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6SHFL¿FDWLRQGCE Mathematics
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8371)
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8372)
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Pure Mathematics (8373)
Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (Additional) (8374)First examination 2014
Pearson Edexcel Level 3 Advanced GCE in Mathematics (9371)
Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9372)
Pearson Edexcel Level 3 Advanced GCE in Pure Mathematics (9373)
Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (Additional) (9374)First examination 2014
For more information on our wide range of support and services for this GCE in Mathematics, Further 0DWKHPDWLFV��3XUH�0DWKHPDWLFV�DQG�)XUWKHU�0DWKHPDWLFV��$GGLWLRQDO��TXDOL¿FDWLRQ��YLVLW�RXU�*&(�ZHEVLWH��ZZZ�HGH[FHO�FRP�JFH�����
Edexcel GCE in Mathematics, Further Mathematics, Pure Mathematics and Further Mathematics (Additional) is GHVLJQHG�IRU�XVH�LQ�VFKRRO�DQG�FROOHJHV��,W�LV�SDUW�RI�D�VXLWH�RI�*&(�TXDOL¿FDWLRQV�RIIHUHG�E\�(GH[FHO�
Key features of the specification(GH[FHO¶V�0DWKHPDWLFV�VSHFL¿FDWLRQ�KDV�
� pathways leading to full Advanced Subsidiary and Advanced GCE in Mathematics, Further Mathematics, Pure Mathematics and Further Mathematics (Additional)
� updated Decision Mathematics 1 and Decision Mathematics 2 units, for a more balanced approach to content
C1 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�
C2 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��WULJRQRPHWU\��H[SRQHQWLDOV�DQG�ORJDULWKPV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�
C3 Algebra and functions; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��QXPHULFDO�PHWKRGV�
C4 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ��YHFWRUV�
Further Pure Mathematics
8QLW 6XPPDU\�RI�XQLW�FRQWHQW
FP1 Series; complex numbers; numerical solution of equations; coordinate V\VWHPV��PDWUL[�DOJHEUD��SURRI�
M1 0DWKHPDWLFDO�PRGHOV�LQ�PHFKDQLFV��YHFWRUV�LQ�PHFKDQLFV��NLQHPDWLFV�RI�D�particle moving in a straight line; dynamics of a particle moving in a straight OLQH�RU�SODQH��VWDWLFV�RI�D�SDUWLFOH��PRPHQWV�
M2 Kinematics of a particle moving in a straight line or plane; centres of mass; ZRUN�DQG�HQHUJ\��FROOLVLRQV��VWDWLFV�RI�ULJLG�ERGLHV�
M4 Relative motion; elastic collisions in two dimensions; further motion of SDUWLFOHV�LQ�RQH�GLPHQVLRQ��VWDELOLW\�
M5 Applications of vectors in mechanics; variable mass; moments of inertia of a ULJLG�ERG\��URWDWLRQ�RI�D�ULJLG�ERG\�DERXW�D�¿[HG�VPRRWK�D[LV�
Statistics
8QLW 6XPPDU\�RI�XQLW�FRQWHQW
S1 Mathematical models in probability and statistics; representation and summary of data; probability; correlation and regression; discrete random YDULDEOHV��GLVFUHWH�GLVWULEXWLRQV��WKH�1RUPDO�GLVWULEXWLRQ�
S2 The Binomial and Poisson distributions; continuous random variables; FRQWLQXRXV�GLVWULEXWLRQV��VDPSOHV��K\SRWKHVLV�WHVWV�
S4 Quality of tests and estimators; one-sample procedures; two-sample SURFHGXUHV�
Decision Mathematics
8QLW 6XPPDU\�RI�XQLW�FRQWHQW
D1 Algorithms; algorithms on graphs; the route inspection problem; critical path DQDO\VLV��OLQHDU�SURJUDPPLQJ��PDWFKLQJV�
D2 Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, dynamic programming; ÀRZV�LQ�QHWZRUNV�
Advanced subsidiary awards comprise three teaching units per DZDUG�
1DWLRQDO�FODVVL¿FDWLRQ�FRGH
&DVK�LQ�FRGH
$ZDUG &RPSXOVRU\�XQLWV 2SWLRQDO�XQLWV
2210 8371 GCE AS Mathematics C1 and C2 M1, S1 or D1
2230 8372 GCE AS Further Mathematics
FP1 Any *
2230 8373 GCE AS Pure Mathematics C1, C2, C3
2230 8374 GCE AS Further Mathematics (Additional)
Any which have not been used for a previous TXDOL¿FDWLRQ
*For GCE AS Further Mathematics, excluded units are C1, C2, C3, C4
**See Appendix C for description of this code and all other codes relevant to this TXDOL¿FDWLRQ
Advanced Subsidiary combinations
Combinations leading to an award in Advanced Subsidiary 0DWKHPDWLFV�FRPSULVH�WKUHH�$6�XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�DQ�award in Advanced Subsidiary Further Mathematics comprise three XQLWV��7KH�FRPELQDWLRQ�OHDGLQJ�WR�DQ�DZDUG�LQ�$GYDQFHG�6XEVLGLDU\�3XUH�0DWKHPDWLFV�FRPSULVHV�XQLWV�&���&��DQG�&��
8371 Advanced Subsidiary Mathematics
Core Mathematics units C1 and C2 plus one of the Applications XQLWV�0���6��RU�'��
8372 Advanced Subsidiary Further Mathematics
Further Pure Mathematics unit FP1 plus two other units (excluding &�±&����6WXGHQWV�ZKR�DUH�DZDUGHG�FHUWL¿FDWHV�LQ�ERWK�$GYDQFHG�GCE Mathematics and AS Further Mathematics must use unit results IURP�QLQH�GLIIHUHQW�WHDFKLQJ�PRGXOHV�
8374 Additional Qualification in Advanced Subsidiary Further Mathematics (Additional)
