© International Baccalaureate Organization 2025
All rights reserved. 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 the prior written permission from the IB. Additionally, the license tied with this
product prohibits 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, whether fee-covered or not, is prohibited and is a
criminal offense.
More information on how to request written permission in the form of a license can be
obtained from https://ibo.org/become-an-ib-school/ib-publishing/licensing/applying-for-a-
license/.
© Organisation du Baccalauréat International 2025
Tous droits réservés. 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
préalable de l’IB. De plus, la licence associée à ce produit interdit toute utilisation 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, moyennant
paiement ou non, est interdite et constitue une infraction pénale.
Pour plus d’informations sur la procédure à suivre pour obtenir une autorisation écrite
sous la forme d’une licence, rendez-vous à l’adresse https://ibo.org/become-an-ib-school/
ib-publishing/licensing/applying-for-a-license/.
© Organización del Bachillerato Internacional, 2025
Todos los derechos reservados. 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 la previa autorización por escrito del
IB. Además, la licencia vinculada a este producto prohíbe el uso 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—, ya sea incluido en tasas o no, está prohibido
y constituye un delito.
En este enlace encontrará más información sobre cómo solicitar una autorización por
escrito en forma de licencia: https://ibo.org/become-an-ib-school/ib-publishing/licensing/
applying-for-a-license/.
,Mathematics: analysis and approaches
Higher level
Paper 3
21 May 2025
Zone A afternoon Zone B afternoon Zone C afternoon
1 hour 15 minutes
Instructions to candidates
y Do not open this examination paper until instructed to do so.
y A graphic display calculator is required for this paper.
y Answer all the questions in the answer booklet provided.
y Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
y A clean copy of the mathematics: analysis and approaches HL formula booklet is required
for this paper.
y The maximum mark for this examination paper is [55 marks].
2225 – 7108
5 pages © International Baccalaureate Organization 2025
, –2– 2225 – 7108
Answer all questions in the answer booklet provided. Please start each question on a new page. Full
marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. 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: 23]
This question asks you to use polynomial functions to model some situations in probability.
Two unbiased tetrahedral (four-sided) dice with faces labelled 1 , 2 , 3 and 4 are thrown and
the scores recorded.
The random variable M denotes the maximum of these two scores.
The probability distribution of M is given in the following table.
m 1 2 3 4
1 3 5 7
P (M = m)
16 16 16 16
(a) Find E (M ) . [2]
An alternative way to represent the probability distribution of M is to use a polynomial
4
function, G , where G t
P M
m
t
m
1
m
.
1 3 5 7
Hence, for the distribution of M , G t
t
t2
t3
t4 .
16 16 16 16
(b) Find G (1) . [1]
(c) (i) Find G′ (t ) . [2]
(ii) Hence, show that G′ (1) = E (M ) . [3]
(This question continues on the following page)
All rights reserved. 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 the prior written permission from the IB. Additionally, the license tied with this
product prohibits 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, whether fee-covered or not, is prohibited and is a
criminal offense.
More information on how to request written permission in the form of a license can be
obtained from https://ibo.org/become-an-ib-school/ib-publishing/licensing/applying-for-a-
license/.
© Organisation du Baccalauréat International 2025
Tous droits réservés. 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
préalable de l’IB. De plus, la licence associée à ce produit interdit toute utilisation 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, moyennant
paiement ou non, est interdite et constitue une infraction pénale.
Pour plus d’informations sur la procédure à suivre pour obtenir une autorisation écrite
sous la forme d’une licence, rendez-vous à l’adresse https://ibo.org/become-an-ib-school/
ib-publishing/licensing/applying-for-a-license/.
© Organización del Bachillerato Internacional, 2025
Todos los derechos reservados. 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 la previa autorización por escrito del
IB. Además, la licencia vinculada a este producto prohíbe el uso 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—, ya sea incluido en tasas o no, está prohibido
y constituye un delito.
En este enlace encontrará más información sobre cómo solicitar una autorización por
escrito en forma de licencia: https://ibo.org/become-an-ib-school/ib-publishing/licensing/
applying-for-a-license/.
,Mathematics: analysis and approaches
Higher level
Paper 3
21 May 2025
Zone A afternoon Zone B afternoon Zone C afternoon
1 hour 15 minutes
Instructions to candidates
y Do not open this examination paper until instructed to do so.
y A graphic display calculator is required for this paper.
y Answer all the questions in the answer booklet provided.
y Unless otherwise stated in the question, all numerical answers should be given exactly or
correct to three significant figures.
y A clean copy of the mathematics: analysis and approaches HL formula booklet is required
for this paper.
y The maximum mark for this examination paper is [55 marks].
2225 – 7108
5 pages © International Baccalaureate Organization 2025
, –2– 2225 – 7108
Answer all questions in the answer booklet provided. Please start each question on a new page. Full
marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanations. 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: 23]
This question asks you to use polynomial functions to model some situations in probability.
Two unbiased tetrahedral (four-sided) dice with faces labelled 1 , 2 , 3 and 4 are thrown and
the scores recorded.
The random variable M denotes the maximum of these two scores.
The probability distribution of M is given in the following table.
m 1 2 3 4
1 3 5 7
P (M = m)
16 16 16 16
(a) Find E (M ) . [2]
An alternative way to represent the probability distribution of M is to use a polynomial
4
function, G , where G t
P M
m
t
m
1
m
.
1 3 5 7
Hence, for the distribution of M , G t
t
t2
t3
t4 .
16 16 16 16
(b) Find G (1) . [1]
(c) (i) Find G′ (t ) . [2]
(ii) Hence, show that G′ (1) = E (M ) . [3]
(This question continues on the following page)
