Advanced Higher Mathematics Course Assessment Specification Valid from August 015 This edition: April 013, version 1.0 This specification may be reproduced in whole or in part for educational purposes provided that no profit is derived from reproduction and that, if reproduced in part, the source is acknowledged. Additional copies of this Course Specification can be downloaded from SQA s website: www.sqa.org.uk. Please refer to the note of changes at the end of this Course Assessment Specification for details of changes from previous version (where applicable). Scottish Qualifications Authority 013 April 013, version 1.0 1
Course outline Course title: SCQF level: Course code: Course assessment code: Advanced Higher Mathematics 7 (3 SCQF credit points) to be advised to be advised The purpose of the Course Assessment Specification is to ensure consistent and transparent assessment year on year. It describes the structure of the Course assessment and the mandatory skills, knowledge and understanding that will be assessed. Course assessment structure Component 1 question paper Total marks 100 marks 100 marks This Course includes eight SCQF credit points to allow additional time for preparation for Course assessment. The Course assessment covers the added value of the Course. Equality and inclusion This Course Assessment Specification has been designed to ensure that there are no unnecessary barriers to assessment. Assessments have been designed to promote equal opportunities while maintaining the intrity of the qualification. For guidance on assessment arrangements for disabled learners and/or those with additional support needs, please follow the link to the Assessment Arrangements web page: www.sqa.org.uk/sqa/14977.html. Guidance on inclusive approaches to delivery and assessment of this Course is provided in the Course Support Notes. April 013 version 1.0
Assessment To gain the award of the Course, the learner must pass all the Units as well as the Course assessment. Course assessment will provide the basis for grading attainment in the Course award. Course assessment SQA will produce and give instructions for the production and conduct of Course assessments based on the information provided in this document. Added value The purpose of the Course assessment is to assess added value of the Course as well as confirming attainment in the Course and providing a grade. The added value for the Course will address the key purposes and aims of the Course, as defined in the Course Rationale. It will do this by addressing one or more of breadth, challenge or application. In this Course assessment, added value will focus on the following: breadth drawing on knowledge and skills from across the Course challenge requiring greater depth or etension of knowledge and skills application requiring application of knowledge and skills in practical or theoretical contets as appropriate This added value consists of: using mathematical reasoning skills to think logically, provide justification and solve problems using a range of comple concepts selecting and applying comple operational skills using reasoning skills to interpret information and to use comple mathematical models effectively communicating solutions in a variety of mathematical contets eplaining and justifying concepts through the idea of rigorous proof thinking creatively To achieve success in the Course, learners must show that they can apply knowledge and skills acquired across the Course to unseen situations. The question paper requires learners to demonstrate aspects of breadth, challenge and application in mathematical contets. The use of a calculator will be permitted. April 013 version 1.0 3
Grading Course assessment will provide the basis for grading attainment in the Course award. The Course assessment is graded A D. The grade is determined on the basis of the total mark for the Course assessment. A learner s overall grade will be determined by their performance across the Course assessment. Grade description for C For the award of Grade C, learners will have demonstrated successful performance in all of the Units of the Course. In the Course assessment, learners will typically have demonstrated successful performance in relation to the mandatory skills, knowledge and understanding for the Course. Grade description for A For the award of Grade A, learners will have demonstrated successful performance in all of the Units of the Course. In the Course assessment, learners will typically have demonstrated a consistently high level of performance in relation to the mandatory skills, knowledge and understanding for the Course. Credit To take account of the etended range of learning and teaching approaches, remediation, consolidation of learning and intration needed for preparation for eternal assessment, si SCQF credit points are available in Courses at National 5 and Higher, and eight SCQF credit points in Courses at Advanced Higher. These points will be awarded when a Grade D or better is achieved. April 013 version 1.0 4
Structure and coverage of the Course assessment The Course assessment will consist of one Component: a question paper. Component 1 question paper The purpose of the question paper is to assess mathematical skills. A calculator may be used. The question paper will sample the skills, knowledge and understanding that are contained in the Further mandatory information on Course coverage section at the end of this Course Assessment Specification. The question paper will consist of a series of short and etended response questions (some of which may be set in contets) that require the application of skills developed in the Course. Learners will be epected to communicate responses clearly and to justify solutions. The paper will have 100 marks. April 013 version 1.0 5
Setting, conducting and marking of assessment Question paper The question paper will be set and marked by SQA and conducted in centres under conditions specified for eternal eaminations by SQA. Learners will complete this in 3 hours. April 013 version 1.0 6
Further mandatory information on Course coverage The following gives details of mandatory skills, knowledge and understanding for the Advanced Higher Mathematics Course. Course assessment will involve sampling the skills, knowledge and understanding. This list of skills, knowledge and understanding also provides the basis for the assessment of Units of the Course. Mathematics (Advanced Higher) Methods in Algebra and Calculus 1.1 Applying algebraic skills to partial fractions Epressing proper rational functions as a sum of partial fractions (denominator of dree at most 3 and easily factorised) Epress a proper rational function as a sum of partial fractions where the denominator is of the type: 7 1 a (linear factors) 6 (b) 5 6 3 3 (irreducible quadratic factor) 3 10 (c) (repeated factor) 69 Reduce an improper rational function to a polynomial and a proper rational function by division or otherwise 3. 1 3 1. Applying calculus skills through techniques of differentiation Differentiating functions using the chain rule Differentiating functions using the product rule Differentiating functions using the quotient rule. f g f g g f g f g f g f f g f g g g Differentiate functions which require more than one application of the above. Know that dy d 1 d dy. April 013 version 1.0 7
Use logarithmic differentiation, recognising when it is appropriate in etended products and quotients y 7 3, 1 and indices involving the variable y. Differentiating inverse trigonometric functions Finding the derivative of functions defined implicitly Finding the derivative of functions defined parametrically Use the technique: y f 1 f y f 1 f y etc 1,. Use implicit differentiation to find the first derivative. Apply differentiation to related rates in problems where the functional relationship is given implicitly and y r where yare, functions of t ; given d dy, find dt dt using d dy y dt dt 0. Apply implicit differentiation to find second derivatives. Use parametric differentiation to find the first derivative. Apply parametric differentiation to motion in a plane. Use parametric differentiation to find the second derivative. Apply differentiation to related rates in problems where the functional relationship is given eplicitly V 1 r h ; given dh 3 dt, find dv dt. Solve practical related rates by first establishing a functional relationship between appropriate variables. 1.3 Applying calculus skills through techniques of intration Intrating epressions using standard results Intrate simple functions on sight e d, 3 f d. Know the intrals 1 1 1, 1. Recognise and intrate epressions of the form f d f d. 3 April 013 version 1.0 8
Intrating by substitution Intrating proper rational functions Intrate where the substitution is given and 3 cos sin, cos d u 6sin d, u cos. 1 4cos Use the substitution at to intrate functions of the form a 1 1, a. Use partial fractions to intrate proper rational functions, where the denominator has two separate or repeated linear factors. Use partial fractions to intrate proper and improper rational functions where the denominator may have: (a) two separate or repeated linear factors (b) three linear factors with constant or non-constant numerator (c) a linear factor and an irreducible quadratic factor of the form a Intrating by parts Use one application of intration by parts sin d. Involve repeated applications of intration by parts 3 e d. Apply intration to the evaluation of areas including intration with respect to y. 1.4 Applying calculus skills to solving differential equations Solving first order differential equations with variables separable Solve equations that can be written in the form dy d g. h y Solving first order linear differential equations using the intrating factor Solve equations by writing linear equations dy a b y g d in the standard form dy P y f d and hence as d P d P d e y e f. d April 013 version 1.0 9
Find general solutions and solve initial value problems : miing problems, such as salt water entering a tank of clear water which is then draining at a given rate growth and decay problems, an alternative method of solution to separation of variables simple electronic circuits Solving second order differential equations Find the general solution of a second order homogeneous ordinary differential equation d y dy 0 a b cy d d with constant coefficients where the roots of the auiliary equation: (a) (b) (c) are real and distinct coincide (are equal) are comple conjugates Solve initial value problems for second order homogeneous ordinary differential equations with constant coefficients d y dy 4 13y 0 with y 1 d d and dy d when 0 Solve second order non-homogeneous ordinary equations with constant coefficients d y dy a b cy f d d using the auiliary equation and particular intral method. April 013 version 1.0 10
Mathematics (Advanced Higher) Applications in Algebra and Calculus 1.1 Applying algebraic skills to the binomial theorem and to comple numbers Epanding epressions Know and use the binomial theorem for rnn., n a b a b r0 r Epand 3 4 n n nr r Find the term in 5 in 3 7 Epand ( u - 3v) 5 Find the term in 7 in 9 Performing operations on comple numbers Perform algebraic operations on comple numbers: equality (equating real and imaginary parts), addition, subtraction, multiplication and division. Evaluating the modulus and argument Evaluate the modulus, argument and conjugate of comple numbers. Find the roots of a quartic when one comple root is given. Solve z i z 1. Solve z z. 1. Applying algebraic skills to sequences and series Finding the general term and summing arithmetic and geometric sequences Apply the rules on sequences and series to find the n th term, sum to n terms (partial sum), limit, sum to infinity (limit to infinity of the sequence of partial sums), common difference, and common ratio of arithmetic and geometric sequences. Using the Maclaurin series epansion to find a stated number of terms of the power series for a simple function Epand power series: e e sin e cos3 1.3 Applying algebraic skills to summation and mathematical proof Applying summation formulae Prove sums of certain series and other straightforward results: n r 1 ( ar b) a n r 1 r n r 1 b April 013 version 1.0 11
n r1 4 i1 n r1 n r1 1 r r n n n 3 1 1 3i 1 1 r( r 1)( r ) n( n 1)( n )( n 3) 4 rr ( ). Using proof by induction Use proof by mathematical induction in simple eamples: 1 + + +... + n = n +1 1, for all n N 8 n is a factor of (4n)! for all n N n < n for all n N 1.4 Applying algebraic and calculus skills to properties of functions Finding the asymptotes of rational functions Find the vertical asymptote of a rational function 4. 1 Find the non-vertical asymptote of a rational function 4. 1 Sketching the graph of a rational function including appropriate analysis of stationary points Sketch graphs of real rational functions using available information, derived from calculus and/or algebraic arguments, on zeros, asymptotes (vertical and non-vertical), critical points, symmetry. Sketch the graph of 4.. 1 NB For rational functions, the dree of the numerator will be less than or equal to three and the denominator will be less than or equal to two. Etrema of functions: the maimum and minimum values of a continuous function f defined on a closed interval [a,b] can occur at stationary points, end points or points where f ' is not defined: reflection in the line y = f() = e, < < f 1 () = ln, > 0 April 013 version 1.0 1
1.5 Applying algebraic and calculus skills to motion and optimisation Applying differentiation to rectilinear motion Find the acceleration of a particle whose displacement s metres from a certain point at time t seconds is given by s = 8 75t + t 3. Applying differentiation to optimisation April 013 version 1.0 13