6WXGHQWV�ZKR�FRPSOHWH�¿IWHHQ�XQLWV�ZLOO�KDYH�DFKLHYHG�WKH�equivalent of the standard of Advanced Subsidiary GCE Further 0DWKHPDWLFV�LQ�WKHLU�DGGLWLRQDO�XQLWV��6XFK�VWXGHQWV�ZLOO�EH�HOLJLEOH�for the award of Advanced Subsidiary GCE Further Mathematics (Additional) in addition to the awards of Advanced GCE Mathematics DQG�$GYDQFHG�*&(�)XUWKHU�0DWKHPDWLFV�
Combinations leading to an award in mathematics must comprise VL[�XQLWV��LQFOXGLQJ�DW�OHDVW�WZR�$��XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�an award in Further Mathematics must comprise six units, including DW�OHDVW�WKUHH�$��XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�DQ�DZDUG�LQ�3XUH�0DWKHPDWLFV�FRPSULVH�XQLWV�&�±&���)3��DQG�RQH�RI�)3��RU�)3��
9371 Advanced GCE Mathematics
Core Mathematics units C1, C2, C3 and C4 plus two Applications XQLWV�IURP�WKH�IROORZLQJ�VL[�FRPELQDWLRQV��0��DQG�0���6��DQG�6���'��DQG�'���0��DQG�6���6��DQG�'���0��DQG�'��
9372 Advanced GCE Further Mathematics
Further Pure Mathematics units FP1, FP2, FP3 and a further three $SSOLFDWLRQV�XQLWV��H[FOXGLQJ�&�±&���WR�PDNH�D�WRWDO�RI�VL[�XQLWV��RU�FP1, either FP2 or FP3 and a further four Applications units �H[FOXGLQJ�&�±&���WR�PDNH�D�WRWDO�RI�VL[�XQLWV��6WXGHQWV�ZKR�DUH�DZDUGHG�FHUWL¿FDWHV�LQ�ERWK�$GYDQFHG�*&(�0DWKHPDWLFV�DQG�Advanced GCE Further Mathematics must use unit results from 12 GLIIHUHQW�WHDFKLQJ�PRGXOHV�
9374 Additional Qualification in Advanced Further Mathematics (Additional)
Students who complete eighteen units will have achieved the equivalent of the standard of Advanced GCE Further Mathematics �$GGLWLRQDO��LQ�WKHLU�DGGLWLRQDO�XQLWV��6XFK�VWXGHQWV�ZLOO�EH�HOLJLEOH�for the award of Advanced GCE Further Mathematics (Additional) in addition to the awards of Advanced GCE Mathematics and Advanced *&(�)XUWKHU�0DWKHPDWLFV�
AO1 UHFDOO��VHOHFW�DQG�XVH�WKHLU�NQRZOHGJH�RI�PDWKHPDWLFDO�IDFWV��FRQFHSWV�DQG�techniques in a variety of contexts
���
AO2 construct rigorous mathematical arguments and proofs through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions, including the construction of extended arguments for handling substantial problems presented in unstructured form
���
AO3 UHFDOO��VHOHFW�DQG�XVH�WKHLU�NQRZOHGJH�RI�VWDQGDUG�PDWKHPDWLFDO�PRGHOV�WR�UHSUHVHQW�situations in the real world; recognise and understand given representations involving standard models; present and interpret results from such models in terms of the original situation, including discussion of the assumptions made and UH¿QHPHQW�RI�VXFK�PRGHOV
���
AO4 comprehend translations of common realistic contexts into mathematics; use the UHVXOWV�RI�FDOFXODWLRQV�WR�PDNH�SUHGLFWLRQV��RU�FRPPHQW�RQ�WKH�FRQWH[W��DQG��ZKHUH�appropriate, read critically and comprehend longer mathematical arguments or examples of applications
��
AO5 use contemporary calculator technology and other permitted resources (such DV�IRUPXODH�ERRNOHWV�RU�VWDWLVWLFDO�WDEOHV��DFFXUDWHO\�DQG�HI¿FLHQWO\��XQGHUVWDQG�ZKHQ�QRW�WR�XVH�VXFK�WHFKQRORJ\��DQG�LWV�OLPLWDWLRQV��*LYH�DQVZHUV�WR�DSSURSULDWH�DFFXUDF\�
Subject criteria 7KH�*HQHUDO�&HUWL¿FDWH�RI�(GXFDWLRQ�LV�SDUW�RI�WKH�/HYHO���SURYLVLRQ��7KLV�VSHFL¿FDWLRQ�LV�EDVHG�RQ�WKH�*&(�$6�DQG�$�/HYHO�6XEMHFW�criteria for Mathematics, which is prescribed by the regulatory DXWKRULWLHV�DQG�LV�PDQGDWRU\�IRU�DOO�DZDUGLQJ�ERGLHV��
7KH�*&(�LQ�0DWKHPDWLFV�HQDEOHV�VWXGHQWV�WR�IROORZ�D�ÀH[LEOH�course in mathematics, to better tailor a course to suit the LQGLYLGXDO�QHHGV�DQG�JRDOV�
� develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected
� recognise how a situation may be represented mathematically and understand the relationship between ‘real-world’ problems and standard and other mathematical models and how these can EH�UH¿QHG�DQG�LPSURYHG
� use mathematics as an effective means of communication
� read and comprehend mathematical arguments and articles concerning applications of mathematics
� DFTXLUH�WKH�VNLOOV�QHHGHG�WR�XVH�WHFKQRORJ\�VXFK�DV�FDOFXODWRUV�and computers effectively, recognise when such use may be inappropriate and be aware of limitations
� develop an awareness of the relevance of mathematics to other ¿HOGV�RI�VWXG\��WR�WKH�ZRUOG�RI�ZRUN�DQG�WR�VRFLHW\�LQ�JHQHUDO
Unit C1 Core Mathematics 1 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics
C1.1 Unit description
Algebra and functions; coordinate geometry in the (x, y) plane; VHTXHQFHV�DQG�VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�
C1.2 Assessment information
Preamble Construction and presentation of rigorous mathematical arguments through appropriate use of precise statements and logical deduction, involving correct use of symbols and appropriate FRQQHFWLQJ�ODQJXDJH�LV�UHTXLUHG��6WXGHQWV�DUH�H[SHFWHG�WR�exhibit correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, µVXI¿FLHQW¶��DQG�QRWDWLRQ�VXFK�DV�∴, ⇒, ⇐ and ⇔�
For this unit, students may QRW have access to any calculating aids, LQFOXGLQJ�ORJ�WDEOHV�DQG�VOLGH�UXOHV�
Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�EHORZ�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV�
Functions to include simple cubic functions and the reciprocal
function y = k
x with x ≠ 0.
Knowledge of the term asymptote LV�H[SHFWHG�
Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a,
y = f(x + a), y = f(ax).
Students should be able to apply one of these transformations to any of the above functions (quadratics, cubics, reciprocal) DQG�VNHWFK�WKH�UHVXOWLQJ�JUDSK��
Given the graph of any function y = f(x) students should be able to VNHWFK�WKH�JUDSK�UHVXOWLQJ�IURP�RQH�RI�WKHVH�WUDQVIRUPDWLRQV�
Equation of a straight line, including the forms y – y
1 = m(x – x
1) and ax + by + c = 0.
7R�LQFOXGH�
(i) the equation of a line through two given points
(ii) the equation of a line parallel (or perpendicular) to a given line through a given SRLQW��)RU�H[DPSOH��WKH�OLQH�perpendicular to the line 3x + 4y = 18 through the point (2, 3) has equation
y – 3 = (x – 2).
Conditions for two straight lines to be parallel or SHUSHQGLFXODU�WR�HDFK�RWKHU�
3 Sequences and series
What students need to learn:
Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form x
n+1 = f(x
n).
Arithmetic series, including the formula for the sum RI�WKH�¿UVW�n�QDWXUDO�QXPEHUV�
The general term and the sum to n�WHUPV�RI�WKH�VHULHV�DUH�UHTXLUHG��The proof of the sum formula VKRXOG�EH�NQRZQ��
The derivative of f(x) as the gradient of the tangent to the graph of y = f (x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; VHFRQG�RUGHU�GHULYDWLYHV�
For example,�NQRZOHGJH�WKDW�
is
the rate of change of y with respect to x��.QRZOHGJH�RI�WKH�FKDLQ�UXOH�LV�QRW�UHTXLUHG��
The notation f ′(x) PD\�EH�XVHG�
Differentiation of xn, and related sums and GLIIHUHQFHV�
For example, for n ≠ 1, the ability to differentiate expressions such
as (2x + 5)(x − 1) and 2 5 3
3 1 2x + x
x−
/ is
expectHG�
Applications of differentiation to gradients, tangents DQG�QRUPDOV�
8VH�RI�GLIIHUHQWLDWLRQ�WR�¿QG�equations of tangents and normals DW�VSHFL¿F�SRLQWV�RQ�D�FXUYH�
Unit C2 Core Mathematics 2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics
C2.1 Unit description
Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�
Calculators Students are expected to have available a calculator with at least
WKH�IROORZLQJ�NH\V��+, −, ×, ÷, ʌ, x2, √x,
1
x, xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG��
Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�EHORZ�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV�
Simple algebraic division; use of the Factor Theorem DQG�WKH�5HPDLQGHU�7KHRUHP�
Only division by (x + a) or (x – a) will EH�UHTXLUHG�
6WXGHQWV�VKRXOG�NQRZ�WKDW�LI f(x) = 0 when x = a, then (x – a) is a factor of f(x)�
Students may be required to factorise cubic expressions such as x3
+ 3x2 – 4 and 6x3
+ 11x2 – x – 6�
Students should be familiar with the terms ‘quotient’ and ‘remainder’ and be able to determine the remainder when the polynomial f(x) is divided by (ax + b)�
2 Coordinate geometry in the (x, y) plane
What students need to learn:
Coordinate geometry of the circle using the equation of a circle in the form (x – a)
2 + (y – b)2 = r2 and
LQFOXGLQJ�XVH�RI�WKH�IROORZLQJ�FLUFOH�SURSHUWLHV�
(i) the angle in a semicircle is a right angle;
(ii) the perpendicular from the centre to a chord bisects the chord;
�LLL��WKH�SHUSHQGLFXODULW\�RI�UDGLXV�DQG�WDQJHQW�
6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�the radius and the coordinates of the centre of the circle given the equation of the circle, and vice YHUVD�
Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics
C3.1 Unit description
Algebra and functions; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��QXPHULFDO�PHWKRGV�
C3.2 Assessment information
Prerequisites and preamble
Prerequisites
$�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��DQG�&���WKHLU�SUHDPEOHV��prerequisites and associated formulae, is assumed and may be WHVWHG�
Preamble
Methods of proof, including proof by contradiction and disproof by FRXQWHU�H[DPSOH��DUH�UHTXLUHG��$W�OHDVW�RQH�TXHVWLRQ�RQ�WKH�SDSHU�ZLOO�UHTXLUH�WKH�XVH�RI�SURRI�
Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�
Calculators Students are expected to have available a calculator with at least
the folloZLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x, , xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG��
Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�EHORZ�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV�
The concept of a function as a one-one or many-one mapping from ! (or a subset of !) to !��The notation f : x and f(x) will be XVHG��
6WXGHQWV�VKRXOG�NQRZ�WKDW fg will mean ‘do g�¿UVW��WKHQ f�¶��
6WXGHQWV�VKRXOG�NQRZ�WKDW�LI�f −1
exists, then f −1
f(x) = ff −1
(x) = x.
