3. Pure Mathematics 3
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3.1 Algebra
3.1.1Modulus equations
• understand the meaning of |x|, sketch the graph of y = |ax + b| and use relations such as |a| = |b| ⇔ a2 = b2 and |x - a| < b ⇔ a - b < x < a + b when solving equations and inequalities Graphs of y = |f(x)| and y = f(|x|) for non-linear functions f are not included. e.g. |3x - 2| = |2x + 7|, 2x + 5 < |x + 1|.
3.1.2Polynomial division
• divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero)
3.1.3Factor and remainder theorems
• use the factor theorem and the remainder theorem e.g. to find factors and remainders, solve polynomial equations or evaluate unknown coefficients. Including factors of the form (ax + b) in which the coefficient of x is not unity, and including calculation of remainders.
3.1.4Partial fractions
• 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)(cx + d)2 - (ax + b)(cx2 + d) Excluding cases where the degree of the numerator exceeds that of the denominator.
3.1.5Binomial expansion
• use the expansion of (1 + x)n, where n is a rational number and x 11. Finding the general term in an expansion is not included. Adapting the standard series to expand e.g. x2 2 1 1 - - `j is included, and determining the set of values of x for which the expansion is valid in such cases is also included.
3.2 Logarithmic and exponential functions
3.2.1Log laws
• understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
3.2.2Exponential and ln
• understand the definition and properties of ex and ln x, including their relationship as inverse functions and their graphs Including knowledge of the graph of y = ekx for both positive and negative values of k.
3.2.3Log equations
• use logarithms to solve equations and inequalities in which the unknown appears in indices e.g. 25x 1, 32 5<x31# -, 34xx 12 1= -+.
3.2.4Linearising relationships
• use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. e.g. y = kxn gives ln y = ln k + n ln x which is linear in ln x and ln y. y = k (ax) gives ln y = ln k + x ln a which is linear in x and ln y.
3.3 Trigonometry
3.3.1Reciprocal trig functions
• 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
3.3.2Trig identities
• 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 - sect an122 /ii + and cosecc ot122 /ii + - the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) - the formulae for sin 2A, cos 2A and tan 2A - the expression of sinc osab ii+ in the forms sinR ! ai^h and cosR ! ai^h. e.g. simplifying cos(x - 30˚) - 3 sin(x - 60˚). e.g. solving tanc ot 4ii+ =, sect an25 2 -ii =, coss in32 1ii+ =.
3.4 Differentiation
3.4.1Advanced differentiation
• use the derivatives of ex, ln x, sin x, cos x, tan x, tan-1 x, together with constant multiples, sums, differences and composites Derivatives of sin-1 x and cos-1 x are not required.
3.4.2Product and quotient rules
• differentiate products and quotients e.g. x x 32 24 - +, x2 ln x, xe1 - x2.
3.4.3Differentiation rules
• find and use the first derivative of a function which is defined parametrically or implicitly. e.g. x = t - e2t, y = t + e2t. e.g. x2 + y2 = xy + 7. Including use in problems involving tangents and normals.
3.5 Integration
3.5.1Extended integration rules
• extend the idea of 'reverse differentiation' to include the integration of eax + b, ax b 1 +, sin(ax + b), cos(ax + b), sec2(ax + b) and xa 1 22+ Including examples such as x23 1 2+.
3.5.2Trig integration
• use trigonometrical relationships in carrying out integration e.g. use of double-angle formulae to integrate sin2 x or cos2(2x).
3.5.3Partial fractions
• integrate rational functions by means of decomposition into partial fractions Restricted to types of partial fractions as specified in topic 3.1 above.
3.5.4An integrand of the form x
• recognise an integrand of the form x k x f f l ^ ^ h h, and integrate such functions e.g. integration of x x 12 +, tan x.
3.5.5Inverse tangent
• recognise when an integrand can usefully be regarded as a product, and use integration by parts e.g. integration of x sin 2x, x2e-x, ln x, x tan-1 x.
3.5.6Integration by substitution
• use a given substitution to simplify and evaluate either a definite or an indefinite integral. e.g. to integrate sin2 2x cos x using the substitution u = sin x.
3.6 Numerical solution of equations
3.6.1Root location
• locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change e.g. finding a pair of consecutive integers between which a root lies.
3.6.2Iterative approximations
• understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation
3.6.3Iteration formulae
• understand how a given simple iterative formula of the form xn + 1 = F(xn) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. Knowledge of the condition for convergence is not included, but an understanding that an iteration may fail to converge is expected.
3.7 Vectors
3.7.1Standard notations for vectors
• use standard notations for vectors, i.e. x yfp, xi + yj, x y z fp, xi + yj + zk, AB, a
3.7.2Vector operations
• carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms e.g. 'OABC is a parallelogram' is equivalent to OB OA OC= +. The general form of the ratio theorem is not included, but understanding that the midpoint of AB has position vector OA OB2 1 +_ i is expected.
3.7.3Position vectors
• calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors In 2 or 3 dimensions.
3.7.4Straight lines
• understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a + tb, and find the equation of a line, given sufficient information e.g. finding the equation of a line given the position vector of a point on the line and a direction vector, or the position vectors of two points on the line.
3.7.5Whether two lines are parallel
• determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists Calculation of the shortest distance between two skew lines is not required. Finding the equation of the common perpendicular to two skew lines is also not required.
3.7.6Scalar product
• use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points. e.g. finding the angle between two lines, and finding the foot of the perpendicular from a point to a line; questions may involve 3D objects such as cuboids, tetrahedra (pyramids), etc. Knowledge of the vector product is not required.
3.8 Differential equations
3.8.1Differential equations
• formulate a simple statement involving a rate of change as a differential equation The introduction and evaluation of a constant of proportionality, where necessary, is included.
3.8.2Differential equations
• find by integration a general form of solution for a first order differential equation in which the variables are separable Including any of the integration techniques from topic 3.5 above.
3.8.3An initial condition to find a
• use an initial condition to find a particular solution
3.8.4Differential equations
• interpret the solution of a differential equation in the context of a problem being modelled by the equation. Where a differential equation is used to model a 'real-life' situation, no specialised knowledge of the context will be required.
3.9 Complex numbers
3.9.1Complex 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 Notations Re z, Im z, |z|, arg z, z* should be known. The argument of a complex number will usually refer to an angle i such that 1 G-rr i, but in some cases the interval 02 1G ri may be more convenient. Answers may use either interval unless the question specifies otherwise.
3.9.2Complex numbers
• carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy For calculations involving multiplication or division, full details of the working should be shown.
3.9.3The result that
• use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs e.g. in solving a cubic or quartic equation where one complex root is given.
3.9.4Complex numbers
• represent complex numbers geometrically by means of an Argand diagram
3.9.5Complex numbers
• carry out operations of multiplication and division of two complex numbers expressed in polar form coss inrr i ei/ii+ i^h Including the results |z1z2| = |z1||z2| and arga rg argzz zz12 12= +__ _ii i, and corresponding results for division.
3.9.6Complex numbers
• find the two square roots of a complex number e.g. the square roots of 5 + 12i in exact Cartesian form. Full details of the working should be shown.
3.9.7Complex numbers
• understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers
3.9.8Complex numbers
• illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram. e.g. |z - a| < k, |z - a| = |z - b|, arg(z - a) = α.