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2. Further Pure Mathematics 2

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  1. 2.1 Hyperbolic functions

    1. 2.1.1Hyperbolic functions

      • understand the definitions of the hyperbolic functions sinh x, cosh x, tanh x, sech x, cosech x, coth x in terms of the exponential function

    2. 2.1.2Hyperbolic functions

      • sketch the graphs of hyperbolic functions

    3. 2.1.3Hyperbolic functions

      • prove and use identities involving hyperbolic functions e.g. cosh2 x - sinh2 x ≡ 1, sinh2x ≡ 2 sinh x cosh x, and similar results corresponding to the standard trigonometric identities.

    4. 2.1.4Hyperbolic functions

      • understand and use the definitions of the inverse hyperbolic functions and derive and use the logarithmic forms

  2. 2.2 Matrices

    1. 2.2.1Matrix equations

      • formulate a problem involving the solution of 3 linear simultaneous equations in 3 unknowns as a problem involving the solution of a matrix equation, or vice versa

    2. 2.2.2Determinants and inverses

      • understand the cases that may arise concerning the consistency or inconsistency of 3 linear simultaneous equations, relate them to the singularity or otherwise of the corresponding matrix, solve consistent systems, and interpret geometrically in terms of lines and planes e.g. three planes meeting in a common point, or in a common line, or having no common points.

    3. 2.2.3Eigen terminology

      • understand the terms 'characteristic equation', 'eigenvalue' and 'eigenvector', as applied to square matrices Including use of the definition Ae = λe to prove simple properties, e.g. that λn is an eigenvalue of An.

    4. 2.2.4Eigenvalues and eigenvectors

      • find eigenvalues and eigenvectors of 2 x 2 and 3 x 3 matrices Restricted to cases where the eigenvalues are real and distinct.

    5. 2.2.5Eigenvalues and eigenvectors

      • express a square matrix in the form QDQ-1, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use this expression e.g. in calculating powers of 2 x 2 or 3 x 3 matrices.

    6. 2.2.6Eigen terminology

      • use the fact that a square matrix satisfies its own characteristic equation. e.g. in finding successive powers of a matrix or finding an inverse matrix; restricted to 2 x 2 or 3 x 3 matrices only.

  3. 2.3 Differentiation

    1. 2.3.1Hyperbolic functions

      • differentiate hyperbolic functions and differentiate sin-1x, cos-1x, sinh-1x, cosh-1x and tanh-1x

    2. 2.3.2An expression for x y d

      • obtain an expression for x y d d 2 2 in cases where the relation between x and y is defined implicitly or parametrically

    3. 2.3.3Maclaurin series

      • derive and use the first few terms of a Maclaurin's series for a function. Derivation of a general term is not included, but successive 'implicit' differentiation steps may be required, e.g. for y = tan x following an initial differentiation rearranged as y′= 1 + y2.

  4. 2.4 Integration

    1. 2.4.1Hyperbolic functions

      • integrate hyperbolic functions and recognise integrals of functions of the form ax 1 22-, xa 1 22+ and xa 1 22-, and integrate associated functions using trigonometric or hyperbolic substitutions as appropriate Including use of completing the square where necessary, e.g. to integrate xx 1 2 +.

    2. 2.4.2And use reduction formulae for the

      • derive and use reduction formulae for the evaluation of definite integrals e.g. sin xxdn 0 2 1 r y, x x1edx n 0 1 -- ^hy. In harder cases hints may be given, e.g. sec xxdn 0 4 1 r y by considering tans ecx xxd d n^h.

    3. 2.4.3How the area under a curve

      • understand how the area under a curve may be approximated by areas of rectangles, and use rectangles to estimate or set bounds for the area under a curve or to derive inequalities or limits concerning sums Questions may involve either rectangles of unit width or rectangles whose width can tend to zero, e.g. ln r n1 1 1ln n r n 1 22+ + = ^h/, 1 n n r x x1 1 1 d r n 1 1 0 1.+ + - - = c ^m h/ y.

    4. 2.4.4Arc length and surface area

      • use integration to find - arc lengths for curves with equations in Cartesian coordinates, including the use of a parameter, or in polar coordinates - surface areas of revolution about one of the axes for curves with equations in Cartesian coordinates, including the use of a parameter. Any questions involving integration may require techniques from Cambridge International A Level Mathematics (9709) applied to more difficult cases, e.g. integration by parts for sin xxedxy, or use of the substitution tantx 2 1=. Surface areas of revolution for curves with equations in polar coordinates will not be required.

  5. 2.5 Complex numbers

    1. 2.5.1de Moivre's theorem

      • understand de Moivre's theorem, for a positive or negative integer exponent, in terms of the geometrical effect of multiplication and division of complex numbers

    2. 2.5.2de Moivre's theorem

      • prove de Moivre's theorem for a positive integer exponent e.g. by induction.

    3. 2.5.3de Moivre's theorem

      • use de Moivre's theorem for a positive or negative rational exponent - to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle - to express powers of sin i and cos i in terms of multiple angles - in the summation of series - in finding and using the nth roots of unity. e.g. expressing cos5 i in terms of cos i or tan5 i in terms of tan i. e.g. expressing sin6 i in terms of cos2 i, cos4 i and cos6 i. e.g. using the 'C + iS ' method to sum series such as 1 sinn r r r n i = eo/.

  6. 2.6 Differential equations

    1. 2.6.1Linear differential equations

      • find an integrating factor for a first order linear differential equation, and use an integrating factor to find the general solution e.g. x y yx2d d 2- =, x x y yxd d 4- =, coth coshx y yx xd d + =.

    2. 2.6.2Linear differential equations

      • recall the meaning of the terms 'complementary function' and 'particular integral' in the context of linear differential equations, and recall that the general solution is the sum of the complementary function and a particular integral

    3. 2.6.3Linear differential equations

      • find the complementary function for a first or second order linear differential equation with constant coefficients For second order equations, including the cases where the auxiliary equation has distinct real roots, a repeated real root or conjugate complex roots.

    4. 2.6.4Linear differential equations

      • recall the form of, and find, a particular integral for a first or second order linear differential equation in the cases where a polynomial or a ebx or a cos px + b sin px is a suitable form, and in other simple cases find the appropriate coefficient(s) given a suitable form of particular integral e.g. evaluate k given that kx cos 2x is a particular integral of sinx y yx42d d 2 2 + =.

    5. 2.6.5Differential substitutions

      • use a given substitution to reduce a differential equation to a first or second order linear equation with constant coefficients or to a first order equation with separable variables e.g. the substitution x = et to reduce to linear form a differential equation with terms of the form ax x y bx x y cyd d d d2 2 2 ++, or the substitution y = ux to reduce x y xy xy d d -= + to separable form.

    6. 2.6.6Initial conditions

      • use initial conditions to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation.