Maths — Pure 3

Underground Mathematics

Cambridge's Underground Mathematics Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Combining Functions, Combining Vectors, Counting & Binomials, Divisibility & Induction, Exponentials & Logarithms, Hyperbolic Functions, Polynomials & Rational Functions, Power Series, Product Rule & Integration by Parts, Thinking about Algebra, Thinking about Functions, Thinking about Geometry, Thinking about Numbers, Exponentials & Logarithms, Calculus of Trigonometry & Logarithms, Thinking about Algebra, Trigonometry: Compound Angles, Trigonometry: Triangles to Functions, Calculus of Trigonometry & Logarithms, Calculus meets Functions, Calculus of Powers, Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Hyperbolic Functions, Polynomials & Rational Functions, Product Rule & Integration by Parts, Thinking about Functions, Thinking about Geometry, Vector Geometry, Calculus meets Functions, Calculus of Powers, Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Circles, Combining Functions, Counting & Binomials, Divisibility & Induction, Geometry of Equations, Hyperbolic Functions, Introducing Calculus, Polynomials & Rational Functions, Product Rule & Integration by Parts, Sequences, Thinking about Functions, Calculus meets Functions, Calculus of Powers, Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Circles, Combining Functions, Counting & Binomials, Exponentials & Logarithms, Geometry of Equations, Hyperbolic Functions, Introducing Calculus, Polynomials & Rational Functions, Product Rule & Integration by Parts, Quadratics, Sequences, Thinking about Algebra, Thinking about Functions, Thinking about Numbers, Trigonometry: Compound Angles, Vector Geometry, Calculus meets Functions, Calculus of Powers, Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Circles, Combining Functions, Combining Vectors, Divisibility & Induction, Exponentials & Logarithms, Geometry of Equations, Hyperbolic Functions, Introducing Calculus, Product Rule & Integration by Parts, Quadratics, Thinking about Algebra, Thinking about Functions, Thinking about Geometry, Thinking about Numbers, Trigonometry: Compound Angles, Trigonometry: Triangles to Functions, Vector Geometry, Calculus meets Functions, Calculus of Powers, Calculus of Trigonometry & Logarithms, Chain Rule & Integration by Substitution, Combining Functions, Counting & Binomials, Divisibility & Induction, Exponentials & Logarithms, Hyperbolic Functions, Introducing Calculus, Product Rule & Integration by Parts, Quadratics, Sequences, Thinking about Algebra, Thinking about Functions, Thinking about Geometry, Thinking about Numbers, Trigonometry: Triangles to Functions, Vector Geometry, Polynomials & Rational Functions, Quadratics, Thinking about Algebra, Exponentials & Logarithms, Counting & Binomials, Calculus meets Functions, Combining Functions, Circles, Vector Geometry, Combining Vectors, Trigonometry: Compound Angles, Thinking about Numbers, Calculus of Powers, Chain Rule & Integration by Substitution, Trigonometry: Triangles to Functions, Product Rule & Integration by Parts, Geometry of Equations and Thinking about Functions collections for Cambridge International A-Level Mathematics — 149 resources mapped across 43 specification points, each with suggestions and worked solutions.

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43 specification points

3.1.1 · understand the meaning of | x |, sketch the graph of y = | ax + b | and use relations such as

3 resources

3.1.2 · | a | = | b | ⇔ a² = b² and | x – a | < b ⇔ a – b < x < a + b when solving equations and inequalities

1 resource

Inequality sets recommendedMany ways problem · Developing

3.1.3 · divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero)

5 resources

3.1.4 · use the factor theorem and the remainder theorem

21 resources

3.1.5 · recall 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 )

5 resources

3.1.6 · use the expansion of (1 + x)ⁿ, where n is a rational number and | x | < 1

3 resources

3.2.1 · understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)

16 resources

3.2.2 · understand the definition and properties of e Including knowledge of the graph of and ln x , including their relationship as inverse functions and their graphs

2 resources

3.2.3 · use logarithms to solve equations and e.g. inequalities in which the unknown appears in indices

14 resources

3.2.4 · use logarithms to transform a given relationship e.g. to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

2 resources

3.3.1 · understand 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

1 resource

Is this product of integrals positive or negative? recommendedReview question · teacher support included

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 tan( A ± B ) forms R sin ^ i ! a h and R cos ^ i ! a h .: • sec i / 1 + tan i and cosec i / 1 + cot i • the expansions of sin( A ± B ) , cos( A ± B ) and • the formulae for sin 2 A , cos 2 A and tan 2 A • the expression of a sin i + b cos i in the

0 resources

3.4.1 · use the derivatives of e , ln x , sin x , cos x , tan x , tan x , together with constant multiples, sums, differences and composites

13 resources

3.4.2 · differentiate products and quotients e.g.

12 resources

3.4.3 · find and use the first derivative of a function e.g. which is defined parametrically or implicitly.

15 resources

3.5.1 · extend 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

10 resources

3.5.2 · use trigonometrical relationships in carrying out integration or

6 resources

3.5.3 · integrate rational functions by means of Restricted to types of partial fractions as specified in decomposition into partial fractions topic 3.1 above. k f l ^ x h e.g. integration of

2 resources

3.5.4 · recognise an integrand of the form , and integrate such functions

0 resources

3.5.5 · recognise when an integrand can usefully be e.g. integration of regarded as a product, and use integration by parts

5 resources

3.5.6 · use a given substitution to simplify and evaluate e.g. to integrate either a definite or an indefinite integral.

7 resources

3.6.1 · locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change

4 resources

3.6.2 · understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation

2 resources

3.6.3 · understand 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.

2 resources

3.7.1 · use standard notations for vectors, i.e. f p , x i + y j , f p y , x i + y j + z k , AB , a

2 resources

3.7.2 · carry 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

11 resources

3.7.3 · calculate the magnitude of a vector, and use In 2 or 3 dimensions. unit vectors, displacement vectors and position vectors

5 resources

3.7.4 · understand 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 resources

3.7.5 · determine 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.

4 resources

3.7.6 · use 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.

4 resources

3.7.7 · Knowledge of the vector product is not required.

0 resources

3.8.1 · formulate a simple statement involving a rate of change as a differential equation

0 resources

3.8.2 · find by integration a general form of solution for a first order differential equation in which the variables are separable

0 resources

3.8.3 · use an initial condition to find a particular solution

0 resources

3.8.4 · interpret the solution of a differential equation in the context of a problem being modelled by the equation.

0 resources

3.9.1 · understand 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

0 resources

3.9.2 · carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + i y

0 resources

3.9.3 · use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs

0 resources

3.9.4 · represent complex numbers geometrically by means of an Argand diagram

0 resources

3.9.5 · carry 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

0 resources

3.9.6 · find the two square roots of a complex number

0 resources

3.9.7 · understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers

0 resources

3.9.8 · illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram

0 resources