let n be the number of bottles in a sample. Find the minimum value of n so that the
regulations are satisfied at least 95 % of the time.
Let p be a positive integer and n an integer larger than 1. Consider the numbers a = np and
b = (n − 1) p .
(a)
Find the greatest common divisor (gcd) of a and b.
(b)
Hence find the gcd of 15 x (8 y + 5) and 24 x (5 y + 3) where x and y are positive integers.
224-258
Turn over
–4–
9.
M04/540/S(1)
Consider the following two pairs of graphs, G1 , G2 and H1 , H 2 shown in Figures 1 and 2.
Determine whether or not the graphs in each pair are isomorphic. If they are isomorphic, copy
and label the second one to show the isomorphism. If the two are not isomorphic, justify your
answer.
A
B
F
Figure 1
C
E
D
G1
G2
P
Q
R
Figure 2
S
T
U
H1
10.
H2
The mean value theorem states that if a function f is continuous over
[ a, a + h ]
differentiable over ] a , a + h [ , then there exists a number θ with 0 < θ < 1 such that
f (a + h) − f (a ) = hf ′(a + θ h) .
Let f ( x) = e x . Find θ in terms of h.
224-258
and

IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M05/5/FURMA/SP1/ENG/TZ0/XX
22057101
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Monday 23 May 2005 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
Ÿ Do not open this examination paper until instructed to do so.
Ÿ Answer all the questions.
Ÿ Unless otherwise stated in the question, all numerical answers must be given exactly or to three
significant figures.
2205-7101
3 pages
–2–
M05/5/FURMA/SP1/ENG/TZ0/XX
Please start each question on a new page. You are advised to show all working, where possible. Where an
answer is wrong, some marks may be given for correct method, provided this is shown by written working.
Solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs
are used to find a solution, you should sketch these as part of your answer.
1.
Given that f ( x) = 3 1 + x , use the Maclaurin series, up to the term in x3 , to find an approximate value of
3
1.2 . Give your answer correct to 5 decimal places.
2.
Prove by mathematical induction that a tree with n vertices has exactly n −1 edges, where n ∈ ¢ + .
3.
Consider the group (G, ×) with the identity element e. Given two elements a and b of the group such
that ab 2 a = b and a 2 = b3 = e , show that
4.
(a)
ab = b 2 a ;
(b)
( ab )
2 2
= e.
The following table shows the number of females and males who are left handed or right handed.
Left Right
Female 43
357
Male
76
524
At the 1 % level of significance, is there evidence of an association between the gender of a person
and whether they are left or right handed?
5.
Find the general solution of the recurrence relation an+2 = 5an+1 − 6an , n ≥ 1 . What are the initial
conditions for the sequence {an } to generate powers of 3?
6.
Determine whether or not the following series is convergent.
1
4 7 10
+ 3 + + 3 + ...
2
4 2 2 2
3
2205-7101
–3–
7.
M05/5/FURMA/SP1/ENG/TZ0/XX
In the triangle ABC, AC < BC , as shown in the following diagram.
The points M, N and P are the midpoints of the sides [BC], [CA] and [AB] respectively. F is the foot
of the perpendicular from C to [AB].
8.
(a)
Show that MP = FN.
(b)
Hence or otherwise show that MNFP is a cyclic quadrilateral.
The weights of a particular type of nail follow a normal distribution with mean 5.2 g and standard
deviation 0.7 g.
(a)
A random sample of 50 nails is taken and the mean weight calculated. Calculate the probability
that this sample mean is less than 5 g.
(b)
A random sample is taken such that the probability of the sample mean exceeding 5.3 g is less
than 0.2. Find the minimum sample size.
9.
An ellipse is given by the parametric equations x = 2 cos t + 3 and y = 3 sin t − 1. Find the coordinates
of the centre and the foci of this ellipse.
10.
Let V be the set of all directed line segments in the plane. The relation ≅ on V×V is defined as
follows.
→
→
→
AB ≅ CD if and only if [AD] and [BC] have a common midpoint, where AB represents the directed
line segment from A to B. Show that ≅ is an equivalence relation.
2205-7101

IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M06/5/FURMA/SP1/ENG/TZ0/XX
22067101
FURTHER MATHEMATICS
STANDARD LEVEL
PAPER 1
Monday 15 May 2006 (afternoon)
1 hour
INSTRUCTIONS TO CANDIDATES
Ÿ Do not open this examination paper until instructed to do so.
Ÿ Answer all the questions.
Ÿ Unless otherwise stated in the question, all numerical answers must be given exactly or correct to
three significant figures.
