- #1

ahsanxr

- 350

- 6

Here is the content which it covers:

Pure Mathematics:

1. Polynomials and rational functions

− recall and use the relations between the roots

and coefficients of polynomial equations, for

equations of degree 2, 3, 4 only;

− use a given simple substitution to obtain an

equation whose roots are related in a simple

way to those of the original equation;

− sketch graphs of simple rational functions,

including the determination of oblique

asymptotes, in cases where the degree of the

numerator and the denominator are at most 2

(detailed plotting of curves will not be required,

but sketches will generally be expected to show

significant features, such as turning points,

asymptotes and intersections with the axes).

2. Polar coordinates

− understand the relations between cartesian and

polar coordinates (using the convention

r [ 0), and convert equations of curves from

cartesian to polar form and vice versa;

− sketch simple polar curves, for 0 Y θ < 2π or

−π < θ Y π or a subset of either of these intervals

(detailed plotting of curves will not be required, but

sketches will generally be expected to show

significant features, such as symmetry, the form of

the curve at the pole and least/greatest values of

r);

− recall the formula ∫ β

α

2 r

2

1

dθ for the area of a sector,

and use this formula in simple cases.

3. Summation of series

− use the standard results forΣr , Σ 2

r , Σ 3

r to

find related sums;

− use the method of differences to obtain the sum of

a finite series, e.g. by expressing the general term

in partial fractions;

− recognise, by direct consideration of a sum to n

terms, when a series is convergent, and find the

sum to infinity in such cases.

4. Mathematical induction

− use the method of mathematical induction to

establish a given result (questions set may involve

divisibility tests and inequalities, for example);

− recognise situations where conjecture based on a

limited trial followed by inductive proof is a useful

strategy, and carry this out in simple cases e.g.

find the nth derivative of xex.

5. Differentiation and integration

− obtain an expression for (d2x/dx2) in cases where the

relation between y and x is defined implicitly or

parametrically;

− derive and use reduction formulae for the

evaluation of definite integrals in simple cases;

− use integration to find

mean values and centroids of two- and threedimensional

figures (where equations are

expressed in cartesian coordinates, including the

use of a parameter), using strips, discs or shells as

appropriate,

arc lengths (for curves with equations in cartesian

coordinates, including the use of a parameter, or in

polar coordinates),

surface areas of revolution about one of the axes

(for curves with equations in cartesian coordinates,

including the use of a parameter, but not for

curves with equations in polar coordinates).

6. Differential equations

− recall the meaning of the terms ‘complementary

function' and ‘particular integral' in the context of

linear differential equations, and recall that the

general solution is the sum of the complementary

function and a particular integral;

− find the complementary function for a second order

linear differential equation with constant

coefficients;

− recall the form of, and find, a particular integral for

a second order linear differential equation

in the cases where a polynomial or

ebx or a cos px + b sin px is a suitable form, and in

other simple cases find the appropriate

coefficient(s) given a suitable form of particular

integral;

− use a substitution to reduce a given differential

equation to a second order linear equation with

constant coefficients;

− use initial conditions to find a particular solution to a

differential equation, and interpret a solution in terms

of a problem modeled by a differential equation.

7. Complex numbers

understand de Moivre's theorem, for a positive integral

exponent, in terms of the geometrical effect of

multiplication of complex numbers;

− prove de Moivre's theorem for a positive integral

exponent;

− use de Moivre's theorem for positive integral exponent

to express trigonometrical ratios of multiple angles in

terms of powers of trigonometrical ratios of the

fundamental angle;

− use de Moivre's theorem, for a positive or negative

rational exponent

in expressing powers of sin θ and cos θ in terms of

multiple angles,

in the summation of series,

in finding and using the nth roots of unity.

