My Math skills,

[pisano]

Pure MAthe INtermediate level: (MATRIC board of studies Malta)


Topics

Notes



1.



Positive and negative rational indices. Laws of indices. Laws of logarithms





Solution of simple equations involving indices and logarithms of the form only. Problems involving change of base will not be set.





Use and manipulation of surds.





2.



Polynomials, rational functions, factor and remainder theorems.

Simple partial fractions.





Problems on partial fractions could include denominators such as



and



3.

The quadratic equation in one variable.

Solution of the quadratic equation by completing the square or the use of the formula.

Nature of roots.

Knowledge of the relation between the roots and the coefficients of a quadratic equation. Forming new equations with roots related to the original. Calculation of expressions such as:



4.

Arithmetic series, finite and infinite geometric series. Pascal's triangle. The binomial expansion for positive integral indices.

The general term and the summation of an arithmetic and geometric progression are included. Knowledge of the notation .



5.

Simple counting problems involving permutations and combinations. Applications to simple problems in probability

The knowledge of probability expected will be limited to the calculation of probabilities arising from simple problems of enumeration of equally likely possibilities, including simple problems involving the probability of the complement of an event and of the union and intersection of two events.

Knowledge of the rules:



is expected

Questions on conditional probability will not be set.

Tree and Venn diagrams may be used.



6.



Plane Cartesian coordinates. Distance between two points and the perpendicular distance from a point to a line. Elementary treatment of lines.

Intersection of a straight line with a curve.







Reduction of a relation to linear form, and graphical determination of the constants



Relations will be limited only to equations of the form:



and





7.

The concept of a function as a mapping: domain and range.

Use of function notations,

e.g.





The exponential and logarithmic functions and their graphs.





The six trigonometric functions

Graphs of functions of the type where is a positive integer. Graphs of and are not required.









Solution of simple trigonometric equations.

Trigonometric equations of the type and where n is a positive integer only.

Quadratic trigonometric equations may be included.

General solutions are not required.





The identity and simple corollaries.

Other trigonometric identities, and the addition theorems are not required.



8.

Simple curve sketching.

Curve sketching of polynomials not higher than the third degree.





Transformations.

Knowledge of the effect of the simple transformations

and .

Transformations of exponential, logarithmic, trigonometric and polynomial functions.

Combinations of transformations will not be required.



Simple inequalities in one variable.

Graphical or algebraic solution of inequalities such as the following :

(i)

(ii)



9.

Radian measure.

Knowledge of the values of cosine, sine, and tangent of , in surd or rational form. Use of the formulae:



.







10.

The derivative as a limit.

Differentiation of sums, products, quotients and composition of functions.

Differentiation of algebraic, trigonometric, exponential and logarithmic functions. Applications of differentiation to gradients, tangents, maxima and minima, points of inflexion and curve sketching.



A rigorous treatment of limits is not expected.



Inverse trigonometric functions are excluded.



Differentiation of implicit and parametric functions is not required.

Second order derivatives are not required.



Simple problems on rates of change.

Problems involving the chain rule of the type may be set.



11.

Integration as the limit of a sum and as the inverse of differentiation.



A rigorous treatment is not required.



The evaluation of integrals by means of standard forms and by partial fractions.

Integrals of the functions

and are required.

Use of the result .





Definite integrals. Application of integration to the calculation of areas.





12.

First order differential equations with separable variables.



Problems requiring the formation of a differential equation will not be set.

13.

The algebra of matrices.

Addition and multiplication.

Distributivity of multiplication over addition. Associativity. The zero matrix and the identity matrix. Non-commutativity of multiplication. The inverse of a matrix.



Students will only be expected to be able to find inverses of 2 X 2 matrices but they should be able to verify that, say, two given 3 X 3 matrices are inverses of each other.



Applications will be limited to linear transformations in the plane.

Students are expected to find the matrices of the following transformations:

a) enlargement,

b) rotation through multiples of ,

c) reflections in the lines and .

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