# Subject content

## 3.1 Algebra (USE1)

### 3.1 Algebra (USE1)

Note that Algebra is not a free-standing qualification in the pilot scheme and no separate FSMQ certificate is available for the unit outside AS and A level Use of Mathematics.
 Before you start this qualification You must be able to: This includes: plot by hand accurate graphs of paired variable data and linear and simple quadratic functions in all four quadrants quadratics of the type $y = ax^2 + bx + c$ recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions quadratics of the type $\space y = kx^2 + c$ fit linear functions to model data pairs calculating gradient and intercept for linear functions rearrange basic algebraic expressions by collecting like terms expanding brackets extracting common factors solve basic equations by exact methods pairs of linear simultaneous equations use power notation positive and negative integers and fractions solve quadratic equations by at least one of the following methods: use of a graphics calculator use of formula $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (which must be memorised) completing the square Solution by factorisation is also required where the quadratic factorises.

### Using calculators and computers

Using calculators and computers

When carrying out calculations, you may find the use of a standard scientific calculator sufficient.

You should learn to use your calculator effectively and efficiently. This will include learning to use:

• memory facilities
• function facilities (eg $\space \mathrm{e}^x, \space \mathrm{\sin} x, \space ...$ )

It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.

Whenever you use a calculator you should record your working as well as the result.

 You should learn to use a graphics calculator or graph plotting software (possibly a spreadsheet) on a computer to: This includes: plot graphs of paired variable data plot graphs of functions use function facilities $e^x, \space \sin x, \space \cos x, \space$etc . trace graphs (if possible) finding intersections of functions with other functions and axes use zoom facilities (if possible) finding significant features of functions such as turning points

### Fitting functions to data

 Fitting functions to data You should: This includes: be familiar with the graphs of quadratic functions of the form $y = ax^2 + bx + c$ $y = (rx - s)(x - t)$ $y = m(x+n)^2 + p$ knowing the general shape, orientation, position etc. of a given quadratic relating the shape and position of a graph of $y = m(x+n)^2 + p \space$ to $\space m, \space n \space$ and $\space p$ relating zeroes of a function $\mathrm f$$(x)$ to roots of the equation $\mathrm f$$(x)=0$ be familiar with the graphs of functions of powers of   $x$ $y = kx^{-2} = \frac{k}{x^2}; \space y = kx^{-1} = \frac{k}{x}; \space y = kx^{\frac{1}{2}} = k \sqrt{x}$ knowing the general shape, orientation and position of such a function be familiar with the graphs of trigonometric functions of the form: $y = A \sin (mx + c)$ $y = A \cos (mx + c)$ knowing the general shape and position of a given trigonometric function using correctly the terms amplitude, frequency and period be familiar with the graphs of exponential functions of the form $y = ka^{mx}$ and $\space y = k$$\mathrm e$$^{mx}$ (m positive or negative) understanding ideas of exponential growth and decay be familiar with graphs of natural logarithmic functions of the form $y = a \ln (bx)$ understanding the logarithmic function as the inverse of the exponential function understand the idea of inverse functions and be able to find graphically the inverse of a function for which you have a graph using reflection in the line $\space y = x$ have an understanding of how geometric transformations can be applied to basic functions. This understanding should assist you when fitting a function to data. Using translation of $\space y = \mbox{f}(x) \space$ by vector $\space \left [\begin{matrix} 0 \\ a \end{matrix} \right ]$ to give $\space y = \mbox{f}(x) + a$ (eg $\space y = \sin x, \space y = 4 + \sin x$) translation of $\space y = \mbox{f}(x) \space$ by vector $\space \left [\begin{matrix} a \\ 0 \end{matrix} \right ]$ to give $\space y = \mbox{f}(x+a)$ (eg $y = \mathrm {sin}x, y = \mathrm {sin}(x + 60^\circ)$) stretch of $\space y = \mbox{f}(x)$ scale factor a with invariant line $x = 0$ , to give $\space y = a\mbox{f}(x)$ (eg $\space y = \mbox{sin} x, y = 5 \mbox{sin} x$) stretch of $\space y = \mbox{f}(x) \space$ scale factor $a$ with an invariant line $\space y = 0$ to give $\space y = \mbox {f}(ax)$ (eg $\space y = \sin x, \space y = \sin 2x \space$ ) being able to describe geometric transformations fully be able to determine parameters of non-linear laws by plotting appropriate linear graphs Applications only in the two cases below $y = ax^2 + b \space$ (plotting $\space y$ against $\space x^2$) $y = ax^b \space$ and $\space y = a^x$ using natural logarithms

