Acet syllabus 1 fac

Actuarial Common Entrance Test (ACET)
ACET - January 2012 Examinations Institute of Actuaries of India
The FAC course
The Foundation Course consists of a set of eight chapters of notes covering the
following ideas:
Chapter 1 Notation
Chapter 2 Numerical Methods I
Chapter 3 Mathematical constants and standard functions
Chapter 4 Algebra
Chapter 5 Numerical Methods II
Chapter 6 Differentiation
Chapter 7 Integration
Chapter 8 Vectors and matrices
We recommend that you work through the sections that you are unsure of, completing the
questions that are given. If you need further practice, there is a Question and Answer
Bank and a Summary Test. These both cover material from all the chapters.
Section 5 of this study guide gives you an index of the topics that are covered in order to
identify quickly which chapters you need to look at.
When you are working through the Core Technical subjects you can continue to use this
course as a reference document if you come across areas of mathematics which you are
still unhappy about.

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1 Syllabus
1.1 Syllabus
This syllabus has been written to help in designing introductory tuition
material for new students. It is not an official syllabus of the profession. The topics are
all required within the Core Technical subjects. From past experience we know that
some students can be a little rusty on mathematical techniques so we have designed a
course to help students brush up on their knowledge.
Unlike other actuarial subjects, there is no official syllabus, Core Reading or formal
examination.
(a) Mathematical Notation, Terminology and Methods
(a)(i) Be familiar with standard mathematical notation and terminology, so as to be
able to understand statements such as the following:
1. , , , , 3: n n n
a b c n n a b c∃ ∈ ≥ + =/
2. 2
( ,0], {0}x x +
∀ ∈ −∞ ∈ ∪
3. { : 1,2,3, }x x= = …
4. x x∈∅ ⇒ = −163
5. “Zero is a non-negative integer; π is a positive real number.”
6. “ f x( ) vanishes as x tends to − ∞ , is not defined when x = 0, but takes
positive values for sufficiently large x .”
(a)(ii) Know the representations and names of the letters of the Greek alphabet that are
commonly used in mathematical, statistical and actuarial work, including in
particular, the following letters:
– lower-case: α, β, γ, δ, ε, θ, κ, λ, μ, ν, π, ρ, σ, τ, φ, χ, ψ, ω
– upper-case: Γ, Δ, Θ, Π, Σ, Φ, Ω
(a)(iii) Understand the meaning of the following commonly used conventions:
– round brackets used to denote negative currency amounts,
– K and m used as abbreviations for “thousand” and “million”
– Δ used to denote the signed magnitude of a change in a quantity,
– “iff” used as an abbreviation for “if and only if”.

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(a)(iv) Understand the concept of a mathematical proof and the meaning of
“necessary”, “sufficient” and “necessary and sufficient” as they are used in
mathematical derivations.
(a)(v) Prove a result using the method of mathematical induction.
(b) Numerical methods
(b)(i) Evaluate numerical expressions using an electronic calculator with the
following features: arithmetic functions ( + − × ÷ ), powers ( yx
) and roots
( yx ), exponential (ex
) and natural log (ln x ) functions. (The following
features are also useful but not essential: factorial function ( n!), combinations
( n rC ), hyperbolic tangent function and its inverse ( tanh x and tanh−1
x ),
fraction mode, at least one memory and an “undo” facility. Statistical and
financial functions are not required.) Students should be able to make efficient
use of memories, brackets and/or the calculator stack.
(b)(ii) Estimate the numerical value of expressions without using a calculator and
apply reasonableness tests to check the result of a calculation.
(b)(iii) Quote answers to a specified or appropriate number of decimal places or
significant figures (using the British convention for representing numbers), and
be able to assess the likely accuracy of the result of a calculation that is based
on rounded or approximated data values.
(b)(iv) Be able to carry out consistent calculations using a convenient multiple of a
standard unit (eg working in terms of £000s).
(b)(v) Express answers, where appropriate, in the form of a percentage (%) or as an
amount per mil (‰).
(b)(vi) Calculate the absolute change, the proportionate change or the percentage
change in a quantity (using the correct denominator and sign, where
appropriate) and understand why changes in quantities that are naturally
expressed as percentages, such as interest rates, are often specified in terms of
“basis points”.
(b)(vii)Calculate the absolute error, the proportionate error or the percentage error in
comparisons involving “actual” versus “expected” values or approximate
versus accurate values (using the correct denominator and sign, where
appropriate).