7KH�PRGXOXV�IXQFWLRQ� 6WXGHQWV�VKRXOG�EH�DEOH�WR�VNHWFK�the graphs of y = ⏐ax + b⏐ and the graphs of y = ⏐f(x)⏐ and y = f(⏐x⏐), given the graph of y = f(x)�
Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax)�
6WXGHQWV�VKRXOG�EH�DEOH�WR�VNHWFK�the graph of, for example, y = 2f(3x), y = f(−x) + 1, given the graph of y = f(x) or the graph of, for example, y = 3 + sin 2x,
Knowledge of secant, cosecant and cotangent and of DUFVLQ��DUFFRV�DQG�DUFWDQ��7KHLU�UHODWLRQVKLSV�WR�VLQH��FRVLQH�DQG�WDQJHQW��8QGHUVWDQGLQJ�RI�WKHLU�JUDSKV�DQG�DSSURSULDWH�UHVWULFWHG�GRPDLQV�
Angles measured in both degrees DQG�UDGLDQV�
Knowledge and use of sec2�ș = 1 + tan
2�ș and
cosec2 ș = 1 + cot
2 ș.
Knowledge and use of double angle formulae; use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B) and of expressions for a cos�ș���E�sin�ș�in the equivalent forms of r cos (ș�± a) or r sin (ș�± a)�
To include application to half DQJOHV��.QRZOHGJH�RI�WKH t (tan
12θ )
formulae will not�EH�UHTXLUHG�
Students should be able to solve equations such as a cos ș�+ b sin ș = c in a given interval, and to prove simple identities such as cos x cos 2x + sin x sin 2x ≡ cos x�
3 Exponentials and logarithms
What students need to learn:
The function ex DQG�LWV�JUDSK� To include the graph of y = eax + b + c.
The function ln x and its graph; ln x as the inverse function of ex�
Differentiation of ex, ln x, sin x, cos x, tan x and their sums and differences.
Differentiation using the product rule, the quotient UXOH�DQG�WKH�FKDLQ�UXOH��
The use of dd d
d
yx x
y
=⎛
⎝⎜
⎞
⎠⎟
1�
Differentiation of cosec x, cot x and sec x�DUH�UHTXLUHG��6NLOO�ZLOO�EH�expected in the differentiation of functions generated from standard forms using products, quotients and composition, such as 2x4
sin x,
e3x
x , cos x2
and tan2 2x�
(J�¿QGLQJ�ddyx for x = sin 3y�
5 Numerical methods
What students need to learn:
Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is FRQWLQXRXV�
Approximate solution of equations using simple iterative methods, including recurrence relations of the form x
n+1 = f(x
n).
Solution of equations by use of iterative procedures for which OHDGV�ZLOO�EH�JLYHQ�
Unit C4 Core Mathematics 4 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics
C4.1 Unit description
Algebra and functions; coordinate geometry in the (x, y) plane; VHTXHQFHV�DQG�VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ��YHFWRUV�
C4.2 Assessment information
Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&���&��DQG�&��DQG�WKHLU�preambles, prerequisites and associated formulae, is assumed and PD\�EH�WHVWHG�
Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�
Calculators Students are expected to have available a calculator with at least
the following�NH\V� +, −, ×, ÷��ʌ��x2, √x,
1
x , xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG��
Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�RYHUOHDI�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV.