2206-7101
4 pages
–2–
M06/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is
shown by written working. All students should therefore be advised to show their working.
1.
[Maximum mark: 9]
The general term of a sequence is given by the formula an =
2.
n 2 + 3n
, n ∈ ¢+ .
2
2n − 1
(a)
Given that lim an = L , where L ∈ ¡ , find the value of L.
[3 marks]
(b)
Find the smallest value of N ∈ ¢ + such that an − L < 10−3 for all n ≥ N .
[6 marks]
n →∞
[Maximum mark: 7]
The following diagram shows a circle, centre O, and a point T outside the circle.
Tangents [TL] and [TM] are drawn to touch the circle at L and M. Let P be any point
on the smaller arc LM. The tangent to the circle at P meets [TL] and [TM] at the
points A and B respectively.
T
A
L
P
B
O
M
$ remains constant.
As P moves around the smaller arc LM, show that AOB
2206-7101
[7 marks]
–3–
3.
4.
M06/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 9]
(a)
Convert the base 5 number 2341 to a decimal number.
[3 marks]
(b)
Show that any number written in base 5 is divisible by 2 if the sum of its digits
is divisible by 2.
[6 marks]
[Maximum mark: 11]
The function f : ¢ + → ¢ + is defined by f ( x) = gcd ( x , 6).
5.
(a)
Find the range of the function f.
[3 marks]
(b)
Show that the function f is periodic and find its period.
[3 marks]
(c)
Find the set of positive integers satisfying f ( x) = 2 .
[5 marks]
[Maximum mark: 12]
0.005 e −0.005 x , x ≥ 0
The function f is defined by f ( x) = 
0,
x<0

(a)
Show that the function f is a probability density function.
(b)
While testing the lifetime of light bulbs, in a sample of 150 light bulbs, the
following frequency distribution is obtained.
lifetime (hours)
number of light bulbs
[0, 100[
[100, 200[
[200, 300[
47
40
35
[4 marks]
[300, +∞ [
28
2
Use a χ test at the 5 % significance level to determine whether or not the
probability distribution defined by f is an appropriate model for the data.
2206-7101
[8 marks]
Turn over
–4–
6.
M06/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 12]
dy 3 x 2 + y 2
Consider the differential equation
where x , y > 0 .
=
dx
xy
(a)
Show that the differential equation is homogeneous.
[2 marks]
(b)
Find the general solution of the differential equation, giving your answer in the
form y 2 = f ( x) .
[7 marks]
Solve the differential equation, given that y = 2 when x = 1 .
[3 marks]
(c)
2206-7101

IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
M07/5/FURMA/SP1/ENG/TZ0/XX
22077101
Further mathematics
STANDARD level
Paper 1
Wednesday 16 May 2007 (afternoon)
1 hour
Instructions to candidates
 Do not open this examination paper until instructed to do so.
 Answer all the questions.
 Unless otherwise stated in the question, all numerical answers must be given exactly or correct to
three significant figures.
2207-7101
3 pages
© IBO 2007
––
M07/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this
is shown by written working. All students should therefore be advised to show their working.
1.
[Maximum mark: 8]
The point P (x , y) moves in such a way that its distance from the point (1 , 0) is three
times its distance from the point (– 1 , 0) .
2.
(a)
Find the equation of the locus of P.
[4 marks]
(b)
Show that this equation represents a circle and state its radius and the coordinates
of its centre.
[4 marks]
[Maximum mark: 8]
Calculate the following limits
2x −1
;
x
[3 marks]
(a)
lim
(b)
(1 + x 2 ) 2 − 1
lim
.
x → 0 ln (1 + x ) − x
x →0
3
3.
4.
[5 marks]
[Maximum mark: 12]
(a)
Show that the set S of numbers of the form 2m × 3n , where m , n ∈  , forms a
M , × }} under multiplication.
group{{S,
[6 marks]
(b)
M , × }} is isomorphic to the group of complex numbers m + ni under
Show that{{S,
addition, where m , n ∈ .
[6 marks]
[Maximum mark: 12]
(a)
Use the Euclidean Algorithm to show that 275 and 378 are relatively prime.
[5 marks]
(b)
Find the general solution to the diophantine equation 275 x + 378 y = 1.
[7 marks]
2207-7101
––
5.
M07/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 9]
Solve the differential equation x
dy
+ 2 y = 1 + x2
dx
given that y = 1 when x = 3 .
6.
