AQA Core 4

 

15.1 Algebra and Functions

Simplification of rational expressions including factorising and cancelling, and algebraic division.

 

Use of the Remainder and Factor Theorems. Expressions of the type 

    (3+2x)       
(x-3(2x2 + 1)

Partial fractions (denominators not more complicated than repeated linear terms). 

15.2 Coordinate Geometry in the (x,y) plane

 

Cartesian and parametric equations of curves and conversion between the two forms.  Knowledge of the curves given by the following parametric equations will be expected: 

x = aat2 , y = 2at for a parabola,  

x = asint, y = bcost for an ellipse or circle (a=b),

x = ct, y = c/t for a rectangular hyperbola.

 

 

Co-ordinate geometry of the circle.  Equation of a circle in the form

(x-a)2+(y-b)2=r2.

e.g.

x=t2 , y=2t;

x = acost, y = bsint;

x = 1/t, y = 3t;

x = t+1/t, y = t-1/t   (x+y)(x-y)=4

 

Candidates will be expected to complete the square to find the centre and radius of a circle.

15.3 Sequences and Series

Binomial series for any rational n. To include (a+x)n, |x|<a

 

Series expansion of rational functions including the use of partial fractions.

Greatest level of difficulty

(2+3x)-2 = 1/4(1+3x/2)-2

 

      9+2x2        

(2x+1)(x-3)2

15.4 Trigonometry

Knowledge and use of double angle formulae. Use of formulae for sin(AB) & cos(AB) and tan(AB) and of expressions for acosx+bsinx in the equivalent forms of rcos(xa) or rsin(xa)

 

Knowledge that sin2x = 2sinxcosx

cos2x = cos2x-sin2x = 2cos2x-1 = 1 - 2sin2x

 

and tan 2x = 2tan2x

                      1-tan2x

 

Solution of trigonometric equations in a given interval

e.g. 2sinx + 3cosx = 1.5

       3sin2x = cosx

 

 

Use in simple identities

Sin3x = sin(2x + x) = sinx(3 4sin2x)

 

Use in integration e.g. ∫cos2xdx

15.5 Exponentials and Logarithms

 

Exponential growth and decay. The use of exponential functions as models.

15.6 Differentiation and Integration

Formation of simple differential equations.  To include the context of growth and decay.

Simple cases of integration using partial fractions. Maximum level of difficulty 

∫         (x+1)        dx     ∫        x2        dx

     (3x-4)(x+3)2               (x+5)(x-3)

Analytical solution of simple first order differential equations with separable variables.

Differentiation of simple functions defined implicitly or parametrically. To include the examples in the co-ordinate geometry section above.

The second derivative of curves defined implicitly or parametrically is not required.

 

Equations of tangents and normals for curves specified implicitly or in parametric form.

 

 

 

 

15.7 Vectors

Vectors in two and three dimensions. Position vectors.

Distinguish between the terms Vector and position vector.

 

 

Magnitude of a vector.

The distance between two points.

 

Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations.

Vector equations of lines. To include the intersection of two straight lines in two and three dimensions and skew lines in three dimensions.

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

 

The co-ordinates of the foot of the perpendicular from a point to a line. The perpendicular distance from a point to a line.