Parametric equations of curves and conversion EHWZHHQ�&DUWHVLDQ�DQG�SDUDPHWULF�IRUPV�
6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�the area under a curve given its SDUDPHWULF�HTXDWLRQV��6WXGHQWV�will not�EH�H[SHFWHG�WR�VNHWFK�a curve from its parametric HTXDWLRQV�
3 Sequences and series
What students need to learn:
Binomial series for any rational n�For ⏐x⏐<
b
a , students should be
able to obtain the expansion of (ax + b)
n, and the expansion of rational functions by decomposition LQWR�SDUWLDO�IUDFWLRQV�
Simple cases of integration by substitution and LQWHJUDWLRQ�E\�SDUWV��7KHVH�PHWKRGV�DV�WKH�UHYHUVH�SURFHVVHV�RI�WKH�FKDLQ�DQG�SURGXFW�UXOHV�UHVSHFWLYHO\�
Except in the simplest of cases the VXEVWLWXWLRQ�ZLOO�EH�JLYHQ��
The integral ��ln x dx�LV�UHTXLUHG�
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General and particular solutions ZLOO�EH�UHTXLUHG�
1XPHULFDO�LQWHJUDWLRQ�RI�IXQFWLRQV� $SSOLFDWLRQ�RI�WKH�WUDSH]LXP�UXOH�to functions covered in C3 and &���8VH�RI�LQFUHDVLQJ�QXPEHU�RI�WUDSH]LD�WR�LPSURYH�DFFXUDF\�DQG�HVWLPDWH�HUURU�ZLOO�EH�UHTXLUHG��Questions will not require more WKDQ�WKUHH�LWHUDWLRQV�
Simpson’s Rule is not�UHTXLUHG�
6 Vectors
What students need to learn:
9HFWRUV�LQ�WZR�DQG�WKUHH�GLPHQVLRQV�
0DJQLWXGH�RI�D�YHFWRU� 6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�D�unit vector in the direction of a, and be familiar with ⏐a⏐�
Algebraic operations of vector addition and multiplication by scalars, and their geometrical LQWHUSUHWDWLRQV�
Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�QLQH�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�
Calculators Students are expected to have available a calculator with at least
the followiQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,
1
x , xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
(YDOXDWLRQ�RI�����GHWHUPLQDQWV� Singular and non-singular PDWULFHV�
Inverse of 2 × 2 matrices. Use of the relation (AB)
–1 = B–1A–1
.
&RPELQDWLRQV�RI�WUDQVIRUPDWLRQV� Applications of matrices to JHRPHWULFDO�WUDQVIRUPDWLRQV�
,GHQWL¿FDWLRQ�DQG�XVH�RI�WKH�matrix representation of single and combined transformations IURP��UHÀHFWLRQ�LQ�FRRUGLQDWH�axes and lines y = x, rotation of multiples of 45° about (0, 0) and enlargement about centre (0, 0), with scale factor, (k ���), where k ∈ !.
The inverse (when it exists) of a given WUDQVIRUPDWLRQ�RU�FRPELQDWLRQ�RI�WUDQVIRUPDWLRQV�
Idea of the determinant as an area scale factor in WUDQVIRUPDWLRQV�
5 Series
What students need to learn:
6XPPDWLRQ�RI�VLPSOH�¿QLWH�VHULHV�� Students should be able to sum series such as
Unit FP2 Further Pure Mathematics 2 GCE Further Mathematics and GCE Pure Mathematics A2 optional unit
FP2.1 Unit description
,QHTXDOLWLHV��VHULHV��¿UVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV��VHFRQG�RUGHU�differential equations; further complex numbers, Maclaurin and 7D\ORU�VHULHV�
FP2.2 Assessment information
Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&���&���&���&��DQG�)3���WKHLU�prerequisites, preambles and associated formulae is assumed and PD\�EH�WHVWHG�
Calculators Students are expected to have available a calculator with at least
WKH�IROORZLQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
1 Inequalities
What students need to learn:
The manipulation and solution of algebraic inequalities and inequations, including those LQYROYLQJ�WKH�PRGXOXV�VLJQ�
Euler’s relation eiș = cos ș + i sin ș. Students should be familiar with
cos θ = 12
(eiθ + e
−iθ ) and
sin ș = 12i
(eiθ − e
−iθ ).
De Moivre’s theorem and its application to trigonometric identities and to roots of a complex QXPEHU�
7R�LQFOXGH�¿QGLQJ�cos Qș and sin Pș in terms of powers of sin ș�and cos�ș�and also powers of sin ș�and cos�ș in terms of multiple DQJOHV��6WXGHQWV�VKRXOG�EH�DEOH�WR�prove De Moivre’s theorem for any integer n�
/RFL�DQG�UHJLRQV�LQ�WKH�$UJDQG�GLDJUDP� Loci such as⏐z − a⏐ = b,
⏐z�í�a⏐ = k⏐z�í�b⏐,
arg (z − a) = ȕ, arg z az b
= −−
β and
regions such as ⏐z − a⏐ ≤ ⏐z − b⏐,
⏐z − a⏐ ≤ b.
Elementary transformations from the z-plane to the w�SODQH��
The formation of the differential HTXDWLRQ�PD\�EH�UHTXLUHG��Students will be expected to obtain particular solutions and DOVR�VNHWFK�PHPEHUV�RI�WKH�IDPLO\�RI�VROXWLRQ�FXUYHV�
First order linear differential equations of the form dd
+ = yx
Py Q where P and Q are functions of x�
Differential equations reducible to the above types by PHDQV�RI�D�JLYHQ�VXEVWLWXWLRQ�
The integrating factor e�Pdx may be TXRWHG�ZLWKRXW�SURRI�
5 Second Order Differential Equations
What students need to learn:
The linear second order differential equation
a yx
b yx
cy x2d
d + d
d + = f( )2 where a, b and c are real
constants and the particular integral can be found by LQVSHFWLRQ�RU�WULDO�
The auxiliary equation may have real distinct, equal or complex URRWV� f(x) will have one of the forms kepx
, A + Bx, p + qx + cx2 or m cos Ȧ[ + n sin Ȧ[.
