SecondaryCode 4037
Additional Mathematics
67 topics and 77 objectives, each one something you should be able to do.
Written by Kindred in its own plain words. Your syllabus is still the official list, so check it for anything exact.
What this subject asks you to be able to do
1.1Functions1 objective
- Understand what a function is, together with its domain and its range (image set); know what one–one and many–one functions are, what an inverse function is, and what is meant by the composition of functions.
1.2Functions1 objective
- Determine the domain and range for functions, including inverse and composite ones.
1.3Functions1 objective
- Recognise function notation and work with it.
1.4Functions1 objective
- Understand how y = |f(x)| relates to y = f(x), where f(x) can be a linear, quadratic, cubic or trigonometric function.
1.5Functions1 objective
- Explain in words why it is not possible to find an inverse for a given function.
1.6Functions1 objective
- Work out a one–one function's inverse.
1.7Functions1 objective
- Form and apply composite functions. Understand that the sequence in which functions are combined matters, so fg is not necessarily equal to gf.
1.8Functions1 objective
- Use rough graph sketches to show how a function relates to its inverse. Understand that reflecting either of the two in the line y = x gives the other.
2.1Quadratic functions1 objective
- Determine, for the quadratic function f : x ↦ ax² + bx + c, its maximum or minimum value, either through completing the square or through differentiating.
2.2Quadratic functions1 objective
- Use the maximum or the minimum value of f(x) to help you sketch y = f(x), or to find its range over a stated domain.
2.3Quadratic functions1 objective
- Know what is needed for f(x) = 0 to give (i) a pair of real roots, (ii) a pair of equal roots or (iii) no real roots, together with the matching conditions on a given line and a given curve for the line to (i) intersect the curve, (ii) be a tangent to it or (iii) not intersect it.
2.4Quadratic functions1 objective
- Solve a quadratic equation and find any real roots it has.
2.5Quadratic functions1 objective
- work out the set of solutions to quadratic inequalities, either by using a graph or by algebra
3.1Factors of polynomials1 objective
- Understand and apply the remainder theorem and the factor theorem.
3.2Factors of polynomials1 objective
- Find factors of polynomials.
3.3Factors of polynomials1 objective
- Solve cubic equations.
4.1Equations, inequalities and graphs1 objective
- Solve the following kinds of equation: • |ax + b| = c (c ⩾ 0) • |ax + b| = cx + d • |ax + b| = |cx + d| • |ax² + bx + c| = d, either algebraically or graphically.
4.2Equations, inequalities and graphs1 objective
- Solve, using a graph or algebra, inequalities having these forms: • k|ax + b| > c (c ⩾ 0) • k|ax + b| ⩽ c (c > 0) • k|ax + b| ⩽ |cx + d| where k > 0 • |ax + b| ⩽ cx + d • |ax² + bx + c| > d • |ax² + bx + c| ⩽ d
4.3Equations, inequalities and graphs1 objective
- Apply a substitution that turns an equation into a quadratic equation, then solve that quadratic in order to solve the equation you started with.
4.4Equations, inequalities and graphs1 objective
- Sketch, for a cubic polynomial written as three linear factors multiplied together, both its graph and the graph of its modulus.
4.5Equations, inequalities and graphs1 objective
- Solve, by means of a graph, any cubic inequality of the type f(x) ⩾ d, f(x) > d, f(x) ⩽ d or f(x) < d, where d is a constant and f(x) is three linear factors multiplied together.
5.1Simultaneous equations1 objective
- Solve a pair of simultaneous equations with two unknowns, using either elimination or substitution.
6.1Logarithmic and exponential functions5 objectives
- Understand how logarithmic functions and exponential functions are connected.
- Know the basic properties and graph shapes of exponential and logarithmic functions, including ex and ln x, and use them.
- Understand that the functions f(x) = ex and g(x) = ln x are inverses of one another.
- Understand how the graphs of exponential and logarithmic functions approach their asymptotes.
- Give the equations of whichever asymptotes exist.
6.2Know and use the laws of logarithms, including change of base of logarithms1 objective
- Understand and apply the laws of logarithms, including the rule for changing the base of a logarithm
6.3Solve equations of the form ax = b1 objective
- Solve for x when ax = b.
