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.

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.

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

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.

FP1.4.1Cartesian equations for the parabola

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.

FP1.4.2Parametric equations for conics

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.

FP1.4.3Focus-directrix property of the parabola

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

FP1.4.4Tangents and normals to conics

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.