Master Cambridge A Level Maths — from first principles to full marks

Builds the algebraic foundations required across the entire 9709 syllabus, from surds to polynomials and rational functions.

Covers the AS-level geometry, trigonometry, and series content that forms a major share of Paper 1 marks.

Develops AS differentiation and integration into the full A2 calculus toolkit, including differential equations and complex functions.

Completes the A2 Pure content by mastering compound-angle identities, 3-D vectors, and iterative numerical techniques.

Covers the full 9709 Statistics component — data representation, probability, distributions, and hypothesis testing.

Covers the full 9709 Mechanics component — kinematics, forces, Newton's laws, energy, and connected-body problems.

e.g. to locate the vertex of the graph of
y = ax2 + bx + c or to sketch the graph
e.g. to determine the number of real roots of the
equation ax2 + bx + c = 0. Knowledge of the term
‘repeated root’ is included.
By factorising, completing the square and using the
formula.
e.g. x + y + 1 = 0 and x2 + y2 = 25,
2x + 3y = 7 and 3x2 = 4 + 4xy.
e.g. x4 – 5x2 + 4 = 0, x x 6 1 0 − + = ,
tan2 x = 1 + tan x.

e.g. range of :x x
1 f 7 for x 1H and
range of :x x x 1 g for R 2 7 ! + . Including the
condition that a composite function gf can only be
formed when the range of f is within the domain of
e.g. finding the inverse of
Sketches should include an indication of the mirror
ncluding use of the terms ‘translation’, ‘reflection’
and ‘stretch’ in describing transformations.
Questions may involve algebraic or trigonometric
functions, or other graphs with given features.
line y = x.
: .
g.

e.g. given two points, or one point and the gradient.
Including calculations of distances, gradients,
midpoints, points of intersection and use of the
relationship between the gradients of parallel and
perpendicular lines.
ncluding use of the expanded form
x 2 + y 2 + 2gx + 2fy + c = 0.
Including use of elementary geometrical properties
of circles, e.g. tangent perpendicular to radius,
angle in a semicircle, symmetry.
Implicit differentiation is not included
.g. to determine the set of values of k for which the
line y = x + k intersects, touches or does not meet
a quadratic curve.

Including calculation of lengths and angles in
triangles and areas of triangles.

Including e.g. y = 3 sin x, y = 1 – cos 2x,
. tan y x 4
1r = + c m
.g. , cos sin 150 2
1 3 4
3
2
1 c − r = = .
No specialised knowledge of these functions is
required, but understanding of them as examples
of inverse functions is expected.
e.g. in proving identities, simplifying expressions
and solving equations.
e.g. solve sin x x 3 2 1 0for 1 1 −r r + = ,
sin cos 3 5 1 0 0 360 for 2 c c

Only an informal understanding of the idea of a limit
is expected.
e.g. includes consideration of the gradient of the
chord joining the points with x coordinates 2 and
(2 + h) on the curve y = x3. Formal use of the
general method of differentiation from first principles
is not required.
.g. find x
y
d
d , given y x2 5 3 = + .
Including connected rates of change, e.g. given the
rate of increase of the radius of a circle, find the rate
of increase of the area for a specific value of one of
the variables.Including use of the second derivative for identifying
maxima and minima; alternatives may be used in
questions where no method is specified.
Knowledge of points of inflexion is not included.

e.g. x x x 2 5 1d 3− + ;^ h , x x
e.g. to find the equation of the curve through (1, –2)
for which . x
y x2 1 d
d = +
Including simple cases of ‘improper’ integrals, such
as x x x x d and d
0
1 2
1
2
1 3--; ; .
A volume of revolution may involve a region not
bounded by the axis of rotation, e.g. the region
between y = 9 – x2 and y = 5 rotated about the
x-axis.

Graphs of y = |f(x)| and y = f(|x|) for non-linear
functions f are not included.
e.g. |3x – 2| = |2x + 7|, 2x + 5 < |x + 1|
e.g. to find factors and remainders, solve
polynomial equations or evaluate unknown
coefficients.
Including factors of the form (ax + b) in which the
coefficient of x is not unity, and including calculation
of remainders.

