CAIE A-Level Mathematics answers, written to the spec
26 topics · specification 9709 · explanations and examiner marking in CAIE’s own language
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This is the full CAIE A-Level Mathematics specification, code 9709, as elid holds it — 26 topics, each with the terminology CAIE marks against. Every explanation and every mark elid gives you is written against this list, not a generic A-level one, because the same question answered in AQA, Edexcel or OCR language loses marks it should have earned.
The CAIE A-Level Mathematics specification, topic by topic
Quadratics
Assessed in Paper 1
Quadratic expressions, completing the square, discriminant, roots of quadratic equations, and quadratic inequalities.
Specification terms elid matches
quadratic
discriminant
completing the square
real roots
no real roots
equal roots
quadratic inequality
vertex
maximum value
minimum value
quadratic formula solution
discriminant b^2-4AC
completing the square method
vertex of parabola
Functions
Assessed in Paper 1
Definition of functions, domain and range, one-to-one functions, composite functions, inverse functions, and graphical transformations.
Specification terms elid matches
function
domain
range
one-to-one function
composite function
inverse function
notation
graph of function
translation
reflection
stretch
scale factor
domain of Definition
range of values
Coordinate Geometry
Assessed in Paper 1
Straight lines, gradient, parallel and perpendicular lines, midpoints, distance between two points, and equations of circles.
Binomial expansion for positive integer powers, arithmetic progressions, and geometric progressions (including sum to infinity).
Specification terms elid matches
binomial expansion
binomial coefficient
arithmetic progression
geometric progression
common difference
common ratio
first term
general term
sum of progression
sum to infinity
sigma notation
nth term of arithmetic progression
sum of arithmetic progression
nth term of geometric progression
Differentiation (P1)
Assessed in Paper 1
Derivative as a rate of change, differentiation of simple functions, chain rule, tangent and normal equations, increasing/decreasing functions, stationary points, and practical optimization.
Specification terms elid matches
Derivative
differentiation
gradient of curve
tangent
normal
increasing function
decreasing function
stationary point
maximum
minimum
rate of change
chain rule
second Derivative
Derivative dy/dx
Integration (P1)
Assessed in Paper 1
Integration as the reverse of differentiation, integration of polynomial and simple composite functions, definite integrals, area under a curve, and volume of revolution.
Specification terms elid matches
integration
integral
definite integral
indefinite integral
constant of integration
area under curve
volume of revolution
trapezium rule
indefinite integral of polynomial
constant of integration C
definite integral limits
area under curve integral
volume of revolution about x-axis
Algebra (P3)
Assessed in Paper 3
Modulus functions, polynomial division, factor and remainder theorems, partial fractions, and binomial expansion for negative or fractional powers.
Specification terms elid matches
modulus function
modulus equation
modulus inequality
algebraic division
factor theorem
remainder theorem
partial fractions
binomial expansion
rational expression
modulus equation |f(x)|=g(x)
polynomial long division
factor theorem f(a)=0
remainder theorem f(a)
partial fractions linear factors
Logarithmic and Exponential Functions
Assessed in Paper 3
Exponential and logarithmic functions, e and ln, laws of logarithms, solving equations, and linearizing non-linear graphs.
Specification terms elid matches
exponential
logarithm
natural log
base e
laws of logs
change of base
linearization
log-log graph
semi-log graph
exponential function e^x
natural logarithm ln(x)
laws of logarithms product
laws of logarithms quotient
laws of logarithms power
Trigonometry (P3)
Assessed in Paper 3
Secant, cosecant, and cotangent, trigonometric identities, addition and double-angle formulas, and R-form expressions.
Specification terms elid matches
Secant
cosecant
cotangent
sec
cosec
cot
trigonometric identity
addition formula
double-angle formula
harmonic form
R-form
trigonometric equation
Secant sec(x)
cosecant cosec(x)
Differentiation (P3)
Assessed in Paper 3
Derivatives of e^x, ln(x), sin(x), cos(x), tan(x), product rule, quotient rule, implicit differentiation, and parametric differentiation.
Specification terms elid matches
differentiation
Derivative of exponential
Derivative of natural log
Derivative of trigonometric function
product rule
quotient rule
implicit differentiation
parametric differentiation
tangent
normal
differentiation of e^x
differentiation of ln(x)
differentiation of sin cos tan
product rule u v' + v u'
Integration (P3)
Assessed in Paper 3
Integration of exponential, logarithmic, and trigonometric functions, integration by substitution, integration by parts, partial fractions, and solving first-order differential equations.
Specification terms elid matches
integration
integral of exponential
integral of reciprocal
integral of trigonometric function
integration by substitution
integration by parts
integration using partial fractions
differential equation
separation of variables
integration of e^ax
integration of 1/(ax+b)
integration of sin cos sec^2
integration by substitution u
integration by parts integral u dv
Numerical Solution of Equations
Assessed in Paper 3
Location of roots by sign change, interval bisection, and iterative formula x_(n+1) = g(x_n) including cobweb/staircase diagrams.
Specification terms elid matches
numerical solution
root of equation
sign change
iteration
iterative formula
convergence
divergence
cobweb diagram
staircase diagram
root Location sign change
iterative formula x_(n+1)
convergence of iteration
cobweb diagram plot
staircase diagram plot
Vectors (P3)
Assessed in Paper 3
Vector coordinates in 3D, vector arithmetic, unit vectors, position vectors, scalar product of two vectors, vector equation of a straight line, and parallel/skew/intersecting lines.
