Unit FP1: Further Pure Mathematics 1

Definition of complex numbers in The meaning of conjugate, modulus, argument, real part, the form a + ib and imaginary part and equality of complex numbers should be known. rcos θ + irsin θ.

Sum, product and quotient of | z z | = | z | | z | 1 2 1 2 complex numbers.; Knowledge of the result arg(z z) = arg z + arg z is not 1 2 1 2 required.

Geometrical representation of complex numbers in the Argand diagram.; Geometrical representation of sums, products and quotients of complex numbers.

Complex solutions of quadratic equations with real coefficients.

Finding conjugate complex roots Knowledge that if z is a root of f(z) = 0 then z * is 1 1 and a real root of a cubic equation also a root. with integer coefficients.

Finding conjugate complex roots For example, and/or real roots of a quartic (i) f(x) = x4 − x3 − 5x2 + 7x + 10 equation with real coefficients.; Given that x = 2 + i is a root of f(x) = 0, use algebra to find the three other roots of f(x) = 0 (ii) g(x) = x4 − x3 + 6x2 + 14x − 20 Given g(1) = 0 and g(−2) = 0, use algebra to solve g(x) = 0 completely.

FP1.2.1Sum of roots and product of roots

Sum of roots and product of roots For the equation ax2 + bx + c = 0, whose roots are α and β, of a quadratic equation. b c then α + β = −, αβ =. a a.

FP1.2.2Manipulation of expressions

Manipulation of expressions Knowledge of the identity α3 + β3 ≡ (α + β)3 − 3αβ(α + β). involving the sum of roots and product of roots.

FP1.2.3Forming quadratic equations with 1 1 1 1 2

Forming quadratic equations with 1 1 1 1 2 For example, with roots α3, β3;,;,; α +, new roots. α β α2 β2 β β +; etc. α.

Equations of the form f(x) = 0 f(x) will involve only functions used in P1 and P2. solved numerically by: For the Newton-Raphson process, the only differentiation (i) interval bisection, required will be as defined in unit P1 and P2. (ii) linear interpolation, (iii) the Newton-Raphson process.

Cartesian equations for the parabola Students should be familiar with the equations: and rectangular hyperbola. c y2 = 4ax or x = at2, y = 2at and xy = c2 or x = ct, y =. t.

Idea of parametric equation for The idea of (at2, 2at) as a general point on the parabola is parabola and rectangular hyperbola. all that is required.

The focus-directrix property of the Concept of focus and directrix and parabola as locus of parabola. points equidistant from focus and directrix.

Tangents and normals to these 1 1 c2 Differentiation of y = 2a2x2, y =. curves. x Parametric differentiation is not required.

Addition and subtraction of matrices.

Multiplication of a matrix by a scalar.

Products of matrices.

Evaluation of 2 × 2 determinants.; Singular and non-singular matrices.

Inverse of 2 × 2 matrices.; Use of the relation (AB)–1 = B–1A–1.

Linear transformations of column The transformation represented by AB is the transformation vectors in two dimensions and their represented by B followed by the transformation matrix representation. represented by A.

Applications of 2 × 2 matrices to Identification and use of the matrix representation of single represent geometrical transformations from: reflection in coordinate axes and transformations. lines y = ±x, rotation through any angle about (0, 0), stretches parallel to the x-axis and y-axis, and enlargement about centre (0, 0), with scale factor k, (k ≠ 0), where k ∈.

Combinations of transformations.; Identification and use of the matrix representation of ℝ combined transformations.

The inverse (when it exists) of a Idea of the determinant as an area scale factor in given transformation or transformations. combination of transformations.

Summation of simple finite series.; Students should be able to sum series such as n n n ∑r, ∑r2, ∑r(r2 + 2). r=1 r=1 r=1 The method of differences is not required.

Construct proofs by mathematical induction for sums of series, divisibility results, general terms of recursively defined sequences and matrix powers.