[9 marks]
[Maximum mark: 11]
The weights, X kg , of male birds of a certain species are normally distributed with
mean 4.5 kg and standard deviation 0.2 kg . The weights, Y kg , of female birds of this
species are normally distributed with mean 2.5 kg and standard deviation 0.15 kg .
(a)
(b)
2207-7101
(i)
Find the mean and variance of 2Y − X .
(ii)
Find the probability that the weight of a randomly chosen male bird is
more than twice the weight of a randomly chosen female bird.
[6 marks]
Two randomly chosen male birds and three randomly chosen female birds are
placed together on a weighing machine for which the recommended maximum
weight is 16 kg . Find the probability that this maximum weight is exceeded.
[5 marks]
M10/5/FURMA/SP1/ENG/TZ0/XX
22107101
Further mathematics
STANDARD level
Paper 1
Thursday 20 May 2010 (afternoon)
1 hour
Instructions to candidates
 Do not open this examination paper until instructed to do so.
 Answer all the questions.
 Unless otherwise stated in the question, all numerical answers must be given exactly or correct
to three significant figures.
2210-7101
4 pages
© International Baccalaureate Organization 2010
–2–
M10/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this
is shown by written working. All students should therefore be advised to show their working.
1.
[Maximum mark: 7]
A university Mathematics Department admits students on the basis of performance
in an entrance examination which is graded ‘excellent’, ‘very good’ or ‘good’.
The students sit their final examination three years later when they are awarded
‘first class’, ‘second class’ or ‘third class’ degrees. The results for a particular group
of students are summarised in the following table.
Excellent
Very good
Good
First class
30
8
5
Second class
14
12
6
Third class
6
5
14
Stating your hypotheses, use an appropriate test to investigate at the 1 % level of
significance whether or not there is an association between performance in the entrance
examination and performance in the final examination. Justify your answer.
2.
[Maximum mark: 10]
Let S be the set of matrices given by
a b 
 c d  ; a , b , c , d ∈  , ad − bc = 1.


The relation R is defined on S as follows. Given A , B ∈ S , ARB if and only if there
exists X ∈ S such that A = BX .
(a)
Show that R is an equivalence relation.
[8 marks]
(b)
The relationship between a , b , c and d is changed to ad − bc = n . State, with
a reason, whether or not there are any non-zero values of n , other than 1,
for which R is an equivalence relation.
[2 marks]
2210-7101
–3–
3.
M10/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 11]
The figure below shows the graph G .
A
B
C
D
E
(i)
Write down the adjacency matrix for G .
(ii)
Find the number of walks of length 4 beginning and ending at B.
(i)
Draw G′, the complement of G .
(ii)
Write down the degrees of all the vertices of G and all the vertices of G′.
(iii) Hence, or otherwise, determine whether or not G and G′ are isomorphic.
(a)
(b)
4.
[5 marks]
[6 marks]
[Maximum mark: 9]
Given that n 2 + 2n + 3 ≡ N (mod 8) , where n ∈  + and 0 ≤ N ≤ 7 , prove that N can
take one of only three possible values.
5.
[Maximum mark: 11]
Given that
2210-7101
dy
π
+ 2 y tan x = sin x , and y = 0 when x = , find the maximum value of y.
dx
3
Turn over
–4–
6.
M10/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 12]
S
C1
Q
C2
R
O
T
P
The figure shows a circle C1 with centre O and diameter [PQ] and a circle C2 which
intersects (PQ) at the points R and S. T is one point of intersection of the two circles
and (OT) is a tangent to C2 .
OR OT
.
=
OT OS
(a)
Show that
(b)
(i)
Show that PR − RQ = 2OR .
(ii)
Show that
(c)
2210-7101
PR − RQ PS − SQ
.
=
PR + RQ PS + SQ
Deduce that P, R, Q, S form a harmonic ratio.
[2 marks]
[6 marks]
[4 marks]
M11/5/FURMA/SP1/ENG/TZ0/XX
22117101
Further mathematics
STANDARD level
Paper 1
Thursday 5 May 2011 (afternoon)
1 hour
Instructions to candidates
 Do not open this examination paper until instructed to do so.
 Answer all questions.
 Unless otherwise stated in the question, all numerical answers must be given exactly or correct
to three significant figures.
2211-7101
3 pages
© International Baccalaureate Organization 2011
–2–
M11/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this
is shown by written working. You are therefore advised to show all working.
1.
[Maximum mark: 11]
(a)
Bottles of iced tea are supposed to contain 500 ml. A random sample of 8 bottles
was selected and the volumes measured (in ml) were as follows:
497.2 , 502.0 , 501.0 , 498.6 , 496.3 , 499.1, 500.1, 497.7 .
(i)
Calculate unbiased estimates of the mean and variance.
(ii)
Test at the 5 % significance level the null hypothesis H 0 : µ = 500 against
the alternative hypothesis H1 : µ < 500 .
[5 marks]
A random sample of size four is taken from the distribution N (60 , 36) .
Calculate the probability that the sum of the sample values is less than 250.
[6 marks]
(b)
2.
[Maximum mark: 15]
(a)
(b)
(i)
Find the range of values of n for which
(ii)
Write down the value of
2211-7101
∞
1
∞
1
x n dx exists.
x n dx in terms of n , when it does exist.
[7 marks]
Find the solution to the differential equation
(cos x − sin x)
∫
∫
given that y = −1 when x =
dy
+ (cos x + sin x) y = cos x + sin x ,
dx
π
.
2
[8 marks]
–3–
3.
M11/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 11]
(a)
Prove that the number 14 641 is the fourth power of an integer in any base greater
than 6.
(b)
For a , b ∈  the relation aRb is defined if and only if
a
= 2k , k ∈  .
b
(i)
Prove that R is an equivalence relation.
(ii)
List the equivalence classes of R on the set {1, 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10} .
4.
[8 marks]
[Maximum mark: 11]
(a)
Prove that if gcd (a , b) = 1 and gcd (a , c) = 1, then gcd (a , bc) = 1.
(b)
(i)
A simple graph has e edges and v vertices, where ν > 2 . Prove that if all
the vertices have degree at least k , then 2e ≥ kv .
(ii)
Hence prove that every planar graph has at least one vertex of degree less
than 6.
5.
[3 marks]
[Maximum mark: 12]
The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths 3 cm
and 9 cm respectively. E is a point on side [AB] such that AE is 3 cm. Side [DE] is
produced to meet the circumcircle of ABCD at point P. Use Ptolemy’s theorem to
calculate the length of chord [AP].
2211-7101
[5 marks]
[6 marks]
M12/5/FURMA/SP1/ENG/TZ0/XX
22127101
Further mathematics
STANDARD level
Paper 1
Friday 4 May 2012 (afternoon)
1 hour
Instructions to candidates
 Do not open this examination paper until instructed to do so.
 Answer all questions.
 Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
 A graphic display calculator is required for this paper.
 A clean copy of the Mathematics HL and Further Mathematics SL information booklet is
required for this paper.
 The maximum mark for this examination paper is [60 marks].
2212-7101
4 pages
© International Baccalaureate Organization 2012
–2–
M12/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of
your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this
is shown by written working. You are therefore advised to show all working.
1.
[Maximum mark: 11]
(a)
The set S1 = {2 , 4 , 6 , 8} and ×10 denotes multiplication modulo 10.
(i)
Write down the Cayley table for {S1 , ×10 }.
(ii)
Show that {S1 , ×10 } is a group.
(iii) Show that this group is cyclic.
[8 marks]
Now consider the group {S2 , ×20 } where S2 = {1, 9 , 11, 19} and ×20 denotes
multiplication modulo 20. Giving a reason, state whether or not {S1 , ×10 } and
{S2 , ×20 } are isomorphic.
[3 marks]
(b)
2.
3.
[Maximum mark: 7]
(a)
Express the number 47502 as a product of its prime factors.
[2 marks]
(b)
The positive integers M , N are such that gcd ( M , N ) = 63 and
lcm ( M , N ) = 47502 . Given that M is even and M < N , find the two possible
pairs of values for M , N .
[5 marks]
[Maximum mark: 13]
(a)
By evaluating successive derivatives at x = 0 , find the Maclaurin series for
ln cos x up to and including the term in x 4 .
(b)
Consider lim
x →0
ln cos x
, where n ∈ .
xn
Using your result from (a), determine the set of values of n for which
(i)
the limit does not exist;
(ii)
the limit is zero;
(iii) the limit is finite and non-zero, giving its value in this case.
2212-7101
[8 marks]
[5 marks]
–3–
4.
M12/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 7]
The diagram below shows a quadrilateral ABCD and a straight line which intersects
(AB), (BC), (CD), (DA) at the points P, Q, R, S respectively.
S
B
A
P
Q
C
D
Using Menelaus’ theorem, show that
5.
R
AP BQ CR DS
×
×
×
= 1.
PB QC RD SA
[Maximum mark: 13]
Bill buys two biased coins from a toy shop.
(a)
The shopkeeper claims that when one of the coins is tossed, the probability of
obtaining a head is 0.6. Bill wishes to test this claim by tossing the coin 250
times and counting the number of heads obtained.
