AQA Core 1

 

12.1 Algebra

Use and manipulation of surds

To include simplification and rationalization of the denominator of  a fraction

√12 + 2√27 = 8√3

   1        = √2 + 1

√2 -1

Algebraic manipulation of polynomials, including expanding brackets and collecting like terms

Quadratic functions and their graphs Reference to the vertex and line of symmetry of the graph

Completing the square

The discriminant

Conditions for real roots, distinct real roots and for no real roots

                

Factorization of quadratic poynomials

Eg 2x2 + x – 6

Solution of quadratic equations

Any method

 

Solution of linear & quadratic inequalities

2x2 + x ³ 6

Simple algebraic division

Quadratic or cubic polynomial divided by a linear term of the form (x+ a) ( x – a)  where a is a small whole number

Use of the remainder theorem

When a quadratic or cubic polynomial f(x) is divided by ( x – a) the remainder is f(a) and, that when f(a) = 0, then ( x – a) is a factor and vice versa

Use of the factor theorem

Greatest level of difficulty

 x3 – 5x2 + 7x – 3

ie always a factor ( x + a) ( x – a ) including cases of three distinct linear factors, repeated linear factors or a quadratic factor which cannot be factorised

 

Graphs of functions: sketching, f(x) notation, but only a general idea of the concept of a function is required. Domain and range are not included linear, quadratic, cubic graphs of circles are included

Knowledge of the effect of translations on graphs and their equations

Applied to quadratic graphs and circles ie y = ( x – a )2 + b as a translation of y = x2 and

(x – a)2 + ( y –b)2 = r2 as a      translation of x2 + y2 = r2

Solving simultaneous equations

               Two linear

                A linear and a quadratic Analytical solution by substitution

 

12.2 Coordinate Geometry

Equation of a straight line, including the forms y – y1 = m( x – x1 ) , ax + by  c = 0 and y = mx + c

Problems using gradients, mid-points and the distance between two points

Conditions for two straight lines to be parallel or perpendicular to each other Product of two perpendicular lines is -1

The equation of a circle in the form

( x – a) 2 + ( y – b ) 2 = r2

Completing the square to ind the centre and radius of a circle

Eg x2 + 4x + y2 -6y – 12 = 0

Coordinate geometry of the circle

Use of the following circle properties is required

i) the angle in a semicircle is a right angle

ii) the perpendicular from the centre to a chord bisects the chord

iii) the tangent to a circle is perpendicular to the radius at its point of contact

 

Intersection of a straight line and a curve

Solutions from intersection points

Applications will be to either circles or graphs of quadratic functions

Interpretation of geometrical implication of equal roots, distinct real toots or no real rots

The equation of the tangent and normal at a given point to the circle

Implicit differentiation is not required. Candidates will be expected to use the coordinates of the centre and a point on the circle or of other appropriate points to find relevant gradients

Proof

Construction and presentation of rigorous mathematical arguments through appropriate use of precise statements and logical deduction.

Not counter examples

Correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, implies’, ‘is implied by’, ‘necessary’, ‘sufficient’, and notation for Û, Ü, or Þ therefore

|X+3|<3|X| * X>3

Replace * by Û, Ü, or Þ

12.3 Differentiation

The derivative of f(x) as the gradient of the tangent to the graph y = f(x) at a point

The notations f’(x) or dy will be used

                                   dx

By first principles is not required but the gradient of the tangent as a limit

 

 

Interpretation as  rate of change

Differentiation of polynomials

A general appreciation only of the derivative when interpreting it is required. Differentiation from first principles will not be tested

Applications of differentiation to gradients, tangents and normals

Applications of differentiation to:

         maxima and minima

         stationary points

         increasing and decreasing functions Questions will not be set requiring the determination of or knowledge of points of inflection

Questions may be set in the form of a practical problems where a function of a single variable has to be optimised

Second order derivatives

Application to determining maxima and minima

12.4 Integration

 

Indefinite integration as the reverse of differentiation

 

Integration of polynomials

Evaluation of definite integrals

Interpretation of the definite integral as the area under the curve

Area between a curve and the x axis

Areas wholly below the x axis, knowledge that the integral will give a negative value.

Areas partially above and below the x axis will not be set