GCSE Geography AQA 8035

The basic processes of algebra

Knowledge and use of basic skills in manipulative algebra including use of the associative, commutative and distributive laws, are expected

Use and manipulation of formulae and expressions

*Rearrange $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$ to make $v$ the subject *

Use of the factor theorem for rational values of the variable for polynomials

*Factorise $x^3 - 2x^2 - 5x + 6$

Show that $2x - 3$ is a factor of $2x^3 - x^2 - 7x + 6$

Solve $x^3 + x^2 - 10x + 8 = 0$

Show that $x - 7$ is a factor of $x^5 - 7x^4 - x + 7$ *

Completing the square

*Work out the values of $a$, $b$ and $c$ such that

$2x^2 + 6x + 7 ≡ a(x + b)^2 + c$ *

Drawing and sketching of functions

Interpretation of graphs

*Graphs could be linear, quadratic, exponential and restricted to no more than 3 domains

Exponential graphs will be of the form $y = ab^x$ and $y = ab^{-x}$, where $a$ and $b$ are rational numbers

Sketch the graph of $y = x^2 - 5x + 6$

Label clearly any points of the intersection with the axes

A function $f$ is defined as

$f(x) = x^2 \quad 0 ⩽ x < 1$$\qquad = 1 \quad 1 ⩽ x < 2$$\qquad = 3 - x \quad 2 ⩽ x < 3$

Draw the graph of $y = f(x)$ on the grid below for values of $x$ from 0 to 3

Given a sketch of $y = ab^{-x}$, and two points, work out the values of $a$ and $b$ *

Solution of linear and quadratic equations

*Solutions of quadratics to include solution by factorisation, by graph, by completing the square or by formula

Problems will be set in a variety of contexts, which result in the solution of linear or quadratic equations *

Algebraic and graphical solution of simultaneous equations in two unknowns, where the equations could both be linear or one linear and one second order

*Solve $4x - 3y = 0$ and $6x + 15y = 13$

Solve $y = x + 2$ and $y^2 = 4x + 5$

Solve $y = x^2$ and $y - 5x = 6$

Solve $xy = 8$ and $x + y = 6$ *

Algebraic solution of linear equations in three unknowns

*Solve

$2x - 5y + 4z = 2$$2x + y + 2z = 4$$x - y - 6z = -4$ *

Solution of linear and quadratic inequalities

*Solve $5(x – 7) > 2(x + 1)$

Solve $x^2 < 9$

Solve $2x^2 + 5x ⩽ 3$ *

Index laws, including fractional and negative indices and the solution of equations

*Express as a single power of $x$

$\sqrt{x^{\frac{1}{2}} × x^{\frac{7}{2}}}$

$\sqrt{\dfrac{x^{\frac{3}{2}} × x^{\frac{7}{2}}}{x^2}}$

Solve $x^{-\frac{1}{2}} = 3$

Solve $\sqrt{x} - \dfrac{10}{\sqrt{x}} = 3 \quad x > 0$ *

Algebraic proof

*Prove $(n + 5)^2 - (n + 3)^2$ is divisible by 4 for any integer value of $n$ *

Definition of a function

*Notation $f(x)$ will be used, e.g. $f(x) = x^2 - 9$ *

Using $n$th terms of sequences

Limiting value of a sequence as $n → ∞$

*Work out the difference between the 16th and 6th terms of the sequence with $n$th term $\dfrac{2n}{n+4}$

Write down the limiting value of $\dfrac{2n}{n+4}$ as $n → ∞$ *

$n$th terms of linear sequences

*A linear sequence starts 180 176 172 ...

By using the nth term, work out which term has value –1000

Work out the $n$th term of the linear sequence $r + s \quad r + 3s \quad r + 5s \quad ...$ *

$n$th terms of quadratic sequences

*Work out the $n$th term of the quadratic sequence

10 16 18 16 ...

Which term has the value 0? *

Domain and range of a function

*Domain may be expressed as, for example, $x > 2$, or "for all x, except $x = 0$" and range may be expressed as $f(x) > -1$ *

Composite functions

*The result of two or more functions, say $f$ and $g$, acting in succession.

$fg(x)$ is $g$ followed by $f$ *

Inverse functions

*The inverse function of $f$ is written $f^{-1}$

Domains will be chosen for $f$ to make $f$ one-one*

Expanding brackets and collecting like terms

*Expand and simplify

$(y^2 - 2y + 3) (2y - 1) - 2(y^3 - 3y^2 + 4y - 2)$*

Expand $(a+b)^n$ for positive integer $n$

*Expand and simplify $(5x + 2)^3$

Use Pascal's triangle to work out the coefficient of $x^3$ in the expansion of $(3 + 2x)^5$ *

Factorising

*Factorise fully $(2x + 3)^2 - (2x - 5)^2$

Factorise $15x^2 - 34xy - 16y^2$

Factorise fully $x^4 - 25x^2$ *

Manipulation of rational expressions:

Use of + – × ÷ for algebraic fractions with denominators being numeric, linear or quadratic

*Simplify $\dfrac{5}{x+2} - \dfrac{3}{2x-1}$

Simplify $\dfrac{x^3 + 2x^2 + x}{x^2 + x}$

Simplify $\dfrac{5x^2 - 14x - 3}{4x^2 - 25} ÷ \dfrac{x-3}{4x^2 + 10x}$ *