7.1Use the equation of a straight line1 objective
- Use straight-line equations.
7.2Know and use the condition for two lines to be parallel or perpendicular1 objective
- Know what condition makes two lines parallel and what condition makes them perpendicular, and use these.
7.3Solve problems involving midpoint and length of a line, including finding and using the equation of a perpendicular bisector1 objective
- Solve problems about a line's length and its midpoint, including finding the equation of the line's perpendicular bisector and using it.
7.4Transform given relationships to and from straight-line form, including determining unknown constants by calculating the gradient or intercept of the transformed graph1 objective
- Transform given relationships into and out of straight-line form, including finding unknown constants by working out the gradient or intercept of the graph after transformation
8.1Know and use the equation of a circle with radius r and centre (a, b)2 objectives
- Understand and apply the equation that describes a circle whose radius is r and whose centre sits at (a, b)
- Identify the coordinates of the centre and the value of the radius for a circle whose equation may be given in any form
8.2Solve problems involving the intersection of a circle and a straight line3 objectives
- Solve problems about where a straight line intersects a circle.
- This also covers working out the points of intersection.
- It also covers telling, for a given straight line, if it touches the circle as a tangent, forms a chord of it, or does not intersect it.
8.3Solve problems involving tangents to a circle2 objectives
- Solve problems about a circle and the tangents to it.
- Working out the equation of a tangent is part of this too.
8.4Solve problems involving the intersection of two circles2 objectives
- Solve problems about where two circles intersect.
- Covers working out the points of intersection and the common chord's equation, and deciding which case applies to the two circles: they intersect, they touch, or they do not intersect.
9.1Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure1 objective
- Solve problems on the length of an arc and the area of a sector of a circle, including knowing about and using radian measure
10.1Know and use the six trigonometric functions of angles of any magnitude1 objective
- Know the six trigonometric functions and use them whatever the magnitude of the angle.
10.2Understand and use the amplitude and period of a trigonometric function, including the relationship between graphs of related trigonometric functions1 objective
- Understand and use a trigonometric function's amplitude and period, including the way the graphs of linked trigonometric functions relate to one another.
10.3Draw and use the graphs of y = a sin bx + c, y = a cos bx + c, y = a tan bx + c1 objective
- Draw the graph of any function of the form y = a sin bx + c, y = a cos bx + c or y = a tan bx + c, and then work with it.
10.4Use the relationships: sin² A + cos² A = 1, sec² A = 1 + tan² A, cosec² A = 1 + cot² A1 objective
- Apply the identities sin² A + cos² A = 1, sec² A = 1 + tan² A and cosec² A = 1 + cot² A
10.5Solve, for a given domain, trigonometric equations involving the six trigonometric functions1 objective
- Solve trigonometric equations that include any of the six trigonometric functions, where the solutions must lie within a stated domain
10.6Prove trigonometric relationships involving the six trigonometric functions1 objective
- Prove identities built from any of the six trigonometric functions.
11.1Recognise the difference between permutations and combinations and know when each should be used1 objective
- Recognise how permutations and combinations differ, and understand when each one is the right one to employ
11.2Know and use the notation n! and the expressions for permutations and combinations of n items taken r at a time1 objective
- Know what n! means, and use it along with the expressions for permutations and for combinations when r items are taken from n.
11.3Solve problems on arrangement and selection using permutations or combinations1 objective
- Solve problems involving the ordering and choosing of items by applying permutations or combinations
12.1Use the binomial theorem for expansion of (a + b)n for positive integer n1 objective
- apply the binomial theorem to expand (a + b)n where n is a positive integer
12.2Use the general term (n r) an–rbr, 0 ⩽ r ⩽ n1 objective
- apply the expression (n r) an–rbr for the general term, where 0 ⩽ r ⩽ n
12.3Recognise arithmetic and geometric progressions and understand the difference between them1 objective
- Recognise arithmetic and geometric progressions, and appreciate in what ways they differ from each other
12.4Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions1 objective
- Apply the nth-term formula and the formula for the sum to n terms when working on problems about arithmetic or geometric progressions.
