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2. Pure Mathematics 2

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  1. 2.1 Algebra

    1. 2.1.1Modulus 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|.

    2. 2.1.2Polynomial 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. 2.1.3Factor 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.

  2. 2.2 Logarithmic and exponential functions

    1. 2.2.1Log laws

      • understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)

    2. 2.2.2Exponential 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. 2.2.3Log equations

      • use logarithms to solve equations and inequalities in which the unknown appears in indices e.g. 25x 1, 32 5x31# 1-, 34xx 12 1= -+.

    4. 2.2.4Linearising 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. 2.3 Trigonometry

    1. 2.3.1Reciprocal 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

    2. 2.3.2Trig 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 !ia^h and cosR !ia^h. e.g. simplifying coss inxx 30 3 60cc- - -^^ hh. e.g. solving tanc ot 4ii+ =, sect an25 2 ii- =, coss in32 1ii+ =.

  4. 2.4 Differentiation

    1. 2.4.1Advanced differentiation

      • use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites

    2. 2.4.2Product and quotient rules

      • differentiate products and quotients e.g. x x 32 24 - +, x2 ln x, xel - x2.

    3. 2.4.3Differentiation 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.

  5. 2.5 Integration

    1. 2.5.1Extended integration rules

      • extend the idea of 'reverse differentiation' to include the integration of eax + b, ax b 1 +, sin(ax + b), cos(ax + b) and sec2(ax + b) Knowledge of the general method of integration by substitution is not required.

    2. 2.5.2Trig integration

      • use trigonometrical relationships in carrying out integration e.g. use of double-angle formulae to integrate sin2 x or cos2(2x).

    3. 2.5.3Trapezium rule

      • understand and use the trapezium rule to estimate the value of a definite integral. Including the use of sketching graphs in simple cases to determine whether the trapezium rule gives an over-estimate or an under-estimate.

  6. 2.6 Numerical solution of equations

    1. 2.6.1Root 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.

    2. 2.6.2Iterative approximations

      • understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation

    3. 2.6.3Iteration 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.