8. Vectors

− use the equation of a plane in any of the forms

ax + by + cz = d or r.n. = p or r = a + λb + μc, and

convert equations of planes from one form to another

as necessary in solving problems;

− recall that the vector product a x b of two vectors can

be expressed either as I a I IbI sin θ n ˆ , where n ˆ is a

unit vector, or in component form as

(a2 b3 – a3 b2) i + (a3 b1 – a1 b3) j + (a1 b2 – a2 b1) k;

− use equations of lines and planes, together with scalar

and vector products where appropriate, to solve

problems concerning distances, angles and

intersections, including

determining whether a line lies in a plane, is parallel to

a plane or intersects a plane, and finding the point of

intersection of a line and a plane when it exists,

finding the perpendicular distance from a point to a

plane,finding the angle between a line and a plane, and the

angle between two planes,

finding an equation for the line of intersection of two

planes,

calculating the shortest distance between two skew

lines,

finding an equation for the common perpendicular to

two skew lines.

9. Matrices and linear spaces

− recall and use the axioms of a linear (vector) space

(restricted to spaces of finite dimension over the field

of real numbers only);

− understand the idea of linear independence, and

determine whether a given set of vectors is dependent

or independent;

− understand the idea of the subspace spanned by a

given set of vectors;

− recall that a basis for a space is a linearly

independent set of vectors that spans the space, and

determine a basis in simple cases;

− recall that the dimension of a space is the number of

vectors in a basis;

− understand the use of matrices to represent linear

transformations from

n →

m;

− understand the terms ‘column space', ‘row space',

‘range space' and ‘null space', and determine the

dimensions of, and bases for, these spaces in simple

cases;

− determine the rank of a square matrix, and use

(without proof) the relation between the rank, the

dimension of the null space and the order of the

matrix;

− use methods associated with matrices and linear

spaces in the context of the solution of a set of linear

equations;

− evaluate the determinant of a square matrix and find

the inverse of a non-singular matrix

(2 x 2 and 3 x 3 matrices only), and recall that the

columns (or rows) of a square matrix are

independent if and only if the determinant is nonzero;

− understand the terms ‘eigenvalue' and ‘eigenvector',

as applied to square matrices;

− find eigenvalues and eigenvectors of 2 x 2 and 3 x 3

matrices (restricted to cases where the eigenvalues

are real and distinct);

− express a matrix in the form QDQ−1, where D is a

diagonal matrix of eigenvalues and Q is a matrix

whose columns are eigenvectors, and use this

expression, e.g. in calculating powers of matrices.

Mechanics:

1. Momentum and impulse

− recall and use the definition of linear

momentum, and show understanding of its

vector nature (in one dimension only);

− recall Newton's experimental law and the

definition of the coefficient of restitution, the

property 0 Y e Y 1, and the meaning of the

terms ‘perfectly elastic' (e = 1) and ‘inelastic'

(e = 0);

− use conservation of linear momentum and/or

Newton's experimental law to solve problems

that may be modeled as the direct impact of two

smooth spheres or the direct or oblique impact

of a smooth sphere with a fixed surface;

− recall and use the definition of the impulse of a

constant force, and relate the impulse acting on

a particle to the change of momentum of the

particle (in one dimension only).

2. Circular motion

− recall and use the radial and transverse

components of acceleration for a particle

moving in a circle with variable speed;

− solve problems which can be modeled by the

motion of a particle in a vertical circle without

loss of energy (including finding the tension in a

string or a normal contact force, locating points

at which these are zero, and conditions for

complete circular motion).

3. Equilibrium of a rigid body under

coplanar forces

− understand and use the result that the effect of

gravity on a rigid body is equivalent to a single

force acting at the centre of mass of the body,

and identify the centre of mass by

considerations of symmetry in suitable cases;

− calculate the moment of a force about a point in

2 dimensional situations only (understanding of

the vector nature of moments is not required);

− recall that if a rigid body is in equilibrium under

the action of coplanar forces then the vector

sum of the forces is zero and the sum of the

moments of the forces about any point is zero,

and the converse of this;

− use Newton's third law in situations involving the

contact of rigid bodies in equilibrium;

− solve problems involving the equilibrium of rigid

bodies under the action of coplanar forces

(problems set will not involve complicated

trigonometry).