12.5Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions1 objective
- Apply the nth-term formula and the formula for the sum to n terms when working on problems about arithmetic or geometric progressions.
13.1Understand and use vector notation1 objective
- Understand how to work with vector notation
13.2Know and use position vectors and unit vectors1 objective
- Know what position vectors and unit vectors are, and work with both kinds.
13.3Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars1 objective
- work out the magnitude of a given vector; carry out addition and subtraction of vectors, and scale a vector by multiplying it by a scalar
13.4Compose and resolve velocities. Determine a resultant vector by adding two or more vectors together1 objective
- Compose velocities into one and resolve a velocity into components. Find the resultant of two or more vectors by adding them.
14.1Understand and use the concepts of differentiation and integration2 objectives
- Understand the ideas behind differentiation and integration, and apply them
- Understand what a derived function is. Only an informal feel for limits is needed, and differentiating from first principles will not be asked for.
14.2Use the notations1 objective
- Apply the following notations: f′(x), f″(x), dy/dx, d²y/dx², δx, δx → 0, dy/dx.
14.3Know and use the derivatives of the standard functions1 objective
- Know the derivatives of these standard functions and use them: xⁿ (where n is any rational number), sin x, cos x, tan x, eˣ and ln x. It covers constant multiples, sums and composite functions, the last of these by the chain rule. For example, differentiate 3x² + 1/x³. Angles in trigonometric functions are always measured in radians.
14.4Differentiate products and quotients of functions1 objective
- differentiate functions in the form of products, and those in the form of quotients.
14.5Use differentiation to find gradients, tangents and normals1 objective
- Apply differentiation to obtain the gradient of a curve, its tangents and its normals.
14.6Use differentiation to find stationary points1 objective
- Apply differentiation to locate stationary points. This does not cover points of inflexion.
14.7Apply differentiation to connected rates of change, small increments and approximations1 objective
- Use differentiation for small increments and approximations, and for connected rates of change.
14.8Apply differentiation to practical problems involving maxima and minima1 objective
- Use differentiation to tackle real-life situations that involve maxima and minima.
14.9Use the first and second derivative tests to discriminate between maxima and minima1 objective
- Apply the first derivative test and the second derivative test to tell maxima apart from minima. Points of inflexion are not part of this objective. Students must fully justify their conclusions. You might also be asked to explain how a maximum point can be told apart from a minimum point. Any correct approach may be used, provided no other one is specified.
14.10Understand integration as the reverse process of differentiation1 objective
- Understand that integration is differentiation done in reverse. Every indefinite integral should be written with an arbitrary constant.
14.11Integrate sums of terms in powers of x, including x⁻¹ and ax + b⁻¹1 objective
- Integrate a sum of terms that are powers of x, including x⁻¹ and ax + b⁻¹. Each indefinite integral should come with an arbitrary constant.
14.12Integrate functions of the form1 objective
- Integrate (ax + b)ⁿ for any rational value of n (n = –1 included), as well as sin (ax + b), cos (ax + b) and sec² (ax + b), and eᵃˣ⁺ᵇ. Angles in trigonometric functions are always measured in radians, and each indefinite integral should come with an arbitrary constant.
14.13Evaluate definite integrals and apply integration to the evaluation of plane areas1 objective
- Evaluate definite integrals, and use integration to work out plane areas: the area enclosed by a curve and a line, the area enclosed by two curves, and an area found by adding two areas together.
14.14Apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration1 objective
- Use differentiation and integration to solve kinematics questions concerning the displacement, velocity and acceleration of a particle travelling along a straight line, where the acceleration is either variable or constant. For instance, suppose a particle has velocity v = 3t² – 30t + 72; work out its acceleration when t = 2.
14.15Make use of the relationships in 14.14 to draw and use the following graphs1 objective
- Make displacement–time, distance–time, velocity–time, speed–time and acceleration–time graphs from the relationships in 14.14, and use them. For instance, a particle travels along a straight line and, at time t, its displacement from a fixed point is s, where s = 3t³ – 10t² + 4t + 8 for 0 ⩽ t ⩽ 3; sketch the graphs of its displacement, its speed and its acceleration against time.
Kindred builds your map from these, so you can see what you've covered and what's still to do.