Students should be familiar with the terms ‘complementary IXQFWLRQ¶�DQG�µSDUWLFXODU�LQWHJUDO¶�
Students should be able to solve equations of the form
d2 y
xd 2 + 4y = sin 2x�
Differential equations reducible to the above types by PHDQV�RI�D�JLYHQ�VXEVWLWXWLRQ�
Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit
FP3.1 Unit description
Further matrix algebra; vectors, hyperbolic functions; differentiation; integration, further coordinate systems
FP3.3 Assessment information
Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&���&���&���&��DQG�)3���WKHLU�prerequisites, preambles and associated formulae is assumed and PD\�EH�WHVWHG�
Calculators Students are expected to have available a calculator with at least
WKH�IROORZLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Unit M1 Mechanics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit
M1.1 Unit description
Mathematical models in mechanics; vectors in mechanics; NLQHPDWLFV�RI�D�SDUWLFOH�PRYLQJ�LQ�D�VWUDLJKW�OLQH��G\QDPLFV�RI�D�particle moving in a straight line or plane; statics of a particle; PRPHQWV�
Calculators Students are expected to have available a calculator with at least
the folORZLQJ�NH\V� +, −, ×, ÷��ʌ� x2, √x,
1
x , xy
, ln x, ex, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DQ\�RWKHU�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
The basic ideas of mathematical modelling as applied LQ�0HFKDQLFV�
Students should be familiar with WKH�WHUPV��SDUWLFOH��ODPLQD��ULJLG�body, rod (light, uniform, non-uniform), inextensible string, smooth and rough surface, light VPRRWK�SXOOH\��EHDG��ZLUH��SHJ��Students should be familiar with the assumptions made in using WKHVH�PRGHOV�
Students may be required to resolve a vector into two components or use a vector GLDJUDP��4XHVWLRQV�PD\�EH�VHW�involving the unit vectors i and j�
Application of vectors to displacements, velocities, DFFHOHUDWLRQV�DQG�IRUFHV�LQ�D�SODQH�
Use of velocity =
change of displacementtime
in the case of constant velocity, and of
acceleration = change of velocity
time
in the case of constant DFFHOHUDWLRQ��ZLOO�EH�UHTXLUHG�
3 Kinematics of a particle moving in a straight line
What students need to learn:
0RWLRQ�LQ�D�VWUDLJKW�OLQH�ZLWK�FRQVWDQW�DFFHOHUDWLRQ� Graphical solutions may be required, including displacement-time, velocity-time, speed-time and acceleration-time JUDSKV��.QRZOHGJH�DQG�XVH�RI�formulae for constant acceleration ZLOO�EH�UHTXLUHG�
4 Dynamics of a particle moving in a straight line or plane
What students need to learn:
7KH�FRQFHSW�RI�D�IRUFH��1HZWRQ¶V�ODZV�RI�PRWLRQ� Simple problems involving constant acceleration in scalar form or as a vector of the form ai + bj.
Simple applications including the motion of two FRQQHFWHG�SDUWLFOHV�
Problems may include
(i) the motion of two connected particles moving in a straight line or under gravity when the forces on each particle are constant; problems involving VPRRWK�¿[HG�SXOOH\V�DQG�RU�pegs may be set;
(ii) motion under a force which FKDQJHV�IURP�RQH�¿[HG�YDOXH�to another, eg a particle hitting the ground;
(iii) motion directly up or down a smooth or rough inclined SODQH�
Calculators Students are expected to have available a calculator with at least
the followiQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DQ\�RWKHU�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
The use of an axis of symmetry will be acceptable where DSSURSULDWH��8VH�RI�LQWHJUDWLRQ�LV�QRW�UHTXLUHG��)LJXUHV�PD\�LQFOXGH�the shapes referred to in the IRUPXODH�ERRN��5HVXOWV�JLYHQ�LQ�WKH�IRUPXODH�ERRN�PD\�EH�TXRWHG�ZLWKRXW�SURRI�
(TXLOLEULXP�RI�ULJLG�ERGLHV� Problems involving parallel and QRQ�SDUDOOHO�FRSODQDU�IRUFHV��Problems may include rods or ladders resting against smooth or rough vertical walls and on VPRRWK�RU�URXJK�JURXQG�
Calculators Students are expected to have available a calculator with at least
the folORZLQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DQ\�RWKHU�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
(QHUJ\�VWRUHG�LQ�DQ�HODVWLF�VWULQJ�RU�VSULQJ� 6LPSOH�SUREOHPV�XVLQJ�WKH�ZRUN�HQHUJ\�SULQFLSOH�LQYROYLQJ�NLQHWLF�energy, potential energy and HODVWLF�HQHUJ\�
3 Further dynamics
What students need to learn:
Newton’s laws of motion, for a particle moving in RQH�GLPHQVLRQ��ZKHQ�WKH�DSSOLHG�IRUFH�LV�YDULDEOH�
The solution of the resulting equations will be consistent with the level of calculus in units C2, &��DQG�&���3UREOHPV�PD\�LQYROYH�the law of gravitation, ie the LQYHUVH�VTXDUH�ODZ�
6LPSOH�KDUPRQLF�PRWLRQ� Proof that a particle moves with simple harmonic motion in a given
situation may be required (ie
showing that = −ω 2x x��
Geometric or calculus methods RI�VROXWLRQ�ZLOO�EH�DFFHSWDEOH��Students will be expected to be familiar with standard formulae, which may be quoted without SURRI�
