Pure Mathematics 2 & 3·Cambridge A-Level

Algebra

Syllabus Learning Objectives

01carry out the process of completing the square for a quadratic polynomial ax2 + bx + c and use a completed square form

02find the discriminant of a quadratic polynomial ax2 + bx + c and use the discriminant

03understand the terms function, domain, range, one-one function, inverse function and composition of functions

04understand that the equation (x – a)2 + (y – b)2 = r 2 represents the circle with centre (a, b) and radius r

05understand the definition of a radian, and use the relationship between radians and degrees

06sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians)

07use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles

08use the notations sin–1x, cos–1x, tan–1x to denote the principal values of the inverse trigonometric relations

09use the identities tan θ ≡ sin θ/cos θ and sin2 θ + cos2 θ ≡ 1

10recognise arithmetic and geometric progressions

11understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations f ′(x), f ″(x), dy/dx, and d2y/dx2 for first and second derivatives

12understand 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

13solve quadratic equations, and quadratic inequalities, in one unknown

14solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic

15recognise and solve equations in x which are quadratic in some function of x.

16identify the range of a given function in simple cases, and find the composition of two given functions

17determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases

18illustrate in graphical terms the relation between a one-one function and its inverse

19understand and use the transformations of the graph of y = f(x) given by y = f(x) + a, y = f(x + a), y = af(x), y = f(ax) and simple combinations of these.

20find the equation of a straight line given sufficient information

21interpret and use any of the forms y = mx + c, y – y1 = m(x – x1), ax + by + c = 0 in solving problems

22use algebraic methods to solve problems involving lines and circles

23understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations.

24use the formulae s = rθ and A = 1/2 r 2θ in solving problems concerning the arc length and sector area of a circle.

25find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).

26use the expansion of (a + b)n, where n is a positive integer

27use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions

28use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

29use the derivative of xn (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule

30apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change

31locate stationary points and determine their nature, and use information about stationary points in sketching graphs.

32solve problems involving the evaluation of a constant of integration

33evaluate definite integrals

34use definite integration to find the area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line or between two curves

35use definite integration to find a volume of revolution about one of the axes.