(i)
State suitable hypotheses for this test.
(ii)
He obtains 140 heads. Find the p-value of this result and determine whether
or not it supports the shopkeeper’s claim at the 5 % level of significance.
(b)
Bill tosses the other coin a large number of times and counts the number of
heads obtained. He correctly calculates a 95 % confidence interval for the
probability that when this coin is tossed, a head is obtained. This is calculated as
[0.35199 , 0.44801] where the end-points are correct to five significant figures.
Determine
(i)
the number of times the coin was tossed;
(ii)
the number of heads obtained.
2212-7101
[6 marks]
[7 marks]
Turn over
–4–
6.
M12/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 9]
(a)
(b)
2212-7101
Using mathematical induction or otherwise, prove that the number (1020) n ,
that is the number 1020 written with base n , is divisible by 3 for all values of n
greater than 2.
Explain briefly why the case n = 2 has to be excluded.
[8 marks]
[1 mark]
M13/5/FURMA/SP1/ENG/TZ0/XX
22137101
Further mathematics
STANDARD level
Paper 1
Monday 20 May 2013 (afternoon)
1 hour
Instructions to candidates
 Do not open this examination paper until instructed to do so.
 Answer all questions.
 Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
 A graphic display calculator is required for this paper.
 A clean copy of the Mathematics HL and Further Mathematics SL information booklet is
required for this paper.
 The maximum mark for this examination paper is [60 marks].
2213-7101
4 pages
© International Baccalaureate Organization 2013
–2–
M13/5/FURMA/SP1/ENG/TZ0/XX
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. In particular, solutions found from a graphic display calculator should be
supported by suitable working, for example, if graphs are used to find a solution, you should sketch these
as part of your answer. Where an answer is incorrect, some marks may be given for a correct method,
provided this is shown by written working. You are therefore advised to show all working.
1.
[Maximum mark: 12]
(a)
(b)
(i)
Use the Euclidean algorithm to find gcd (6750, 144) .
(ii)
Express your answer in the form 6750r + 144 s where r , s ∈ .
Consider the base 15 number CBA, where A, B, C represent respectively the
digits ten, eleven, twelve.
(i)
Write this number in base 10.
(ii)
Hence express this number as a product of prime factors, writing the factors
in base 4.
2.
[6 marks]
[6 marks]
[Maximum mark: 12]
G is a group. The elements a , b ∈ G , satisfy a 3 = b 2 = e and ba = a 2b , where e is the
identity element of G .
(a)
Show that (ba ) 2 = e .
[3 marks]
(b)
Express (bab) −1 in its simplest form.
[3 marks]
Given that a ≠ e ,
(c)
2213-7101
(i)
show that b ≠ e ;
(ii)
show that G is not Abelian.
[6 marks]
–3–
3.
M13/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 12]
(a)
A triangle T has sides of length 3, 4 and 5.
(i)
Find the radius of the circumscribed circle of T .
(ii)
Find the radius of the inscribed circle of T .
(b)
[6 marks]
A triangle U has sides of length 4, 5 and 7.
(i)
Show that the orthocentre, H, of U lies outside the triangle.
(ii)
Show that the foot of the perpendicular from H to the longest side divides
it in the ratio 29:20.
4.
[6 marks]
[Maximum mark: 13]
(a)
Find the general solution of the differential equation
for x < 1 .
(b)
(i)
2213-7101
(ii)
2
Show that the solution y = f ( x) that satisfies the condition f (0) =
f ( x) =
(1 − x ) ddyx = 1 + xy ,
arcsin x +
1 − x2
Find lim f ( x) .
x→−1
π
2.
[7 marks]
π
is
2
[6 marks]
Turn over
–4–
5.
M13/5/FURMA/SP1/ENG/TZ0/XX
[Maximum mark: 11]
Let X k be independent normal random variables, where E ( X k ) = µ
Var ( X k ) = k , for k = 1, 2, …
and
(−1) k +1
Xk .
k
k =1
6
The random variable Y is defined by Y = ∑
(a)
(b)
2213-7101
(i)
Find E(Y ) in the form pµ , where p ∈ .
(ii)
Find k if Var ( X k ) < Var (Y ) < Var ( X k +1 ) .
[5 marks]
A random sample of n values of Y was found to have a mean of 8.76.
(i)
Given that n = 10 , determine a 95 % confidence interval for µ .
(ii)
The width of the confidence interval needs to be halved. Find the
appropriate value of n .
[6 marks]