#### A1. QUADRATIC FUNCTIONS

- Finding the maximum or minimum value of a quadratic function using the method of completing the square
- Conditions for y = ax
^{2}+ bx + c to be always positive (or always negative) - Using quadratic functions as models

#### A2. EQUATIONS & INEQUALITIES

- Conditions for a quadratic equation to have:

(i) two real roots

(ii) two equal roots

(iii) no real roots

and related conditions for a given line to:

(i) intersect a given curve

(ii) be a tangent to a given curve

(iii) not intersect a given curve - Solving simultaneous equations in two variables by substitution, with one of the equations being a linear equation
- Solving quadratic inequalities, and representing the solution on the number line

#### A3. SURDS

- Four operations on surds, including rationalising the denominator
- Solving equations involving surds

#### A4. POLYNOMIALS & PARTIAL FRACTIONS

- Multiplication and division of polynomials
- Use of remainder and factor theorems, including factorising polynomials and solving cubic equations
- Use of:

- a^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2})

- a^{3}– b^{3 }= (a – b)(a^{2}+ ab + b^{2}) - Partial fractions with cases where the denominator is no more
complicated than:

- (ax + b) (cx + d)

- (ax + b) (cx + d)^{2}

- (ax + b) (x^{2}+ c^{2})

#### A5. BINOMIAL EXPANSIONS

- Use of the Binomial Theorem for positive integer n
- Use of the notations n! and \( \binom {n} {r} \)
- Use of the general term \( \binom {n} {r} \)a
^{n-r}b^{r}, 0 ⩽ r ⩽ n

(knowledge of the greatest term and properties of the coefficients is not required)

#### A6. EXPONENTIAL & LOGARITHMIC FUNCTIONS

- Exponential and logarithmic functions a
^{x}, e^{x}, log_{a}x, In x and their graphs, including

- laws of logarithms

- equivalence of y = a^{x}and log_{a}x = y

- change of base of logarithms - Simplifying expressions and solving simple equations involving exponential and logarithmic functions
- Using exponential and logarithmic functions as models

#### C1. DIFFERENTIATION

- Derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point
- Derivative as rate of change
- Use of standard notations

f'(x), f''(x), \( \frac{dy}{dx} \), \( \frac{d^2y}{dx^2} [= \frac{d}{dx} ( \frac{dy}{dx})] \) - Derivatives of x
^{n}, for any rational n, sin x, cos x, tan x, e^{x}, and In x together with constant multiples, sums and differences - Derivatives of products and quotients of functions
- Use of Chain Rule
- Increasing and decreasing functions
- Stationary points (maximum and minimum turning points and stationary points of inflexion)
- Use of second derivative test to discriminate between maxima and minima
- Apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems

#### C2. INTEGRATION

- Integration as the reverse of differentiation
- Integration of x
^{n}for any rational n, sin x, cos x, sec^{2}x and e^{x}, together with constant multiples, sums and differences - Integration of (ax + b)
^{n}for any rational n, sin (ax + b), cos (ax + b) and e^{(ax + b)} - Definite integral as area under a curve
- Evaluation of definite integrals
- Finding the area of a region bounded by a curve and line(s) (excluding area of region between 2 curves)
- Finding areas of regions below the x-axis
- Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line

#### G1. TRIGONOMETRIC FUNCTIONS, IDENTITIES & EQUATIONS

- Six trigonometric functions for angles of any magnitude (in degrees or radians)
- Principal values of sin
^{–1}x, cos^{–1}x, tan^{–1}x - Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or (\( \frac{ \pi }{6} , \frac{ \pi }{4} , \frac{ \pi }{3} \))
- Amplitude, periodicity and symmetries related to sine and cosine functions
- Graphs of y = a sin (bx) + c, y = a sin (\frac{x}{b}) + c, y = a cos (bx) + c, y = a cos (\( \frac{x}{b} \) + c and y = a tan (bx), where a is real, b is a positive integer and c is an integer.
- Use of:

- \( \frac{sinA}{cosA} = tanA \), \( \frac{cosA}{sinA} = cotA \), sin^{2}A + cos^{2}A = 1, sec^{2}A = 1 + tan^{2}A, cosec^{2}A = 1 + cot^{2}A

- the expansions of sin(A \( \pm \) B) , cos(A \( \pm \) B) and tan(A \( \pm \) B)

- the formulae for sin2A , cos2A and tan2A

- the expression of*acos*+*\( \theta \)**bsin*in the form*\( \theta \)**Rcos(\( \theta \) \( \pm \) \( \alpha \))*or*Rsin(**\( \theta \)*\( \pm \) \( \alpha \)) - Simplification of trigonometric expressions
- Solution of simple trigonometric equations in a given interval (excluding general solution)
- Proofs of simple trigonometric identities
- Using trigonometric functions as models

#### G2. COORDINATE GEOMETRY IN 2 DIMENSIONS

- Condition for two lines to be parallel or perpendicular
- Midpoint of line segment
- Area of rectilinear figure
- Coordinate geometry of circles in the form:

– (x-a)^{2}+ (y-b)^{2}= r^{2}

– x^{2}+ y^{2}+2gx + 2fy + c = 0

(excluding problems involving two circles) - Transformation of given relationships, including y = ax
^{n}and y = kb^{x}, to linear form to determine the unknown constants from a straight line graph

#### G3. PROOFS IN PLANE GEOMETRY

- Use of:

- properties of parallel lines cut by a transversal, perpendicular and angle bisectors, triangles, special quadrilaterals and circles ∗ • 1

- congruent and similar triangles ∗

- midpoint theorem

- tangent-chord theorem (alternate segment theorem)