### Interpreting models

 Interpretingmodels You should learn to: This includes: understand how functions can be used to model real datathe limitations that a function may have when used to model data (e.g. being valid over a restricted range) find and use intercepts of functions with axes and other functions to make predictions about the real situation investigated find local maximum and minimum points and understand in terms of the real situation their physical significance calculate and understand gradient at a point on the graph of a function using tangents drawn by hand using the zoom and trace facilities of a graphics calculator or computer software if possible use and understand the correct units in which to measure rates of change interpret and understand gradients in terms of their physical significance identify trends of changing gradients and their significance both for functions that you know and curves drawn to fit data

### Using algebraic techniques

 Using algebraic techniques You should learn to: This includes: rearrange any quadratic function into the forms $y = ax^2 + bx + c$ $y = a(x + b)^2 + c$ quadratics expressed in the form $y = (ax+ b)(x + c)$ find maximum and minimum turning points of quadratics by completing the square i.e. expressing in the form $y = a(x + b)^2 + c$ solve polynomial equations of the form $ax^n = b$ solve trigonometric equations of the form: $A \sin (mx + c) = k$ $A \cos (mx + c) = k$ solve exponential equations of the form $\space A \exp (mx + c) = k$ understand how logarithms can be used to represent numbers using natural logarithms know and use the laws of logarithms $\log(ab) = \log a + \log b$ $\log( \frac{a}{b} ) = \log a - \log b$ $\log a^n = n \log a$ use logarithms to convert equations to logarithmic form for example $y = ka^{mx} \space$ gives $\log y = \log k + mx \log a$ use logarithms to solve equations $a^x = b \space$ using natural logarithms

## 3.2 FSMQ Data Analysis (9993)

### Data Analysis (9993)

 You should learn: Including: Statistical diagrams Box and whisker plotBack-to-back stem and leaf diagramHistogramCumulative frequency diagram Grouping of dataIdeas of symmetry, skew and multi-modal distributions. Measures of skewness are not required. Measures of location and spread Mean ( $\space \bar{x} \space)$ , median, modeUpper and lower quartilesPercentilesRange and inter-quartile rangeStandard deviation ( $\sigma_n \space$ and $\space \sigma_{n-1}$ )Outliers Comparing and contrasting data setsUsing a calculator to find $\space \bar{x}, \space \sigma_n \space$ and $\space \sigma_{n-1}$ Bivariate data Scatter diagramsIdeas of positive, negative and no correlationPearson's product moment correlation coefficient (r)Regression lines and the equation of the line of best fit Use of mean valuesUsing a calculator to find r and regression line coefficients. Interpretation of these resultsUnderstanding that correlation does not imply causationUnderstanding that r is only a measure of linear correlation Normal distribution Features of a normal distribution; to include continuous data, symmetry and 2/3rds and 95% rulesStandard normal distribution with mean 0 and standard deviation 1Use of tables to find probabilities and expected frequencies Understanding how a theoretical distribution can be a model for a real population

## 3.3 FSMQ Dynamics (9995)

### Prior Learning

 Prior learning Candidates will need knowledge of the following. Trigonometry: Use of Sin, Cos and Tan (but not the Sine or Cosine rules) Algebra: Collection of like terms and solution of linear equations such as Solution of a quadratic equation by at least one of the following methods: use of a graphics calculator use of formula (which must be memorised) completing the squareSolution by factorisation will be acceptable where the quadratic factorises.

### Formulae

Formulae   Candidates should learn the following formulae which may be required to answer questions.
Constant Acceleration Formulae
 $v^2 = u^2 + 2as$
Weight $W = mg$
Momentum Momentum = $mv$
Newton's Second Law $F=ma$ or Force = rate of change of momentum
Friction

No knowledge of calculus is required in this unit.

