Time

Specification

Notes

Page and Exercise

Other Sources

• Algebra 1:

Indices:

Index notation

Laws of indices for all rational exponents

Multiplying and dividing expressions involving indices (inc. brackets e.g. (3x 2 ) 3 )

 

Surds:

Manipulation of surds

Rationalising the denominator

 

Polynomials:

Definition and algebraic manipulation (+, -, x), including an understanding of f(x) notation

 

 

 

 

Zero, negative and fractional indices could be used

Use and equivalence of a m/n and n √ a m should be known

Look at e.g. 4 x = 8 x-1 (see June 2001 paper)

 

 

 

 

 

 

To include expanding brackets, collecting like terms and factorising expressions (common factors, double brackets, difference of two squares, …) of degree n ≤ 3 e.g. x³ + 4x² + 3x, x³ - 4x² + 3

 

 

Ex 8A p208 q1-50

Ex 1A p2

 

R.Ex1 p92 q2,19,36,42,48…

 

 

 

Ex 1B p5

 

 

Ex1C p9, ex 1D p13

 

 

Ex 17A p406

(Ex 18A p412)

 

 

 

 

 

 

Ex17B p409

(18B p415)

 

Ex4A p119

(Ex4a p124)

• Algebra 2:

Quadratics:

Solving quadratic equations by:

-factorisation

-completing the square

-the quadratic equation formula

 

The discriminant of a quadratic function

 

 

 

Use Waldo “Factorising (Harder) Quadratics”

 

 

Formula NOT in formula booklet

 

Use Waldo “The Discriminant” to relate to graphs.

 

 

Ex8B p213 q21-47

 

Ex1I p40

 

Ex1D p18

Ex1E p23

 

 

Ex1F p27 q1-3

 

Graphs:

Graphs of quadratic functions, including how the discriminant effects the location

Sketching curves defined by simple equations, to include simple cubic functions and the reciprocal y=k/x with x ≠ 0 (knowledge of term “asymptote” is required)

Geometrical interpretation of algebraic solution of equations.

Use of intersection points of graphs to solve equations.

 

Transforming graphs – knowing effect on y=f(x) of y=af(x), y=f(x)+a, y=f(x+a), y=f(ax)

 

Focus on axes intersection and max/min point by completing the square

 

 

 

Covered at GCSE-Revise

 

 

 

Apply one of these to quadratics, cubics and reciprocal graphs, sketching the result (axes intersections to be labelled on all exam questions for marks) (Combinations of these is in C3)

 

Ex 1H p36

 

Ex1H p34 q1-3

 

 

 

 

 

 

 

 

 

 

Ex3A p86

(ex3A p91)

 

 

 

 

 

 

Inequalites:

Quadratic inequalities in one variable

 

Simultaneous Equations:

Analytical solution by substitution

 

Check linear (ax+b>cx+d).  Extend to px²+qx+r ≥ 0, px²+qx+r<ax+b

 

Check 2xlinear by elimination

e.g. one equation is linear and the other quadratic

 

Ex1K p45

 

 

Ex1J p42

 

Ex1L p37

 

 

Ex 1J p41

• Coordinate Geometry in the (x,y) plane:

The distance between two points

The midpoint of a line joining two points

The gradient of a line joining two points

Where does a line cross the axis?

The equation of a straight line:

 

-in the form y – y 1 = m(x – x 1 ) (using 1 point and gradient)

or using two points

 

Parallel and Perpendicular Lines

 

← (Background information – check here)

← (Background information – check here)

← (Background information – check here)

Crosses x-axis when y = 0 and vice versa

Answers may be required in form ax + by + c = 0

← (Background information – check here)

Use y=mx+c to read off gradient and y-intercept

 

y – y 1 )/(y 2 – y 1 ) = (x – x 1 )/(x 2 – x 1 ) could be used

 

e.g. find the equation of the line parallel (or perpendicular) to the line 3x + 4y = 18 through the point (2,3)

 

Ex8c p220 q35-53

Ex3B p90

Ex 8C p220 q1-20

 

 

 

Ex8C p220 q21-34

Ex3A p84

 

 

Ex 3B p90

 

 

 

 

Ex5A p133(p137)

Ex5B p136(p140)

Ex5C p141(p145)

 

 

 

 

Ex5D p146(p150)

 

 

Ex5C p141(p145)

 

 

 

5 hours

• Sequences and Series:

Sequences, including those given by a formula for the n th term AND those generated by a simple relation of the form x n+1 =f(x n ) i.e. recurrence relations

 

Use of the sigma notation

 

Arithmetic Series

General term & sum to n terms

The sum of the first n natural numbers

 

Mixed examples, inc. use of sigma notation

 

Discuss terms convergent, divergent, oscillating, periodic.

 

 

Understanding of Σ notation will be expected.

 

The general term and the sum to n terms of the series are required.  The proof of the sum formula should be known.  U n =a+(n-1)d and S n =½n(a+l)=½n[2a+(n-1)d} are given in formula booklet (still learn!)

 

 

Ex4A p111

 

Ex3A p38 P2 text

 

 

 

 

Ex4B p118

Ex4Dp126(select appropriate questions)

 

Ex9A p235q1-3(p234)

 

 

Ex9B p241(p240)

 

 

Ex9C p245(p244)

9 hours

• Differentiation:

The derivative of f(x) as the gradient of the tangent to the graph y=f(x) at a point (investigate numerically?); the gradient of the tangent as a limit; interpretation as a rate of change.

Algebraic differentiation of x n , where n is rational (inc. differentiation term by term)

 

Second order derivatives

 

Applications of differentiation to gradients, tangents and normals

 

i.e. knowing that dy/dx (or f’(x) or d(f(x)/dx)is the rate of change of y with respect to x.

 

 

 

 

e.g. (2x – 5)(x – 1); (x² + 5x – 3)/3x ½ .  Chain rule not required. Not in formula booklet (x n ⇒ nx n-1 )

Notation: d²y/dx² or f’’(x) or d²(f(x)/dx²

 

Use of differentiation to find equations of tangents and normals at specific points on a curve

 

Ex5A p137 q1,2

 

 

 

 

 

Ex5A p137 q2--

 

 

 

Ex5B p147q10,11

 

 

 

 

 

 

 

 

 

Ex6A p163(164)

 

Ex6A p163 q11---

(p164)

Ex6B p170 (p171)

 

 

5 hours

• Integration:

INDEFINITE integration as the reverse of differentiation.  

Integration of x n

 

Given f’(x) and a point on the curve, candidates should be able to find an equation of the curve in the form y=f(x)

 

Remember that a constant of integration is required. Not in the formula booklet (x n ⇒ 1/(n+1) x n+1 +c, n ≠ -1

 

 

E.g. coping with expressions such as ½x² - 3x -½ , (x+2)²/x ½

 

 

 

 

Ex6A p156

 

 

 

Ex6B p161 q1

 

 

 

 

Ex7A p193 q1-9

(p192)

 

 

Ex7A p193 q10---