Including knowledge of the graph of y = ekx for both
positive and negative values of k.
y = kxn gives ln y = ln k + n ln x which is linear in
ln x and ln y
y = k (ax) gives ln y = ln k + x ln a which is linear in
x and ln y.

e.g. x
x
3 2
2 4−

• , x2 ln x, xel – x2.
  e.g. x = t – e2t , y = t + e2t.
  e.g. x2 + y2 = xy + 7

Knowledge of the general method of integration by
substitution is not required
e.g. use of double-angle formulae to integrate sin2 x
or cos2(2x).
Including use of sketch graphs in simple cases
to determine whether the trapezium rule gives an
over-estimate or an under-estimate.

Knowledge of the condition for convergence is not
included, but an understanding that an iteration
may fail to converge is expected.

Graphs of y = |f(x)| and y = f(|x|) for non-linear
functions f are not included
e.g. |3x – 2| = |2x + 7|, 2x + 5 < |x + 1|
e.g. to find factors and remainders, solve
polynomial equations or evaluate unknown
coefficients.
Including factors of the form (ax + b) in which the
coefficient of x is not unity, and including calculation
of remainders.
Excluding cases where the degree of the numerator
exceeds that of the denominator
Finding the general term in an expansion is not
included.
− −
2 2
1 1
Adapting the standard series to expand
e.g. x
` j
is included, and determining the set
of values of x for which the expansion is valid in
such cases is also included.

Including knowledge of the graph of y = ekx for both
positive and negative values of k.
e.g.
y = kxn gives ln y = ln k + n ln x which is linear in
ln x and ln y.
y = k (ax) gives ln y = ln k + x ln a which is linear in
x and ln y.

e.g. solving tan cot 4 i i + = , sec tan 2 5 2 −i i= ,
cos sin 3 2 1 i i + = .

Derivatives of sin–1 x and cos–1 x are not required.
e.g. x
x
−
2 4
+
3 2
, x2 ln x, xe1 – x2.
e.g. x = t – e2t , y = t + e2t.
e.g. x2 + y2 = xy + 7.
Including use in problems involving tangents and
normals

Derivatives of sin–1 x and cos–1 x are not required.e.g. x
x
−
2 4
+
3 2
, x2 ln x, xe1 – x2.
e.g. x = t – e2t , y = t + e2t.
e.g. x2 + y2 = xy + 7.
Including use in problems involving tangents and normals.

e.g. finding a pair of consecutive integers between
which a root lies.

e.g. ‘OABC is a parallelogram’ is equivalent to
= +
OB OA OC
.
The general form of the ratio theorem is not
included, but understanding that the midpoint of
AB has position vector OA OB
2
1
_
In 2 or 3 dimensions.
+
i
is expected.
e.g. finding the equation of a line given the position
vector of a point on the line and a direction vector,
or the position vectors of two points on the line.
Calculation of the shortest distance between two
skew lines is not required. Finding the equation of
the common perpendicular to two skew lines is
also not required.
e.g. finding the angle between two lines, and finding
the foot of the perpendicular from a point to a line;
questions may involve 3D objects such as cuboids,
tetrahedra (pyramids), etc.
Knowledge of the vector product is not required.

The introduction and evaluation of a constant of
proportionality, where necessary, is included.
Including any of the integration techniques from
topic 3.5 above

Notations Re z, Im z, |z|, arg z, z* should be known.
The argument of a complex number will usually
refer to an angle i such that
1 G
i
−r r
, but in
some cases the interval 0 2
G
i
1
r
may be more
convenient. Answers may use either interval unless
the question specifies otherwise.

Calculations are always required, not approximate
solutions by scale drawing.
Solutions by resolving are usually expected, but
equivalent methods (e.g. triangle of forces, Lami’s
Theorem, where suitable) are also acceptable;
these other methods are not required knowledge,
and will not be referred to in questions.
Terminology such as ‘about to slip’ may be used to
mean ‘in limiting equilibrium’ in questions.
e.g. the force exerted by a particle on the ground
is equal and opposite to the force exerted by the
ground on the particle

Including direct impact of two bodies where the
bodies coalesce on impact.
Knowledge of impulse and the coefficient of
restitution is not required

W = mg. In this component, questions are mainly
numerical, and use of the approximate numerical
value 10 (m s–2) for g is expected
Including, for example, motion of a particle on a
rough plane where the acceleration while moving
up the plane is different from the acceleration while
moving down the plane
e.g. particles connected by a light inextensible
string passing over a smooth pulley, or a car towing
a trailer by means of either a light rope or a light and tow-bar