AQA AS Maths MPC1 Core 1 Revision notes

Pure Core 1

 Introduction

Candidates will be required to demonstrate:

a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. 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 such as ∴ , ⇒, ⇐ and ⇔ .

Candidates are not allowed to use a calculator in the assessment unit for this module.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Quadratic Equations

ax 2 +bx 2 +c=0   has roots −b±(b 2 −4ac)  √  2a   

Circles

A circle, centre (a,b)and radius r, has equation
(x−a) 2 +y−b) 2 =r 2  

Differentiation

function
ax n  

f(x)+g(x) 

derivative
anx n−1       n is a whole number

f'(x)+g'(x) 

Integration

ax n  

f'(x)+g'(x) 

a n+1  x n+1 +c      n is a whole number

f(x)+g(x)+c 

Area

Area under a curve =∫ b a ydx(y⩾0) 

 Algebra

Use and manipulation of surds.

To include simplification and rationalisation of the denominator of a fraction.

Eg√12+2√27=8√3 ; 1 √2−1  =√2+1 ; 2√3+√2 3√2+√3  =√6 3   

Quadratic functions and their graphs.

To include reference to the vertex and line of symmetry of the graph.

The discriminant of a quadratic function.

To include the conditions for equal roots, for distinct real roots and for no real roots

Factorisation of quadratic polynomials.

Eg factorisation of2x 2 +x−6 

Completing the square.

Egx 2 +6x−1=(x+3) 2 −10 ; 2x 2 −6x+2=2(x−1.5) 2 −2.5 

Solution of quadratic equations.

Use of any of factorisation,−b±b 2 −4ac  √  2a    or
completing the sqaure will be accepted.

Simultaneous equations, e.g. one linear and one quadratic, analytical solution by substitution.

Solution of linear and quadratic inequalities.

Eg2x 2 +x⩾g 

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

Simple algebraic division.

Applied to a quadratic or a cubic polynomial divided by a linear term of the form(x+a) or (x−a)  where a is a small whole number. Any method will be accepted, e.g. by inspection, by equating coefficients or by formal division e.gx 3 −x 2 −5x+2 x+2   

Use of the Remainder Theorem.

Knowledge that when a quadratic or cubic polynomialf(x)  is divided
by(x−a)  the remainder isf(a)  and, that whenf(a)=0 , then(x−a)  is a factor and vice versa.

Use of the Factor Theorem.

Greatest level of difficulty as indicated byx 3 −5x 2 +7x−3  i.e. a
cubic always with a factor(x+a) or (x−a)  where a is a small whole number but including the cases of three distinct linear factors, repeated linear factors or a quadratic factor which cannot be factorized in the real numbers.

Graphs of functions;…