4. Rotation of a rigid body

− understand and use the definition of the moment of

inertia of a system of particles about a fixed axis as

Σ 2

mr , and the additive property of moment of

inertia for a rigid body composed of several parts

(the use of integration to find moments of inertia will

not be required);

− use the parallel and perpendicular axes theorems

(proofs of these theorems will not be required);

− recall and use the equation of angular motion

C = Iθ&& for the motion of a rigid body about a fixed

axis (simple cases only, where the moment C

arises from constant forces such as weights or the

tension in a string wrapped around the

circumference of a flywheel; knowledge of couples

is not included and problems will not involve

consideration or calculation of forces acting at the

axis of rotation);

− recall and use the formula 2

2

1

Ιω for the kinetic

energy of a rigid body rotating about a fixed axis;

− use conservation of energy in solving problems

concerning mechanical systems where rotation of a

rigid body about a fixed axis is involved.

5. Simple harmonic motion

− recall a definition of SHM and understand the

concepts of period and amplitude;

− use standard SHM formulae in the course of

solving problems;

− set up the differential equation of motion in

problems leading to SHM, recall and use

appropriate forms of solution, and identify the

period and amplitude of the motion;

− recognise situations where an exact equation of

motion may be approximated by an SHM equation,

carry out necessary approximations (e.g. small

angle approximations or binomial approximations)

and appreciate the conditions necessary for such

approximations to be useful.

Statistics:

6. Further work on distributions

− use the definition of the distribution function as a

probability to deduce the form of a distribution

function in simple cases, e.g. to find the distribution

function for Y, where Y = X

3 and X has a given

distribution;

− understand conditions under which a geometric

distribution or negative exponential distribution may

be a suitable probability model;

− recall and use the formula for the calculation of

geometric or negative exponential probabilities;

− recall and use the means and variances of a

geometric distribution and negative exponential

distribution.

7. Inference using normal and

t-distributions

− formulate hypotheses and apply a hypothesis test

concerning the population mean using a small

sample drawn from a normal population of

unknown variance, using a t-test;

− calculate a pooled estimate of a population

variance from two samples (calculations based on

either raw or summarised data may be required);

− formulate hypotheses concerning the difference of

population means, and apply, as appropriate,

a 2-sample t-test,

a paired sample t-test,

a test using a normal distribution

(the ability to select the test appropriate to the

circumstances of a problem is expected);

− determine a confidence interval for a population

mean, based on a small sample from a normal

population with unknown variance, using a

t-distribution;

− determine a confidence interval for a difference of

population means, using a t-distribution, or a

normal distribution, as appropriate.

8. χ2–tests (chi-square) − fit a theoretical distribution, as prescribed by a

given hypothesis, to given data (questions will not

involve lengthy calculations);

− use a χ

2-test, with the appropriate number of

degrees of freedom, to carry out the corresponding

goodness of fit analysis (classes should be

combined so that each expected frequency is at

least 5);

− use a χ

2-test, with the appropriate number of

degrees of freedom, for independence in a

contingency table (Yates’ correction is not required,

but classes should be combined so that the

expected frequency in each cell is at least 5).

9. Bivariate data

− understand the concept of least squares,

regression lines and correlation in the context of a

scatter diagram;

− calculate, both from simple raw data and from

summarised data, the equations of regression lines

and the product moment correlation coefficient, and

appreciate the distinction between the regression

line of y on x and that of x on y;

− recall and use the facts that both regression lines

pass through the mean centre ( x

, y ) and that the

product moment correlation coefficient r and the

regression coefficients b1, b2 are related by

r

2 = b1b2;

− select and use, in the context of a problem, the

appropriate regression line to estimate a value, and

understand the uncertainties associated with such

estimations;

− relate, in simple terms, the value of the product

moment correlation coefficient to the appearance of

the scatter diagram, with particular reference to the

interpretation of cases where the value of the

product moment correlation coefficient is close to

+1, −1 or 0;

− carry out a hypothesis test based on the product

moment correlation coefficient.

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Thanks for the help.