Mathematics (Advanced Higher) Geometry, Proof and Systems of Equations 1.1 Applying algebraic skills to matrices and systems of equations Using Gaussian elimination to solve a 3 3 system of linear equations Performing matri operations of addition, subtraction and multiplication Calculating the determinant of a matri Finding the inverse of a matri Using matrices to carry out geometric transformations in the plane Find the solution to a system of equations A = b, where A is a 3 3 matri and where the solution is unique. Show that a system of equations is inconsistent, ie it has no solutions. Show that a system of equations is redundant, ie it has an infinite number of solutions. Compare the solutions of related systems of two equations in two unknowns and recognise ill-conditioning. Perform matri operations (at most order 3): addition, subtraction, multiplication. Know and apply the properties of matri addition and multiplication: A B B A AB BA (in general) A B C AB AC (distributivity) ABC ABC (associativity) Know and apply key properties of the transpose, determinant and inverse: A A A B A B AB BA 1 1 1 AB B A det AB det Adet B Find the determinant of a matri and a 3 3 matri without the aid of CAS. Find the inverse of a matri. Know the relationship of the determinant to invertibility. Use EROs to determine the inverse of a 3 3 matri. The transformations should include rotations, reflections and dilatations. April 013 version 1.0 14
1. Applying algebraic and geometric skills to vectors Calculating a vector product Finding the equation of a line in three dimensions Finding the equation of a plane Find a vector product in three dimensions. Evaluate a. b c. Find the equation of a line in parametric form, given a point on the line and a direction vector. Find the equation of a line in symmetric and vector form, given suitable defining information. Find the angle between two lines in three dimensions. Determine whether or not two lines intersect and, where possible, find the point of intersection. Find the equation of a plane in Cartesian form, given a normal to and a point in the plane. Find the equation of a plane in parametric and vector form, given suitable defining information. Find the point of intersection of a plane with a line which is not parallel to the plane. Determine the intersection of or 3 planes. Find the angle between a line and a plane or between planes. 1.3 Applying geometric skills to comple numbers Converting comple numbers from Cartesian to polar form Plotting a comple number on an Argand diagram Convert a given comple number from Cartesian to polar form and vice-versa. Know and use de Moivre s theorem with inter and fractional indices epand cos isin 4 Apply de Moivre s theorem to multiple angle trigonometric formulae epress sin 5 in terms of sin. Apply de Moivre s theorem to find the nth roots of unity solve 6 z 1 Plot comple numbers as points in the comple plane. Interpret geometrically certain equations or inequalities in the comple plane z i z, z a b 1.4 Applying algebraic skills to number theory Using Euclid s algorithm to find the greatest common divisor of two positive inters Using Euclid s algorithm to find the greatest common divisor of two positive inters ie use the division algorithm repeatedly. Epress the greatest common divisor (of two positive inters) as a linear combination of the two. Use the division algorithm to epress inters in bases other than ten. Know and use the Fundamental Theorem of Arithmetic. April 013 version 1.0 15
1.5 Applying algebraic and geometric skills to methods of proof Disproving a conjecture by providing a countereample Using indirect proof Disprove a conjecture by providing a counter-eample. For eample, for all real values of a and b, a b a b 0 0 A counter-eample is a 3, b 4. Know and be able to use the symbols (there eists) and (for all). Write down the nation of a statement. Prove an unconditional statement by contradiction irrational. is Prove an unconditional statement by contradiction if a and b are real then a b ab. Use further proof by contradiction. For eample, if, y R such that + y is irrational then at least one of,y is irrational; if m,n are inters and mn = 100, then either m 10 or n 10. Use proof by contrapositive. Reasoning skills.1 Interpreting a situation where mathematics can be used and identifying a valid straty. Eplaining a solution and relating it to contet Can be attached to any operational skills to require analysis of a situation. Can be attached to any operational skills to require eplanation of the solution given. April 013 version 1.0 16
Administrative information Published: April 013 (version 1.0) Superclass: to be advised History of changes to Course Assessment Specification Course details Version Description of change Authorised by Date Scottish Qualifications Authority 013 This specification may be reproduced in whole or in part for educational purposes provided that no profit is derived from reproduction and that, if reproduced in part, the source is acknowledged. Additional copies of this Unit can be downloaded from SQA s website at www.sqa.org.uk. Note: You are advised to check SQA s website (www.sqa.org.uk) to ensure you are using the most up-to-date version of the Course Specification. April 013 version 1.0 17