A1. QUADRATIC FUNCTIONS
- Finding the maximum or minimum value of a quadratic function using the method of completing the square
- Conditions for y = ax2 + 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:
- a3 + b3 = (a + b)(a2 – ab + b2)
- a3 – b3 = (a – b)(a2+ ab + b2) - Partial fractions with cases where the denominator is no more
complicated than:
- (ax + b) (cx + d)
- (ax + b) (cx + d)2
- (ax + b) (x2 + c2)
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 \( \binom {n} {r} \)an-rbr, 0 ⩽ r ⩽ n
(knowledge of the greatest term and properties of the coefficients is not required)
A6. EXPONENTIAL & LOGARITHMIC FUNCTIONS
- Exponential and logarithmic functions ax
, ex
, logax, In x and their
graphs, including
- laws of logarithms
- equivalence of y = ax and loga 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 xn , for any rational n, sin x, cos x, tan x, ex , 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 xn for any rational n, sin x, cos x, sec2 x and ex, 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–1x, cos–1x, tan–1x
- 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:
- \( \frac{sinA}{cosA} = tanA \), \( \frac{cosA}{sinA} = cotA \), sin2A + cos2A = 1, sec2A = 1 + tan2A, cosec2A = 1 + cot2A
- 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\( \theta \) in the form 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 = r2
– x2 + y2 +2gx + 2fy + c = 0
(excluding problems involving two circles) - Transformation of given relationships, including y = axn and y = kbx, 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)