Maths — Pure 3

Mathematics 9709 — specification explorer

All 43 specification statements from the official Cambridge International specification, as printed.

Paper/unit applicability has not been derived for this course yet — the statements below are the parsed specification, as printed.

1. Pure Mathematics

1. Algebra

3.1.1understand the meaning of |x|, sketch the graph of y = |ax + b| and use relations such as
3.1.2|a| = |b| ⇔ a² = b² and |x – a| < b ⇔ a – b < x < a + b when solving equations and inequalities
3.1.3divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero)
3.1.4use the factor theorem and the remainder theorem
3.1.5recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than ( ax + b )( cx + d )( ex + f ), ( ax + b )( cx + d )², ( ax + b )( cx² + d )
3.1.6use the expansion of (1 + x)ⁿ, where n is a rational number and |x| < 1

2. Logarithmic and exponential functions

3.2.1understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
3.2.2understand the definition and properties of e Including knowledge of the graph of and ln x , including their relationship as inverse functions and their graphs
3.2.3use logarithms to solve equations and e.g. inequalities in which the unknown appears in indices
3.2.4use logarithms to transform a given relationship e.g. to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

3. Trigonometry

3.3.1understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude
3.3.2
use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of:
- sec²θ≡1+tan²θand cosec²θ≡1+cot²θ
- the expansions of sin(A ± B), cos(A ± B) and tan(A ± B)
- the formulae for sin 2A, cos 2A and tan 2A
- the expression of asinθ+bcosθ in the forms Rsin(θ±A) and Rcos(θ±A).
- e.g. simplifying cos(x – 30˚)–3sin(x –60˚). e.g. solving tanθ+cotθ=4, 2sec²θ−tanθ=5, 3cosθ+2sinθ = 1.

4. Differentiation

3.4.1use the derivatives of e , ln x , sin x , cos x , tan x , tan x , together with constant multiples, sums, differences and composites
3.4.2differentiate products and quotients e.g.
3.4.3find and use the first derivative of a function e.g. which is defined parametrically or implicitly.

5. Integration

3.5.1extend the idea of ‘reverse differentiation’ to Including examples such as ax + b 1 include the integration of e , , ax + b sin( ax + b ) , cos( ax + b ) , sec ( ax + b ) and 2 2 x + a
3.5.2use trigonometrical relationships in carrying out integration or
3.5.3
integrate rational functions by means of decomposition into partial fractions:
- Restricted to types of partial fractions as specified in topic 3.1 above.
3.5.4recognise an integrand of the form , and integrate such functions
3.5.5recognise when an integrand can usefully be e.g. integration of regarded as a product, and use integration by parts
3.5.6use a given substitution to simplify and evaluate e.g. to integrate either a definite or an indefinite integral.

6. Numerical solution of equations

3.6.1locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change
3.6.2understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation
3.6.3understand how a given simple iterative formula of the form x = F( x ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

7. Vectors

3.7.1use standard notations for vectors, i.e. f p , x i + y j , f p y , x i + y j + z k , AB , a
3.7.2carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms The general form of the ratio theorem is not included, but understanding that the midpoint of
3.7.3calculate the magnitude of a vector, and use In 2 or 3 dimensions. unit vectors, displacement vectors and position vectors
3.7.4understand the significance of all the symbols e.g. finding the equation of a line given the position used when the equation of a straight line is vector of a point on the line and a direction vector, or expressed in the form r = a + t b , and find the the position vectors of two points on the line. equation of a line, given sufficient information
3.7.5determine whether two lines are parallel, Calculation of the shortest distance between two intersect or are skew, and find the point of skew lines is not required. Finding the equation of intersection of two lines when it exists the common perpendicular to two skew lines is also not required.
3.7.6use formulae to calculate the scalar product of e.g. finding the angle between two lines, and finding two vectors, and use scalar products in problems the foot of the perpendicular from a point to a line; involving lines and points. questions may involve 3D objects such as cuboids, tetrahedra (pyramids), etc.
3.7.7Knowledge of the vector product is not required.

8. Differential equations

3.8.1formulate a simple statement involving a rate of change as a differential equation
3.8.2find by integration a general form of solution for a first order differential equation in which the variables are separable
3.8.3use an initial condition to find a particular solution
3.8.4interpret the solution of a differential equation in the context of a problem being modelled by the equation.

9. Complex numbers

3.9.1understand the idea of a complex number, recall the meaning of the terms real part, imaginary The argument of a complex number will usually part, modulus, argument, conjugate, and use the refer to an angle fact that two complex numbers are equal if and only if both real and imaginary parts are equal
3.9.2carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + i y
3.9.3use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs
3.9.4represent complex numbers geometrically by means of an Argand diagram
3.9.5carry out operations of multiplication and division of two complex numbers expressed in i i polar form r ^ cos i + i sin i h / r e
3.9.6find the two square roots of a complex number
3.9.7understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers
3.9.8illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram