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.

Use fp1.1.1 - definition of complex numbers to connect the rule to the data and decision in the question.

This matters because fp1.1.1 - definition of complex numbers determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.1 - definition of complex numbers to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.1 - Definition of complex numbers is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.1.2 - sum, product and quotient of complex numbers to connect the rule to the data and decision in the question.

This matters because fp1.1.2 - sum, product and quotient of complex numbers determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.2 - sum, product and quotient of complex numbers to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.2 - Sum, product and quotient of complex numbers is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.1.3 - argand diagrams and complex operations to connect the rule to the data and decision in the question.

This matters because fp1.1.3 - argand diagrams and complex operations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.3 - argand diagrams and complex operations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.3 - Argand diagrams and complex operations is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Complex solutions of quadratic equations with real coefficients.

Use fp1.1.4 - complex solutions of quadratic equations to connect the rule to the data and decision in the question.

This matters because fp1.1.4 - complex solutions of quadratic equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.4 - complex solutions of quadratic equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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.

Use fp1.1.5 - finding conjugate complex roots to connect the rule to the data and decision in the question.

This matters because fp1.1.5 - finding conjugate complex roots determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.5 - finding conjugate complex roots to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.5 - Finding conjugate complex roots is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.1.6 - finding conjugate complex roots to connect the rule to the data and decision in the question.

This matters because fp1.1.6 - finding conjugate complex roots determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.6 - finding conjugate complex roots to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.6 - Finding conjugate complex roots is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.2.1 - sum of roots and product of roots to connect the rule to the data and decision in the question.

This matters because fp1.2.1 - sum of roots and product of roots determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.2.1 - sum of roots and product of roots to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.2.1 - Sum of roots and product of roots is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.2.2 - manipulation of expressions to connect the rule to the data and decision in the question.

This matters because fp1.2.2 - manipulation of expressions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.2.2 - manipulation of expressions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.2.2 - Manipulation of expressions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.2.3 - forming quadratic equations with 1 1 1 1 2 to connect the rule to the data and decision in the question.

This matters because fp1.2.3 - forming quadratic equations with 1 1 1 1 2 determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.2.3 - forming quadratic equations with 1 1 1 1 2 to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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.

Use fp1.3.1 - equations of the form f(x) = 0 f(x) will involve only to connect the rule to the data and decision in the question.

This matters because fp1.3.1 - equations of the form f(x) = 0 f(x) will involve only determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.3.1 - equations of the form f(x) = 0 f(x) will involve only to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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.

Use fp1.4.1 - cartesian equations for the parabola to connect the rule to the data and decision in the question.

This matters because fp1.4.1 - cartesian equations for the parabola determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.4.1 - cartesian equations for the parabola to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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.

Use fp1.4.2 - parametric equations for conics to connect the rule to the data and decision in the question.

This matters because fp1.4.2 - parametric equations for conics determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.4.2 - parametric equations for conics to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

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

Use fp1.4.3 - focus-directrix property of the parabola to connect the rule to the data and decision in the question.

This matters because fp1.4.3 - focus-directrix property of the parabola determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.4.3 - focus-directrix property of the parabola to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.4.3 - Focus-directrix property of the parabola is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.4.4 - tangents and normals to conics to connect the rule to the data and decision in the question.

This matters because fp1.4.4 - tangents and normals to conics determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.4.4 - tangents and normals to conics to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.4.4 - Tangents and normals to conics is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Addition and subtraction of matrices.

Use fp1.5.1 - addition and subtraction of matrices to connect the rule to the data and decision in the question.

This matters because fp1.5.1 - addition and subtraction of matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.5.1 - addition and subtraction of matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.5.1 - Addition and subtraction of matrices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Multiplication of a matrix by a scalar.

Use fp1.5.2 - multiplication of a matrix by a scalar to connect the rule to the data and decision in the question.

This matters because fp1.5.2 - multiplication of a matrix by a scalar determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.5.2 - multiplication of a matrix by a scalar to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.5.2 - Multiplication of a matrix by a scalar is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Products of matrices.

Use fp1.5.3 - products of matrices to connect the rule to the data and decision in the question.

This matters because fp1.5.3 - products of matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.5.3 - products of matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.5.3 - Products of matrices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.5.4 - evaluation of 2 × 2 determinants to connect the rule to the data and decision in the question.

This matters because fp1.5.4 - evaluation of 2 × 2 determinants determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.5.4 - evaluation of 2 × 2 determinants to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.5.4 - Evaluation of 2 × 2 determinants is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.5.5 - inverse of 2 × 2 matrices to connect the rule to the data and decision in the question.

This matters because fp1.5.5 - inverse of 2 × 2 matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.5.5 - inverse of 2 × 2 matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.5.5 - Inverse of 2 × 2 matrices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.6.1 - linear transformations of column to connect the rule to the data and decision in the question.

This matters because fp1.6.1 - linear transformations of column determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.6.1 - linear transformations of column to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.6.1 - Linear transformations of column is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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 ∈.

Use fp1.6.2 - applications of 2 × 2 matrices to connect the rule to the data and decision in the question.

This matters because fp1.6.2 - applications of 2 × 2 matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.6.2 - applications of 2 × 2 matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.6.2 - Applications of 2 × 2 matrices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.6.3 - combinations of transformations to connect the rule to the data and decision in the question.

This matters because fp1.6.3 - combinations of transformations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.6.3 - combinations of transformations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.6.3 - Combinations of transformations is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.6.4 - inverse transformations and determinant scale factor to connect the rule to the data and decision in the question.

This matters because fp1.6.4 - inverse transformations and determinant scale factor determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.6.4 - inverse transformations and determinant scale factor to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.6.4 - Inverse transformations and determinant scale factor is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use fp1.7.1 - summation of simple finite series to connect the rule to the data and decision in the question.

This matters because fp1.7.1 - summation of simple finite series determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.7.1 - summation of simple finite series to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.7.1 - Summation of simple finite series is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use fp1.8.1 - proof by mathematical induction to connect the rule to the data and decision in the question.

This matters because fp1.8.1 - proof by mathematical induction determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.8.1 - proof by mathematical induction to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.8.1 - Proof by mathematical induction is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.