Specification terms elid matches
vector
3D vectors
magnitude
unit vector
position vector
displacement vector
scalar product
dot product
vector equation of line
intersecting lines
parallel lines
skew lines
3D position vector r
vector magnitude |v|
Complex Numbers
Assessed in Paper 3
Definition of complex numbers, real and imaginary parts, complex arithmetic, conjugate, Argand diagram, modulus-argument form, and locus problems.
Specification terms elid matches
complex number
imaginary number
real part
imaginary part
complex arithmetic
conjugate
Argand diagram
modulus
argument
modulus-argument form
locus
imaginary unit i
complex number z=a+ib
real part Re(z)
Forces and Equilibrium (M1)
Assessed in Paper 4
Forces as vectors, coplanar forces, resolving forces, friction, coefficient of friction, and static equilibrium.
Specification terms elid matches
force
vector force
coplanar forces
resolving forces
friction
coefficient of friction
limiting friction
equilibrium
normal reaction
resultant force
inclined plane
coplanar forces resolving
resolving Components F cos theta
limiting friction F_max=mu R
Kinematics of Motion in a Straight Line (M1)
Assessed in Paper 4
Motion in a straight line, displacement, velocity, acceleration, constant acceleration (SUVAT) equations, displacement-time/velocity-time graphs, and variable acceleration using calculus.
Specification terms elid matches
kinematics
motion in straight line
displacement
velocity
acceleration
SUVAT
constant acceleration
displacement-time graph
velocity-time graph
variable acceleration
calculus in mechanics
SUVAT constant acceleration
displacement-time graph slope
velocity-time graph area
Momentum (M1)
Assessed in Paper 4
Definition of momentum, impulse of a force, conservation of linear momentum, and direct collisions.
Specification terms elid matches
momentum
impulse
conservation of momentum
collision
mass
velocity
resultant force
change in momentum
linear momentum p=mv
impulse of force I=F t
principle of conservation of momentum
direct collision particles
coalescing colliding bodies
change in momentum delta p
Newton's Laws of Motion (M1)
Assessed in Paper 4
Newton's three laws of motion, F = ma, connected particles, pulleys, motion of a lift, and towing problems.
Specification terms elid matches
Newton's laws
Newton's first law
Newton's second law
Newton's third law
equation of motion
resultant force
mass
acceleration
F=ma
connected particles
pulley
tension
thrust
Newton's first law inertia
Energy, Work and Power (M1)
Assessed in Paper 4
Work done by a force, kinetic energy, gravitational potential energy, conservation of energy, work-energy principle, power, and P = Fv.
Specification terms elid matches
work done
force
displacement
kinetic energy
potential energy
gravitational potential energy
conservation of energy
work-energy principle
power
P=Fv
driving force
resistance
work done by force W=F s cos theta
kinetic energy KE=1/2 m v^2
Representation of Data (S1)
Assessed in Paper 5
Histograms, cumulative frequency graphs, box-and-whisker plots, stem-and-leaf diagrams, mean, standard deviation, and variance.
Specification terms elid matches
representation of data
histogram
cumulative frequency
box-and-whisker plot
stem-and-leaf diagram
class boundary
frequency density
mean
standard deviation
variance
mode
median
quartile
outlier
Permutations and Combinations (S1)
Assessed in Paper 5
Factorials, permutations, combinations, arrangement of objects in a line, and selection of objects.
Specification terms elid matches
permutations
combinations
nPr
nCr
factorial
arrangement
selection
order matters
order does not matter
identical objects
constraints
permutation nPr formula
combination nCr formula
factorial n!
Probability (S1)
Assessed in Paper 5
Sample space, event probability, addition law, multiplication law, mutually exclusive events, independent events, conditional probability, and tree diagrams.
Specification terms elid matches
probability
Sample space
addition law
multiplication law
mutually exclusive
independent
conditional probability
Venn diagram
tree diagram
probability of event P(A)
addition law P(A U B)
multiplication law P(A n B)
mutually exclusive P(A n B)=0
independent events P(A n B)=P(A)P(B)
Discrete Random Variables (S1)
Assessed in Paper 5
Discrete random variables, probability distribution tables, expectation E(X), variance Var(X), and the Binomial distribution.
Specification terms elid matches
Discrete random variable
probability distribution
expectation
E(X)
variance
Var(X)
Binomial distribution
number of trials
probability of success
mean of binomial
variance of binomial
Discrete probability distribution table
expectation E(X) = sum x p
variance Var(X) = E(X^2) - [E(X)]^2
The Normal Distribution (S1)
Assessed in Paper 5
Normal distribution, bell curve symmetry, standard Normal variable Z, use of Normal tables, and Normal approximation to the Binomial distribution.
Specification terms elid matches
Normal distribution
bell curve
standard Normal variable
Z-score
Normal table
mean
variance
standard deviation
standardising
approximation
continuity correction
Normal distribution N(mu, sigma^2)
standard Normal variable Z=(X-mu)/sigma
Normal distribution tables phi(z)
How the papers are weighted
elid uses these weightings when it decides which topics deserve your revision time first.
Paper
Share of the qualification
Paper 1
30%
Paper 3
30%
Paper 4
20%
Paper 5
20%
Marked the way CAIE marks it
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