Oscillations of a particle attached to the end of an HODVWLF�VWULQJ�RU�VSULQJ�
Oscillations will be in the direction RI�WKH�VWULQJ�RU�VSULQJ�RQO\�
Calculators Students are expected to have available a calculator with at least
WKH�IROORZLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DQ\�RWKHU�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
Relative motion of two particles, including relative GLVSODFHPHQW�DQG�UHODWLYH�YHORFLW\�
Problems may be set in vector form and may involve problems of interception or closest approach including the determination of course required for closest DSSURDFK�
2 Elastic collisions in two dimensions
What students need to learn:
Oblique impact of smooth elastic spheres and a VPRRWK�VSKHUH�ZLWK�D�¿[HG�VXUIDFH�
3 Further motion of particles in one dimension
What students need to learn:
Resisted motion of a particle moving in a straight OLQH�
The resisting forces may include the forms a + bv and a + bv2
where a and b are constants and v is the VSHHG�
'DPSHG�DQG�RU�IRUFHG�KDUPRQLF�PRWLRQ� The damping to be proportional to WKH�VSHHG��6ROXWLRQ�RI�WKH�UHOHYDQW�differential equations will be H[SHFWHG�
4 Stability
What students need to learn:
Finding equilibrium positions of a system from FRQVLGHUDWLRQ�RI�LWV�SRWHQWLDO�HQHUJ\�
Positions of stable and unstable HTXLOLEULXP�RI�D�V\VWHP�
Unit M5 Mechanics 5 GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional)A2 optional unit
M5.1 Unit description
Applications of vectors in mechanics; variable mass; moments of LQHUWLD�RI�D�ULJLG�ERG\��URWDWLRQ�RI�D�ULJLG�ERG\�DERXW�D�¿[HG�VPRRWK�D[LV�
M5.2 Assessment information
Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�0���0���0��DQG�0��DQG�WKHLU�SUHUHTXLVLWHV�DQG�DVVRFLDWHG�IRUPXODH��WRJHWKHU�ZLWK�D�NQRZOHGJH�of scalar and vector products, and of differential equations as VSHFL¿HG�LQ�)3���LV�DVVXPHG�DQG�PD\�EH�WHVWHG�
Calculators Students are expected to have available a calculator with at least
the followLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
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0RPHQW�RI�D�IRUFH�XVLQJ�D�YHFWRU�SURGXFW� The moment of a force F about O�LV�GH¿QHG�DV�r�× F, where r is the position vector of the point of application of F�
The analysis of simple systems of forces in three GLPHQVLRQV�DFWLQJ�RQ�D�ULJLG�ERG\�
The reduction of a system of forces acting on a body to a single force, single couple or a couple and a force acting through a VWDWHG�SRLQW�
2 Variable mass
What students need to learn:
0RWLRQ�RI�D�SDUWLFOH�ZLWK�YDU\LQJ�PDVV� Students may be required to derive an equation of motion from ¿UVW�SULQFLSOHV�E\�FRQVLGHULQJ�WKH�change in momentum over a small WLPH�LQWHUYDO�
Unit S1 Statistics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit
S1.1 Unit description
Mathematical models in probability and statistics; representation and summary of data; probability; correlation and regression; discrete random variables; discrete distributions; the Normal GLVWULEXWLRQ�
Calculators Students are expected to have available a calculator with at least
the followinJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,
1
x, xy
, ln x, ex, x!, sine, cosine and
tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�
Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�
Drawing of histograms, stem and leaf diagrams or box plots will not be the direct focus of examination TXHVWLRQV�
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Data may be discrete, continuous, JURXSHG�RU�XQJURXSHG��8QGHUVWDQGLQJ�DQG�XVH�RI�FRGLQJ�
Measures of dispersion — variance, standard GHYLDWLRQ��UDQJH�DQG�LQWHUSHUFHQWLOH�UDQJHV�
Simple interpolation may EH�UHTXLUHG��,QWHUSUHWDWLRQ�of measures of location and GLVSHUVLRQ�
6NHZQHVV��&RQFHSWV�RI�RXWOLHUV� 6WXGHQWV�PD\�EH�DVNHG�WR�illustrate the location of outliers RQ�D�ER[�SORW��$Q\�UXOH�WR�LGHQWLI\�RXWOLHUV�ZLOO�EH�VSHFL¿HG�LQ�WKH�TXHVWLRQ�
6XP�DQG�SURGXFW�ODZV� Use of tree diagrams and Venn GLDJUDPV��6DPSOLQJ�ZLWK�DQG�ZLWKRXW�UHSODFHPHQW�
4 Correlation and regression
What students need to learn:
6FDWWHU�GLDJUDPV��/LQHDU�UHJUHVVLRQ� Calculation of the equation of a linear regression line using WKH�PHWKRG�RI�OHDVW�VTXDUHV��Students may be required to draw this regression line on a scatter GLDJUDP�
Explanatory (independent) and response �GHSHQGHQW��YDULDEOHV��$SSOLFDWLRQV�DQG�LQWHUSUHWDWLRQV��
8VH�WR�PDNH�SUHGLFWLRQV�ZLWKLQ�WKH�range of values of the explanatory variable and the dangers of H[WUDSRODWLRQ��'HULYDWLRQV�ZLOO�QRW�EH�UHTXLUHG��9DULDEOHV�RWKHU�WKDQ�x and y�PD\�EH�XVHG��/LQHDU�FKDQJH�RI�YDULDEOH�PD\�EH�UHTXLUHG�
The probability function and the cumulative GLVWULEXWLRQ�IXQFWLRQ�IRU�D�GLVFUHWH�UDQGRP�YDULDEOH�
Simple uses of the probability function p(x) where p(x) = P(X = x).
