Semester 1:

Quadratics

• carry out the process of completing the square for a quadratic polynomial ax2 + bx + c, and use this form, e.g. to locate the vertex of the graph of y = ax2 + bx + c or to sketch the graph;
• find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant, e.g. To determine the number of real roots of the equation ax2 + bx + c = 0;
• solve quadratic equations, and linear and quadratic inequalities, in one unknown;
• solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic;
• recognise and solve equations in x which are quadratic in some function of x, e.g. x4 − 5x2 +4 = 0.

Logarithmics

• understand the relationship between logarithms and indices, and use the laws of logarithms (including change of base);
• understand the definition and properties of ex and lnx , including their relationship as inverse functions and their graphs;
• use logarithms to solve equations of the form ax = b and similar inequalities;
• use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

Algebra

• divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero);
• use the factor theorem and the remainder theorem, e.g. to find factors, solve polynomial equations or evaluate unknown coefficients,
• recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than (ax + b)(cx + d)(ex + f ) ,  (ax + b)2(cx + d) , ( ax + b )( x2 + c2 ), and where the degree of the numerator does not exceed that of the denominator;
• use the expansion of (1+ x)n  , where n is a rational number and x < 1 (finding a general term is not included, but adapting the standard series to expand e.g. ( 2 - ½ x )-1 is included).

Functions

• understand the terms function, domain, range, one-one function, inverse function and composition of functions;
• identify the range of a given function in simple cases, and find the composition of two given functions;
• determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases;
• illustrate in graphical terms the relation between a one-one function and its inverse.

Differentiation

• understand the idea of the gradient of a curve, and use the notations f′(x) , f′′(x) , dx/dy, d2x/dy2 (the technique of differentiation from first principles is not required);
• use the derivative of xn (for any rational n), together with constant multiples, sums, differences of functions, and of composite functions using the chain rule;
• apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change);
• locate stationary points, and use information about stationary points in sketching graphs (the ability to distinguish between maximum points and minimum points is required, but identification of points of inflexion is not included);
• use the derivatives of ex  , lnx , sin x , cos x , tan x , together with constant multiples, sums, differences and composites;
• differentiate products and quotients;
• find and use the first derivative of a function which is defined parametrically or implicitly.

Circular measure

• understand the definition of a radian, and use the relationship between radians and degrees;
• use the formulae s = rθ and A = ½ r2θ in solving problems concerning the arc length and sector area of a circle.

Trigonometry

• sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians);
• use the exact values of the sine, cosine and tangent of 30º, 45º, 60º, and related angles, e.g. Cos150º;
• use the notations sin-1x, cos−1x, tan-1x to denote the principal values of the inverse trigonometric relations;
• use the identities tanθ ≡ sinθ / cosθ  and sin2θ + cos2θ ≡ 1;
• find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included);
• understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude;
• use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of
  • sec2θ  ≡ 1+ tan2θ  and cosec2θ  ≡ 1+ cot2θ  ,
  • the expansions of sin(A ± B) , cos(A ± B) and tan(A ± B) ,
  • the formulae for sin2A , cos2A and tan2A ,
  • the expressions of asinθ + b cosθ in the forms Rsin(θ ±α) and Rcos(θ ±α) .

Semester 2:

Vectors

• understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb ;
• determine whether two lines are parallel, intersect or are skew;
• find the angle between two lines, and the point of intersection of two lines when it exists;
• understand the significance of all the symbols used when the equation of a plane is expressed in either of the forms ax + by + cz = d or (r − a).n = 0 ;
• use equations of lines and planes to solve problems concerning distances, angles and intersections, and in particular find the equation of a line or a plane, given sufficient information, determine whether a line lies in a plane, is parallel to a plane, or intersects a plane, and find the point of intersection of a line and a plane when it exists, find the line of intersection of two non-parallel planes, find the perpendicular distance from a point to a plane, and from a point to a line, find the angle between two planes, and the angle between a line and a plane.

Complex numbers

• understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal;
• carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in cartesian form x + iy ;
• use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs;
• represent complex numbers geometrically by means of an Argand diagram;
• carry out operations of multiplication and division of two complex numbers expressed in polar form  r(cos θ + isin θ  ) ≡ r ei θ  ;
• find the two square roots of a complex number;
• understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers;
• illustrate simple equations and inequalities involving complex numbers by means of Ioci in an Argand diagram, e.g. |z − a| < k ,| z − a| = |z − b| , arg(z − a) = α .

Integration

• understand integration as the reverse process of differentiation, and integrate (ax + b)n (for any rational n except −1), together with constant multiples, sums and differences;
• solve problems involving the evaluation of a constant of integration, e.g. to find the equation of the curve through (1, − 2) for which dy/dx = 2x + 1;
• evaluate definite integrals (including simple cases of ‘improper’ integrals;
• use definite integration to find the area of a region bounded by a curve and lines parallel to the axes, or between two curves, a volume of revolution about one of the axes;
• extend the idea of ‘reverse differentiation’ to include the integration of e(ax+b) , (ax + b)-1 , sin(ax + b) , cos(ax + b);
• use trigonometrical relationships (such as double-angle formulae) to facilitate the integration of functions such as cos2x;
• use the trapezium rule to estimate the value of a definite integral, and use sketch graphs in simple cases to determine whether the trapezium rule gives an overestimate or an under-estimate.

Differential equations

• formulate a simple statement involving a rate of change as a differential equation, including the introduction if necessary of a constant of proportionality;
• find by integration a general form of solution for a first order differential equation in which the variables are separable;
• use an initial condition to find a particular solution;
• interpret the solution of a differential equation in the context of a problem being modelled by the equation.

Permutations and combinations

• understand the terms permutation and combination, and solve simple problems involving selections;
• solve problems about arrangements of objects in a line, including those involving repetition (e.g. the number of ways of arranging the letters of the word ‘NEEDLESS’), restriction (e.g. the number of ways several people can stand in a line if 2 particular people must — or must not — stand next to each other).

Probability

• evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g. for the total score when two fair dice are thrown), or by calculation using permutations or combinations;
• use addition and multiplication of probabilities, as appropriate, in simple cases;
• understand the meaning of exclusive and independent events, and calculate and use conditional probabilities in simple cases, e.g. situations that can be represented by means of a tree diagram.

Foundation in Science : Maths – Assessment Scheme

All papers

Consist of structured questions. All questions must be answered.

Paper 1 & 2

In each paper, 5 points are allocated for presentation of method and presentation of results.