###### Preparatory Certificates

#### Sample Tests

These sample tests give an approximate idea of the style and level of question you might be asked in your UPC entrance tests.

Click below to access online tests for **UPCSE**:

Click below to access online sample tests for **UPCH**:

## > Mathematics

*The mathematics course will provide you with an
in-depth introduction to pure mathematics.*

Mathematics Syllabus |

If you decide to study mathematics on the UPCSE
course, it is important that you are fully familiar with the topics
listed in the “Essential Background Knowledge in Mathematics” section
outlined below. This may involve the need for private study and/or
revision over the summer vacation before the UPCSE course begins.

External examiners have stated that the standards
reached by students are very high. In Mathematics and in Physics, the
quality of work was in many cases in excess of high-grade A-level.

#### Essential Background Knowledge in Mathematics

**Algebra**

You should be completely familiar with the
arithmetic of the integers, of fractions and of real numbers, laws of
indices for integer and rational exponents, and the manipulation of
surds. You need to be able to solve linear equations in two variables,
and should be familiar with geometric interpretations of these. You
also need to be competent with manipulation of algebraic expressions
and the solution of quadratic equations by completing the square,
solution by discriminant, and the graph of a quadratic.

**Trigonometry**

You should have knowledge of the elementary
trigonometric functions of sin/cos/tan. You should be able to solve
problems using trigonometry and be able to use the sine and cosine
rules appropriately. You should be familiar with measuring angles in
degrees and radians.

**Geometry**

You should be familiar with Pythagoras’s Theorem,
with the solution of problems involving ratio and proportion
(including similar triangles) and links between length, area and
volume of similar figures. You need also to be familiar with the
perimeter and area of a circle, and the volume of a cone and a sphere,
the properties of a circle, and co-ordinate geometry and elementary
2D vectors.

#### Key Topics in Mathematics

A general outline of the topics that will be covered is given below.

**Algebraic Topics**

Algebraic identities, inequalities and functions; partial
fractions; quadratic equations; logarithms; remainder theorem;
Pascal’s triangle; arithmetic and geometric series and their sums to n
terms and sum to infinity of the convergent geometric series.

**Functions**

Mappings; domains and ranges; exponential and log functions;
inverse functions; representing a function as a curve; curve sketching
and even/odd/periodic functions; finding zeros, asymptotes,
symmetries, maxima and minima of function. The modulus function.

**Trigonometric functions and identities**

Sine and cosine rules; trigonometric functions, their
relationships and identities; graphical representations; periodic
properties and symmetries of trigonometric functions; solution to
trigonometric equations; hyperbolic functions and their identities.

**Calculus**To cover both differentiation and integration. Work on
differentiation will include geometrical interpretation, derivatives
of standard functions, differentiation of the sum, product and
quotient of functions, derivatives of simple functions defined
implicitly or parametrically. The derivative of the composition of two
functions and its applications, along with applications to gradients,
tangents, normals, maxima and minima.

Work on integration will include geometric interpretation as area under a curve, the fundamental Theorem of Calculus. Also covered will be the integration of standard functions, techniques of integration, evaluation of definite integrals, evaluation of areas under a curve or between two curves, and numerical appropriations of definite integrals

**Vectors**

Work on vectors will focus on 2D and 3D vectors, algebraic
properties of addition, scalar multiplication and their geometrical
properties; Distance between two points; equations of lines and
planes; Direction Ratios and direction cosines; Scalar and vector
products.

**Complex Numbers**

Imaginary numbers; algebraic properties of complex numbers;
complex roots of quadratic equations; argand diagrams and
modulus/argument form of complex numbers; cube and nth roots of unity;
DeMoivre’s theorem; exponential form of complex numbers; relationships
between hyperbolic, trigonometric and exponential functions.

**Matrices**

Column vectors; general matrix arithmetic; transformations in 2D;
determinants; inverse matrices; solution to simultaneous equations;
basic gaussian elimination.

**Numerical Applications**

Numerical methods for solving integration problems: Trapezium and
Simpson’s rule, small increments and rates of change; numerical
solutions to algebraic functions: graphical methods, interval
bisection.

**Differential Equations**

First order differential equations and integrating factor; second
order linear differential equations with constant coefficients,
general solutions and particular integrals.

Page last modified on 18 sep 13 13:53 by Martin L White