Use of the cumulative distribution IXQFWLRQ��
F(x0 ) = P(X���x
0) = p( )x
x x≤∑
0
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the variance of X.
Knowledge and use of
E(aX + b) = aE(X) + b,
Var (aX + b) = a2 Var (X).
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6 The Normal distribution
What students need to learn:
The Normal distribution including the mean, variance and use of tables of the cumulative distribution IXQFWLRQ�
Knowledge of the shape and the symmetry of the distribution LV�UHTXLUHG��.QRZOHGJH�RI�WKH�probability density function is QRW�UHTXLUHG��'HULYDWLRQ�RI�WKH�mean, variance and cumulative distribution function is not UHTXLUHG��,QWHUSRODWLRQ�LV�QRW�QHFHVVDU\��4XHVWLRQV�PD\�LQYROYH�the solution of simultaneous HTXDWLRQV�
Unit S2 Statistics 2 GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit
S2.1 Unit description
The Binomial and Poisson distributions; continuous random YDULDEOHV��FRQWLQXRXV�GLVWULEXWLRQV��VDPSOHV��K\SRWKHVLV�WHVWV�
S2.2 Assessment information
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For the continuous random variable X having probability density function f(x),
P(a < X ≤ b) = f( ) dx xa
b
∫ .
f(x) = .
1 The Binomial and Poisson distributions
What students need to learn:
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Students will be expected to use the additive property of the Poisson distribution — eg if the number of events per minute ∼ Po(λ) then the number of events per 5 minutes ∼ Po(5λ).
The mean and variance of the binomial and Poisson GLVWULEXWLRQV�
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The probability density function and the cumulative distribution function for a continuous random YDULDEOH��
Use of the probability density function f(x), where
P(a < X ≤ b) = f( ) dx xa
b
∫ .
Use of the cumulative distribution function
F(x0) = P(X ≤ x
0) = f( ) dx x
x
−∞∫0
.
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3 Continuous distributions
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Use of the Normal distribution as an approximation to the binomial distribution and the Poisson distribution, with the application of the continuity FRUUHFWLRQ�
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Unit S3 Statistics 3 GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit
S3.1 Unit description
Combinations of random variables; sampling; estimation, FRQ¿GHQFH�LQWHUYDOV�DQG�WHVWV��JRRGQHVV�RI�¿W�DQG�FRQWLQJHQF\�tables; regression and correlation
S3.2 Assessment information
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Unit D1 Decision Mathematics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit
D1.1 Unit description
Algorithms; algorithms on graphs; the route inspection problem; FULWLFDO�SDWK�DQDO\VLV��OLQHDU�SURJUDPPLQJ��PDWFKLQJV�
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In a list containing N items the ‘middle’ item has position [12 (N + 1)] if N is odd [
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2 Algorithms on graphs
A graph G consists of points (vertices or nodes) which are connected by lines (edges or arcs).
A subgraph of G is a graph, each of whose vertices belongs to G and each of whose edges belongs to G.
If a graph has a number associated with each edge (usually called its weight) then the graph is called a
weighted graph or network.
The degree or valency of a vertex is the number of edges incident to it. A vertex is odd (even) if it has odd
(even) degree.
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Two vertices are connected if there is a path between them. A graph is connected if all its vertices are
connected.
If the edges of a graph have a direction associated with them they are known as directed edges and the graph
is known as a digraph.
A tree is a connected graph with no cycles.
A spanning tree of a graph G is a subgraph which includes all the vertices of G and is also a tree.
A minimum spanning tree (MST) is a spanning tree such that the total length of its arcs is as small as
possible. (MST is sometimes called a minimum connector.)
A graph in which each of the n vertices is connected to every other vertex is called a complete graph.
4 Critical path analysis
The WRWDO�ÀRDW F(i, j) of activity (i, j��LV�GH¿QHG�WR�EH�F(i, j) = lj – e
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6 Matchings
A bipartite graph consists of two sets of vertices X and Y. The edges only join vertices in X to vertices in Y,
not vertices within a set. (If there are r vertices in X and s vertices in Y then this graph is Kr,s
.)
A matching is the pairing of some or all of the elements of one set, X, with elements of a second set, Y. If
every member of X is paired with a member of Y the matching is said to be a complete matching.
Unit D2 Decision Mathematics 2 GCE Mathematics, GCE AS and GCE Further Mathematics, GCE AS and GCE Further Mathematics (Additional) A2 optional unit
D2.1 Unit description
Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, G\QDPLF�SURJUDPPLQJ��ÀRZV�LQ�QHWZRUNV
D2.2 Assessment information
Prerequisites and Preamble
Prerequisites
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optimal solution.
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cells is less than (m + n – 1).
In the transportation problem:
The shadow costs Ri , for the ith row, and K
j , for the jth column, are obtained by solving R
i + K
j = C
ij for
occupied cells, taking R1 = 0 arbitrarily.
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ij = C
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vertex may be revisited.
For three vertices A, B and C, the triangular inequality is ‘length AB ≤ length AC + length CB’.
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4 Further linear programmingThe simplex tableau for the linear programming problem:
Maximise P = 14x + 12y + 13z,
Subject to 4x + 5y + 3z������
5x + 4y + 6z������
will be written as
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s 5 4 6 0 1 24
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where r and s are slack variables.
5 Game theoryA two-person game is one in which only two parties can play.
A zero-sum game is one in which the sum of the losses for one player is equal to the sum of the gains for the
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