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Unit P4: Pure Mathematics 4

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  1. P4.1 - Proof

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

  2. P4.2 - Algebra and functions

    1. 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.

  3. P4.3 - Coordinate geometry in the (x, y) plane

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

  4. P4.4 - Binomial expansion

    1. 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.

  5. P4.5 - Differentiation

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

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

  6. P4.6 - Integration

    1. P4.6.1Evaluation of volume of revolution

      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.

    2. P4.6.2Integration by substitution and parts

      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.

    3. P4.6.3Simple cases of integration

      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).

    4. P4.6.4Analytical solution of simple first

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

    5. P4.6.5Use integration to find the area

      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.

  7. P4.7 - Vectors

    1. Vectors in two and three dimensions.

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

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

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

    5. 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.

    6. 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.

    7. 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.