Unit FP3: Further Pure Mathematics 3

Definition of the six hyperbolic For example, cosh x = 1 (ex + e−x), functions in terms of exponentials.; Graphs and properties of the 1 2 sech x = =. hyperbolic functions. cosh x ex + e−x Students should be able to derive and use simple identities such as cosh2 x − sinh2 x ≡ 1 and cosh2 x + sinh2 x ≡ cosh 2x and to solve equations such as a cosh x + b sinh x = c.

Use fp3.1.1 - definition of the six hyperbolic to connect the rule to the data and decision in the question.

This matters because fp3.1.1 - definition of the six hyperbolic determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.1.1 - definition of the six hyperbolic to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.1.1 - Definition of the six hyperbolic is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Understand inverse hyperbolic functions, their graphs, properties and logarithmic forms, including arsinh x = ln(x + √(1 + x²)).

Use fp3.1.2 - inverse hyperbolic functions to connect the rule to the data and decision in the question.

This matters because fp3.1.2 - inverse hyperbolic functions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.1.2 - inverse hyperbolic functions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.1.2 - Inverse hyperbolic functions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Cartesian and parametric equations Extension of work from FP1. for the ellipse and hyperbola.; Students should be familiar with the equations: x2 y2 + = 1; x = a cos t, y = b sin t. a2 b2 x2 y2 − = 1; x = a sec t, y = b tan t; a2 b2 x = a cosh t, y = b sinh t.

Use fp3.2.1 - cartesian and parametric equations to connect the rule to the data and decision in the question.

This matters because fp3.2.1 - cartesian and parametric equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.2.1 - cartesian and parametric 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.

The focus-directrix properties of For example, students should know that, for the ellipse, the ellipse and hyperbola, including b2 = a2(1 − e2), the foci are (ae, 0) and (−ae, 0) and the the eccentricity. equations of the directrices are a a x = + and x = −. e e.

Use fp3.2.2 - focus-directrix properties to connect the rule to the data and decision in the question.

This matters because fp3.2.2 - focus-directrix properties determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.2.2 - focus-directrix properties to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.2.2 - Focus-directrix properties is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Tangents and normals to these The condition for y = mx + c to be a tangent to these curves curves. is expected to be known.

Use fp3.2.3 - tangents and normals to these to connect the rule to the data and decision in the question.

This matters because fp3.2.3 - tangents and normals to these determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.2.3 - tangents and normals to these to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.2.3 - Tangents and normals to these is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Simple loci problems.

Use fp3.2.4 - simple loci problems to connect the rule to the data and decision in the question.

This matters because fp3.2.4 - simple loci problems determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.2.4 - simple loci problems to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.2.4 - Simple loci problems is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Differentiation of hyperbolic cosh2x For example, tanh 3x, x sinh2 x,. functions and expressions involving (x +1) them.

Use fp3.3.1 - differentiation of hyperbolic cosh2x to connect the rule to the data and decision in the question.

This matters because fp3.3.1 - differentiation of hyperbolic cosh2x determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.3.1 - differentiation of hyperbolic cosh2x to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.3.1 - Differentiation of hyperbolic cosh2x is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Differentiation of inverse functions, For example, arcsin x + x (1 – x2), 1 artanh x2. including trigonometric and 2 hyperbolic functions.

Use fp3.3.2 - differentiation of inverse functions, to connect the rule to the data and decision in the question.

This matters because fp3.3.2 - differentiation of inverse functions, determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.3.2 - differentiation of inverse functions, to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.3.2 - Differentiation of inverse functions, is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Integration of hyperbolic functions and expressions involving them.

Use fp3.4.1 - integration of hyperbolic functions to connect the rule to the data and decision in the question.

This matters because fp3.4.1 - integration of hyperbolic functions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.1 - integration of hyperbolic functions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.4.1 - Integration of hyperbolic functions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Integration of inverse trigonometric ∫ ∫ For example, arsinh x dx, arctan x dx. and hyperbolic functions.

Use fp3.4.2 - integration of inverse trigonometric to connect the rule to the data and decision in the question.

This matters because fp3.4.2 - integration of inverse trigonometric determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.2 - integration of inverse trigonometric to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.4.2 - Integration of inverse trigonometric is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Integration using hyperbolic and To include the integrals of trigonometric substitutions. 1 1 1 1,,, (a2 + x2) (a2 − x2) (a2 + x2) (x2 − a2).

Use fp3.4.3 - integration to connect the rule to the data and decision in the question.

This matters because fp3.4.3 - integration determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.3 - integration to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.4.3 - Integration is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use of substitution for integrals In more complicated cases, substitutions will be given. involving quadratic surds.

