Maths YMA01 — specification explorer

1. Proof

• 1.1

  Proof by contradiction bra and functions:

  • Including proof of the irrationality of 2 and the infinity of primes, and application to unfamiliar proofs.
  P4

1. Algebra and functions

• 2.1

  Decompose rational functions into partial fractions (denominators not more complicated than repeated linear terms). Coordinate geometry in the (x, y) plane:

  • Partial fractions to include denominators such as (ax + b)(cx + d)(ex + f) and (ax + b)(cx + d)². 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.
  P4

1. Coordinate geometry in the ( x , y ) plane

• 3.1Parametric equations of curves and conversion between cartesian and parametric forms.P4

1. Binomial expansion

• 4.1

  Binomial Series for any rational n. Differentiation:

  • For |x| < b⁄a, students should be able to obtain the expansion of (ax + b)ⁿ, and the expansion of rational functions by decomposition into partial fractions.
  P4

1. Differentiation

• 5.1Differentiation of simple functions defined implicitly or parametrically. The finding of equations of tangents and normals to curves given parametrically or implicitly is required.P4
• 5.2Formation of simple differential equations. Questions involving connected rates of change may be set.P4

1. Integration

• 6.1

  Evaluation of volume of revolution.:

  • π∫y²dx is required, but not π∫x²dy. Students should be able to find a volume of revolution, given parametric equations.
  P4
• 6.2

  Simple cases of integration by substitution and integration by parts. Understand these methods as the reverse processes of the chain and product rules respectively.:

  • Students will be expected to use a substitution to find, e.g. ∫x x−2dx The substitution will be given in more complicated integrals. The integral ∫lnxdx is required. More than one application of integration by parts may be required, for example, ∫x²eˣdx, ∫eˣsinxdx.
  P4
• 6.3Simple cases of integration using Integration of rational expressions such as those arising partial fractions. 2 3 from partial fractions, e.g. , 2 . 3 x + 5 ( x − 1) Note that the integration of other rational expressions, such as x and 2 is also required x 2 + 5 (2 x − 1) 4 ( see P3 section 5.2 ).P4
• 6.4Analytical solution of simple first order differential equations with separable variables. General and particular solutions will be required.P4
• 6.5Use integration to find the area under a curve given its parametric equations. Students should be able to find the area under a curve given its parametric equations. Students will not be expected to sketch a curve from its parametric equations.P4

1. Vectors

• 7.1Vectors in two and three dimensions.P4
• 7.2

  Magnitude of a vector.:

  • Students should be able to find a unit vector in the direction of a, and be familiar with |a|.
  P4
• 7.3Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations.P4
• 7.4

  Position vectors.:

  • OB − OA = AB = b − a
  P4
• 7.5The distance between two points. The distance d between two points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) is given by d 2 = ( x 1 – x 2 ) 2 + ( y 1 – y 2 ) 2 + ( z 1 – z 2 ) 2P4
• 7.6

  Vector equations of lines.:

  • To include the forms r = a + t b and r = c + t (d – c) Conditions for two lines to be parallel, intersecting or skew.
  P4
• 7.7

  The scalar product. Its use for calculating the angle between two lines.:

  • Students should know that for OA = a = a₁i + a₂j + a₃k and OB = b = b₁i + b₂j + b₃k then a.b = a₁b₁ + a₂b₂ + a₃b₃ and a.b cos∠AOB = a b Students should know that if a.b = 0, and a and b are non- zero vectors, then a and b are perpendicular.
  P4