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(b)(viii)Determine the units of measurement (dimensions) of a quantity and understand
the advantages of using dimensionless quantities in certain situations.
(b)(ix) Use linear interpolation to find an approximate value for a function or the
argument of a function when the value of the function is known at two
neighbouring points.
(b)(x) Apply simple iterative methods, such as the bisection method or the Newton-
Raphson method, to solve non-linear equations.
(b)(xi) Carry out simple calculations involving vectors, including the use of row/column
vectors and unit vectors, addition and subtraction of vectors, multiplication of a
vector by a scalar, scalar multiplication (“dot product”) of two vectors,
determining the magnitude and direction of a vector, finding the angle between
two vectors and understanding the concept of orthogonality.
(b)(xii)Carry out calculations involving matrices, including transposition of a matrix,
addition and subtraction of matrices, multiplication of a matrix by a scalar,
multiplication of two appropriately sized matrices, calculating the determinant
of a matrix, calculating and understanding the geometrical interpretation of
eigenvectors and eigenvalues, finding the inverse of a 2 2× matrix and using
matrices to solve systems of simultaneous linear equations.
(c) Mathematical Constants and Standard Functions
(c)(i) Be familiar with the mathematical constants π and e .
(c)(ii) Understand and apply the definitions and basic properties of the functions
xn
(where n may be negative or fractional), cx
(where c is a positive constant),
exp( )x [ = ex
], and ln x [ = loge x or log x ].
(c)(iii) Sketch graphs of simple functions involving the basic functions in (c)(ii) by
identifying key points, identifying and classifying turning points, considering the
sign and gradient, and analysing the behaviour near 0, 1, ± ∞ or other critical
values.
(c)(iv) Simplify and evaluate expressions involving the functions x (absolute value),
[ ]x (integer part), max( )… and min( )… , and understand the concept of a
bounded function. [The notation ( )x − +
100 will also be used as an
abbreviation for max( , )x −100 0 .]

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(c)(v) Simplify and evaluate expressions involving the factorial function n! for non-
negative integer values of the argument and the gamma function Γ( )x for
positive integer and half-integer values of the argument.
(c)(vi) Understand the concept of a complex number and be able to simplify
expressions involving i = −1 , including calculating the complex conjugate.
(c)(vii) Calculate the modulus and argument of a complex number, represent a complex
number on an Argand diagram or in polar form ( z rei
= θ
).
(c)(viii) Apply Euler’s formula e iiθ
θ θ= +cos sin and use the basic properties of the
sine and cosine functions to simplify expressions involving complex numbers,
including determining the real and imaginary parts of an expression.
(c)(ix) Understand the correspondence between the factors of a polynomial expression
and the roots of a polynomial equation and appreciate that a polynomial
equation of degree n with real coefficients will, in general, have n roots
consisting of conjugate pairs and/or real values.
(d) Algebra
(d)(i) Manipulate algebraic expressions involving powers, logs, polynomials and
fractions.
(d)(ii) Solve simple equations, including simultaneous equations (not necessarily
linear) by rearrangement, substitution, cancellation, expansion and
factorisation.
(d)(iii) Solve an equation that can be expressed as a quadratic equation (with real
roots) by factorisation, by “completing the square” or by applying the quadratic
formula, and identify which of the roots is appropriate in a particular context.
(d)(iv) Solve inequalities (“inequations”) in simple cases and understand the concept of
a “strict” or “weak” inequality.
(d)(v) State and apply the arithmetic-geometric mean inequality, and know the
conditions under which equality holds.
(d)(vi) Understand and apply the Σ and Π notation for sums and products, including
sums over sets (eg Σ
i≥0
) and repeated sums.

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(d)(vii)Calculate the sum of a series involving finite arithmetic or geometric
progressions or non-terminating geometric progressions using the formulae:
ΣAP a n d a ln n= + − +2 2
2 1( ( ) ) ( )or ,
ΣGP
a r
r
n
=
−
−
( )1
1
and
Σ∞ =
−
GP
a
r1
,
and be able to determine when a non-terminating geometric series converges.
(d)(viii) Apply the formulae: k n n
k
n
=
∑ = +
1
1
2
1( ) and k n n n
k
n
2
1
1
6
1 2 1
=
∑ = + +( )( ) .
(d)(ix) Solve simple first or second order difference equations (recurrence relations),
including applying boundary conditions, by inspection or by means of an
auxiliary equation.
(d)(x) Recognise and apply the binomial expansion of expressions of the form ( )a b n
+
where n is a positive integer, and ( )1+ x p
for any real value of p and, in the
latter case, determine when the series converges.
(e) Calculus
(e)(i) Understand the concept of a limit (including limits taken from one side) and
evaluate limits in simple cases using standard mathematical notation, including
the use of “order” notation O x( ) and o x( ) , and the sup/ lub and inf / glb
functions (considered as generalisations of max and min ).
(e)(ii) Understand the meaning of a derivative as the rate of change of a function when
its argument is varied (in particular, for functions dependent on t , the time
measured from a specified reference point), including the interpretation of a
derivative as the gradient of a graph.
(e)(iii) Differentiate the standard functions xn
, cx
, ex
and ln x .
(e)(iv) Evaluate derivatives of sums, products (using the product rule), quotients (using
the quotient rule) and “functions of a function” (using the chain rule).
(e)(v) Understand the concept of a higher-order (repeated) derivative and be familiar
with the mathematical notation used to denote such quantities.

