Unit P4: Pure Mathematics 4

Proof by contradiction Including proof of the irrationality of 2 and the infinity of primes, and application to unfamiliar proofs.

Use proof by contradiction to connect the rule to the data and decision in the question.

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

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

Boundary: Proof by contradiction is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Decompose rational functions into Partial fractions to include denominators such as partial fractions (denominators not (ax + b)(cx + d)(ex + f) and (ax + b)(cx + d)2. more complicated than repeated linear terms).; The degree of the numerator may equal or exceed the degree of the denominator.; Applications to integration, differentiation and series expansions.; Quadratic factors in the denominator such as (x2 + a), a > 0, are not required.

Use decompose rational functions into partial fractions to connect the rule to the data and decision in the question.

This matters because decompose rational functions into partial fractions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply decompose rational functions into partial fractions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Decompose rational functions into Partial fractions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Parametric equations of curves and conversion between cartesian and parametric forms.

Use parametric equations of curves and conversion between to connect the rule to the data and decision in the question.

This matters because parametric equations of curves and conversion between determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply parametric equations of curves and conversion between 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.

Binomial Series for any rational n. b For | x | <, students should be able to obtain the expansion a of (ax + b)n, and the expansion of rational functions by decomposition into partial fractions.

Use binomial series for any rational n to connect the rule to the data and decision in the question.

This matters because binomial series for any rational n determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply binomial series for any rational n to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Binomial Series for any rational n is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Differentiation of simple functions The finding of equations of tangents and normals to curves defined implicitly or given parametrically or implicitly is required. parametrically.

Use differentiation of simple functions to connect the rule to the data and decision in the question.

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

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

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

Formation of simple differential Questions involving connected rates of change may be set. equations.

Use formation of simple differential to connect the rule to the data and decision in the question.

This matters because formation of simple differential determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply formation of simple differential to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Formation of simple differential is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Evaluation of volume of revolution. π ∫ y2 dx is required, but not π ∫ x2 dy.; Students should be able to find a volume of revolution, given parametric equations.

Use evaluation of volume of revolution to connect the rule to the data and decision in the question.

This matters because evaluation of volume of revolution determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply evaluation of volume of revolution to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Evaluation of volume of revolution is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Simple cases of integration by Students will be expected to use a substitution to find, e.g. substitution and integration by ∫ x x − 2 dx parts.; Understand these methods as the reverse processes of the chain The substitution will be given in more complicated and product rules respectively. integrals. ∫ The integral ln x dx is required.; More than one application of integration by parts may be required, for example, ∫ x2ex dx, ∫ ex sin x dx.

Use integration by substitution and parts to connect the rule to the data and decision in the question.

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

Example: apply integration by substitution and parts to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Integration by substitution and parts is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Simple cases of integration using Integration of rational expressions such as those arising partial fractions. 2 3 from partial fractions, e.g.,. 3x +5 (x −1)2 Note that the integration of other rational expressions, such x 2 as and is also required x2 +5 (2x −1)4 (see P3 section 5.2).

Use simple cases of integration to connect the rule to the data and decision in the question.

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

Example: apply simple cases of integration to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Simple cases of integration is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Analytical solution of simple first General and particular solutions will be required. order differential equations with separable variables.

Use analytical solution of simple first to connect the rule to the data and decision in the question.

This matters because analytical solution of simple first determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply analytical solution of simple first to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Analytical solution of simple first is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use integration to find the area Students should be able to find the area under a curve given under a curve given its parametric its parametric equations.; Students will not be expected to equations. sketch a curve from its parametric equations.

Use use integration to find the area to connect the rule to the data and decision in the question.

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

Example: apply use integration to find the area to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Use integration to find the area is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Vectors in two and three dimensions.

Use vectors in two and three dimensions to connect the rule to the data and decision in the question.

This matters because vectors in two and three dimensions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply vectors in two and three dimensions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Vectors in two and three dimensions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Magnitude of a vector.; Students should be able to find a unit vector in the direction of a, and be familiar with | a |.

Use magnitude of a vector to connect the rule to the data and decision in the question.

This matters because magnitude of a vector determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply magnitude of a vector to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Magnitude of a vector is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Perform vector addition and scalar multiplication and interpret both operations geometrically.

Use vector addition and scalar multiplication to connect the rule to the data and decision in the question.

This matters because vector addition and scalar multiplication determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply vector addition and scalar multiplication to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Vector addition and scalar multiplication is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Position vectors.; OB − OA = AB = b − a.

Use position vectors to connect the rule to the data and decision in the question.

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

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

Boundary: Position vectors is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

The distance between two points.; The distance d between two points (x, y, z) and (x, y, z) is given by 1 1 1 2 2 2 d 2 = (x – x)2 + (y – y)2 + (z – z)2 1 2 1 2 1 2.

Use distance between two points to connect the rule to the data and decision in the question.

This matters because distance between two points determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply distance between two points to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: distance between two points is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Vector equations of lines.; To include the forms r = a + tb and r = c + t(d – c) Conditions for two lines to be parallel, intersecting or skew.

Use vector equations of lines to connect the rule to the data and decision in the question.

This matters because vector equations of lines determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply vector equations of lines 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.

Use the scalar product a·b = a1b1 + a2b2 + a3b3 and a·b = |a||b| cos θ to calculate angles between lines and identify perpendicular non-zero vectors.

Use scalar product to connect the rule to the data and decision in the question.

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

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

Boundary: Scalar product is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.