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1. Pure Mathematics 1

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  1. 1.1 Quadratics

    1. 1.1.1Completing the square

      • carry out the process of completing the square for a quadratic polynomial ax2 + bx + c and use a completed square form e.g. to locate the vertex of the graph of y = ax2 + bx + c or to sketch the graph.

    2. 1.1.2Quadratic discriminant

      • 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. Knowledge of the term 'repeated root' is included.

    3. 1.1.3Quadratic equations

      • solve quadratic equations, and quadratic inequalities, in one unknown By factorising, completing the square and using the quadratic formula.

    4. 1.1.4Simultaneous equations

      • solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic e.g. x + y + 1 = 0 and x2 + y2 = 25, 2x + 3y = 7 and 3x2 = 4 + 4xy.

    5. 1.1.5Quadratic-form equations

      • recognise and solve equations in x which are quadratic in some function of x. e.g. x4 - 5x2 + 4 = 0, xx61 0-+ =, tan2 x = 1 + tan x.

  2. 1.2 Functions

    1. 1.2.1Function terminology

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

    2. 1.2.2Range and composition

      • identify the range of a given function in simple cases, and find the composition of two given functions e.g. range of: x x 1f 7 for x 1H and range of: xx 1g 27 + for x R!. Including the condition that a composite function gf can only be formed when the range of f is within the domain of g.

    3. 1.2.3One-one and inverse functions

      • determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases e.g. finding the inverse of: x x23 4h 27 -+^h for <x 2 3 -

    4. 1.2.4One-one and inverse functions

      • illustrate in graphical terms the relation between a one-one function and its inverse Sketches should include an indication of the mirror line y = x.

    5. 1.2.5Graph transformations

      • understand 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. Including use of the terms 'translation', 'reflection' and 'stretch' in describing transformations. Questions may involve algebraic or trigonometric functions, or other graphs with given features.

  3. 1.3 Coordinate geometry

    1. 1.3.1Straight lines

      • find the equation of a straight line given sufficient information e.g. given two points, or one point and the gradient.

    2. 1.3.2Line equation forms

      • interpret and use any of the forms y = mx + c, y - y1 = m(x - x1), ax + by + c = 0 in solving problems Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines.

    3. 1.3.3Circle equation

      • understand that the equation (x - a)2 + (y - b)2 = r 2 represents the circle with centre (a, b) and radius r Including use of the expanded form x 2 + y 2 + 2gx + 2fy + c = 0.

    4. 1.3.4Lines and circles

      • use algebraic methods to solve problems involving lines and circles Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry. Implicit differentiation is not included.

    5. 1.3.5Graph intersections

      • understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. e.g. to determine the set of values of k for which the line y = x + k intersects, touches or does not meet a quadratic curve.

  4. 1.4 Circular measure

    1. 1.4.1Radians

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

    2. 1.4.2Arc length and sector area

      • use the formulae sr= i and Ar 2 1 2= i in solving problems concerning the arc length and sector area of a circle. Including calculation of lengths and angles in triangles and areas of triangles.

  5. 1.5 Trigonometry

    1. 1.5.1Trig graphs

      • sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians) Including e.g. y = 3 sin x, y = 1 - cos 2x,.tany x 4 1 r= +`j

    2. 1.5.2Exact trig values

      • use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles e.g.,coss in1503 22 1 4 3 2 1c - r==.

    3. 1.5.3Inverse trig notation

      • use the notations sin-1x, cos-1x, tan-1x to denote the principal values of the inverse trigonometric relations No specialised knowledge of these functions is required, but understanding of them as examples of inverse functions is expected.

    4. 1.5.4Trig identities

      • use the identities cos sin tan/i i i and sinc os 122 /ii+ e.g. in proving identities, simplifying expressions and solving equations.

    5. 1.5.5Trig equations

      • find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included). e.g. solve 3sin 21 0x += for x<<rr-, 3sin 5cos 102 =ii-- for 0° ⩽ i ⩽ 360°.

  6. 1.6 Series

    1. 1.6.1Binomial expansion

      • use the expansion of (a + b)n, where n is a positive integer Including the notations n rfp and n!. Knowledge of the greatest term and properties of the coefficients are not required.

    2. 1.6.2Progressions

      • recognise arithmetic and geometric progressions

    3. 1.6.3Progression formulae

      • use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions Including knowledge that numbers a, b, c are 'in arithmetic progression' if 2b = a + c (or equivalent) and are 'in geometric progression' if b2 = ac (or equivalent). Questions may involve more than one progression.

    4. 1.6.4Geometric convergence

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

  7. 1.7 Differentiation

    1. 1.7.1Gradient as limit

      • understand 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), d d x y, and d d x y 2 2 for first and second derivatives Only an informal understanding of the idea of a limit is expected. e.g. includes consideration of the gradient of the chord joining the points with x coordinates 2 and (2 + h) on the curve y = x3. Formal use of the general method of differentiation from first principles is not required.

    2. 1.7.2Differentiation rules

      • use 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 e.g. find x y d d, given yx 253= +.

    3. 1.7.3Differentiation applications

      • apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change Including connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of the area for a specific value of one of the variables.

    4. 1.7.4Stationary points

      • locate stationary points and determine their nature, and use information about stationary points in sketching graphs. Including use of the second derivative for identifying maxima and minima; alternatives may be used in questions where no method is specified. Knowledge of points of inflexion is not included.

  8. 1.8 Integration

    1. 1.8.1Basic 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 e.g. xx x25 1 d3 - +;^h, x x23 1 d2+> ^h.

    2. 1.8.2Constant of integration

      • solve problems involving the evaluation of a constant of integration e.g. to find the equation of the curve through (1, -2) for which d d 21.x y x=+

    3. 1.8.3Definite integrals

      • evaluate definite integrals Including simple cases of 'improper' integrals, such as dxx 0 1 2 1- $ and 3 dxx2- 1;.

    4. 1.8.4Area and volume

      • use 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 - a volume of revolution about one of the axes. A volume of revolution may involve a region not bounded by the axis of rotation, e.g. the region between y = 9 - x2 and y = 5 rotated about the x-axis.