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(e)(vi) Use differentiation to find the maximum or minimum value of a function over a
specified range (including the application of a monotone function, such as the
natural log function, to simplify the calculation) and determine the nature of
stationary points.
(e)(vii) Understand the meaning of a partial derivative and how to express a partial
derivative in standard mathematical notation, and be able to evaluate partial
derivatives in simple cases. Find extrema of functions of two variables.
(e)(viii)Use the method of Lagrangian multipliers.
(e)(ix)Understand the meaning of an indefinite integral as the anti-derivative of a
function and the meaning of a definite integral as the limit of a sum of
infinitesimal elements, including the interpretation of a definite integral as the
area under a graph.
(e)(x) Integrate the standard functions xn
, cx
and ex
.
(e)(xi) Evaluate indefinite and definite integrals by inspection, by identifying and
applying an appropriate substitution, by integration by parts, by using simple
partial fractions or by a combination of these methods.
(e)(xii) Determine when a definite integral converges.
(e)(xiii)Understand the meaning of a multiple integral and how to express a multiple
integral in standard mathematical notation, and be able to evaluate a double
integral as a repeated integral in simple cases, including determining the
correct limits of integration. Swap the order of integration.
(e)(xiv)Apply the trapezium rule to find the approximate value of an integral.
(e)(xv)State and apply Taylor series and Maclaurin series in their simplest form,
including using these to determine the approximate change in a function when
the argument is varied by a small amount. (Knowledge of the error terms is not
required.)
(e)(xvi)Recognise and apply the Taylor series expansions for ex
and ln( )1+ x and, in
the latter case, determine when the series converges.
(e)(xvii)Solve simple ordinary first and second order differential equations, including
applying boundary conditions, by direct integration (which may involve a
function of the dependent variable), by separation of variables or by applying an
integrating factor.

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ACET - January 2012 Examination Institute of Actuaries of India
(e)(xvii)Differentiate expressions involving definite integrals with respect to a
parameter, including cases where the limits of integration are functions of the
parameter.
(g) General
(g)(i) Be familiar with the currency systems of the United Kingdom (pounds and pence
sterling), the United States (dollars and cents), the European monetary system
(Euros and cents) and other major economies, and be able to interpret and write
down currency amounts using these systems.
(g)(ii) Be familiar with the Gregorian calendar, including determining when a
specified year is a leap year, the concepts of calendar years, quarters and tax
years, and the abbreviations commonly used to represent dates in the United
Kingdom, Europe and the United States.
(g)(iii) Understand the distinction between “expression”/“equation”/“formula” and
“term”/“factor”.
(g)(iv) Understand the meaning of the words “gross”, and “net”.
(g)(v) Be able to spell the following words correctly: actuarial, appropriately,
basically, benefit, benefiting, bias(s)ed, calendar, cancelled, commission,
consensus, correlation, cyclically, deferred, definitely, formatted, fulfil, gauge,
hierarchy, immediately, independence, instalment (British spelling), lose, loose,
millennium, necessary, occasion, occurred/occurring, offered, orthogonal, paid,
particularly, pensioner, precede, proceed, receive, referred, relief, seize,
separate, similarly, specifically, supersede, targeted, theorem, until, yield.
(g)(vi) Be able to determine the correct member of word pairs according to context: eg
affect/effect, principal/principle, dependant/dependent.
(g)(vii)Be able to distinguish between the singular and plural forms of words of Latin
or Greek origin, including the following: criterion/criteria, formula/formulae,
analysis/analyses. [The word “data” may be treated as singular or plural,
according to the preferences of individual authors/speakers.]
(g)(viii)Be familiar with commonly used Latin expressions and abbreviations such as
“per annum”, “vice versa”, “status quo”, “pro rata”, “ie”, “eg”, “cf”, “sic”
and “stet”.

Acet syllabus 1 fac

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