Use fp3.4.4 - use of substitution for integrals to connect the rule to the data and decision in the question.

This matters because fp3.4.4 - use of substitution for integrals determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.4 - use of substitution for integrals to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.4.4 - Use of substitution for integrals is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Derive and use simple reduction formulae for integrals, including powers of sine.

Use fp3.4.5 - reduction formulae to connect the rule to the data and decision in the question.

This matters because fp3.4.5 - reduction formulae determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.5 - reduction formulae 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 calculation of arc length and The equation of the curve may be given in cartesian or the area of a surface of revolution. parametric form.; Equations in polar form will not be set.

Use fp3.4.6 - the to connect the rule to the data and decision in the question.

This matters because fp3.4.6 - the determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.4.6 - the to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.4.6 - The is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

The vector product a × b and the The interpretation of | a × b | as an area and a. b × c as a triple scalar product a. b × c. volume.

Use fp3.5.1 - vector product and scalar triple product to connect the rule to the data and decision in the question.

This matters because fp3.5.1 - vector product and scalar triple product determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.5.1 - vector product and scalar triple product to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.5.1 - Vector product and scalar triple product is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use of vectors in problems Students may be required to use equivalent cartesian forms involving points, lines and planes. also.; The equation of a line in the form Applications to include (r − a) × b = 0. (i) distance from a point to a plane, (ii) line of intersection of two planes, (iii) shortest distance between two skew lines.

Use fp3.5.2 - use of vectors in problems to connect the rule to the data and decision in the question.

This matters because fp3.5.2 - use of vectors in problems determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.5.2 - use of vectors in problems to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.5.2 - Use of vectors in problems is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

The equation of a plane in the Students may be required to use equivalent cartesian forms forms also. r.n = p, r = a + sb + tc.

Use fp3.5.3 - equation of a plane to connect the rule to the data and decision in the question.

This matters because fp3.5.3 - equation of a plane determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.5.3 - equation of a plane 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.

Linear transformations of column Extension of work from FP1 to 3 dimensions. vectors in two and three dimensions and their matrix representation.

Use fp3.6.1 - linear transformations of column to connect the rule to the data and decision in the question.

This matters because fp3.6.1 - 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 fp3.6.1 - linear transformations of column to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.1 - Linear transformations of column is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Combination of transformations.; The transformation represented by AB is the transformation Products of matrices. represented by B followed by the transformation represented by A.

Use fp3.6.2 - combination of transformations to connect the rule to the data and decision in the question.

This matters because fp3.6.2 - combination of transformations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.2 - combination of transformations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.2 - Combination of transformations is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Transpose of a matrix.; Use of the relation (AB)T = BTAT.

Use fp3.6.3 - transpose of a matrix to connect the rule to the data and decision in the question.

This matters because fp3.6.3 - transpose of a matrix determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.3 - transpose of a matrix to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.3 - Transpose of a matrix is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Evaluation of 3 × 3 determinants.; Singular and non-singular matrices.

Use fp3.6.4 - evaluation of 3 × 3 determinants to connect the rule to the data and decision in the question.

This matters because fp3.6.4 - evaluation of 3 × 3 determinants determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.4 - evaluation of 3 × 3 determinants to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.4 - Evaluation of 3 × 3 determinants is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Inverse of 3 × 3 matrices.; Use of the relation (AB)−1 = B−1A−1.

Use fp3.6.5 - inverse of 3 × 3 matrices to connect the rule to the data and decision in the question.

This matters because fp3.6.5 - inverse of 3 × 3 matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.5 - inverse of 3 × 3 matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.5 - Inverse of 3 × 3 matrices 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 given transformation or combination of transformations.

Use fp3.6.6 - inverse transformations and combinations to connect the rule to the data and decision in the question.

This matters because fp3.6.6 - inverse transformations and combinations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.6 - inverse transformations and combinations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.6 - Inverse transformations and combinations is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Eigenvalues and eigenvectors of Normalised vectors may be required. 2 × 2 and 3 × 3 matrices.

Use fp3.6.7 - eigenvalues and eigenvectors to connect the rule to the data and decision in the question.

This matters because fp3.6.7 - eigenvalues and eigenvectors determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.7 - eigenvalues and eigenvectors to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.7 - Eigenvalues and eigenvectors is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Reduction of symmetric matrices to Students should be able to find an orthogonal matrix P such diagonal form. that PTAP is diagonal.

Use fp3.6.8 - reduction of symmetric matrices to connect the rule to the data and decision in the question.

This matters because fp3.6.8 - reduction of symmetric matrices determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp3.6.8 - reduction of symmetric matrices to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP3.6.8 - Reduction of symmetric matrices is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.