### Mathematical Modelling

 Mathematical Modelling Use of assumptions in simplifying reality. Candidates are expected to use mathematical models to solve problems by making assumptions to create a simple model of a real situation. Candidates are expected to use experimental or investigational methods to explore how the mathematical model used relates to the actual situation. Mathematical analysis of models. Modelling will include the appreciation that: it is appropriate at times to treat relatively large moving bodies as point masses; the friction law is experimental; the force of gravity can be assumed to be constant only under certain circumstances. Interpretation and validity of models. Candidates should be able to comment on the modelling assumptions made when using terms such as particle, light, inextensible string, smooth surface and motion under gravity. Candidates should be familiar with the use of the words; light, smooth, rough, inextensible, thin and uniform. Refinement and extension of models.

### Vectors

 Vectors Understanding of a vector; its magnitude and direction.Addition and subtraction of two vectors.Multiplication of a vector by a scalar.Addition and subtraction of quantities using vectors.Magnitude and direction of quantities represented by a vector.Candidates may work with the i, j notation or column vectors, but questions will be set using the column vector notation.

### Kinematics in One and Two Dimensions

Kinematics in One and Two Dimensions
Displacement, speed, velocity, acceleration. Understanding the difference between displacement and distance.
Understanding the difference between velocity and speed.
Sketching and interpreting kinematics graphs. Use of gradients and area under graphs to solve problems
The use of Calculus is NOT required for this unit.
Knowledge and use of constant acceleration equations.
 $v^2 = u^2 + 2as$
Application of vectors in two dimensions to represent position, velocity or acceleration, including the use of unit vectors i and j. Candidates may work with the i, j notation or column vectors, but questions will be set using the column vector notation.
Vertical motion under gravity.
Average speed and average velocity.
Magnitude and direction of quantities represented by a vector.
Finding position, velocity, speed and acceleration of a particle moving in two dimensions with constant acceleration.

The solution of problems such as when a particle is at a specified position or velocity, or finding position, velocity or acceleration at a specified time

Use of constant acceleration equations in vector form, for example,
$\mathbf{v} = \mathbf{u} + \mathbf{a}t$.

### Forces

 Forces Drawing force diagrams, identifying forces present and clearly labelling diagrams. Candidates should distinguish between forces and other quantities such as velocity, that they might show on a diagram. Force of gravity (Newton's Universal Law not required). The acceleration due to gravity , , will be taken as . Friction, limiting friction, coefficient of friction and the relationship of $F = \mu R$ Tensions in strings and rods. Knowledge that the resultant force is zero if a body is in equilibrium. Find the unknown forces on bodies that are at rest or moving with constant velocity. Candidates will not be expected to resolve forces or find the components of forces.Candidates will not be expected to use the triangle of forces.

### Momentum

 Momentum Concept of momentum Momentum as a vector in one or two dimensions. (Resolving velocities is not required.) Momentum = $mv$ The principle of conservation of momentum applied to two particles for direct impacts in one dimension. Knowledge of Newton's law of restitution is not required.

### Newtonâ€™s Laws of Motion

 Newton's Laws of Motion Newton's three laws of motion. Problems may be set in one or two dimensions and may include the use of vectors. Simple applications of the above to the linear motion of a particle of constant mass. Application of Newton's second law to particles moving with constant acceleration. Candidates will be expected to find the acceleration of a body if the forces acting are specified, or unknown forces if the acceleration is given. Use of as a model for dynamic friction.

### Projectiles

 Projectiles Motion of a particle moving freely under uniform gravity in a vertical plane. Candidates will be expected to state and use equations of the form and . Candidates should be aware of any assumptions they are making. Calculate range, time of flight and maximum height. Formulae for the range, time of flight and maximum height should not be quoted in examinations. Inclined plane and problems involving resistance will not be set. The use of the identity will not be required. Candidates may be expected to find initial speeds or angles of projection. Modification of equations to take account of the height of release.

## 3.4 FSMQ Mathematical Principles for Personal Finance (9996)

### Mathematical Principles for Personal Finance (9996)

The content of this unit covers three areas: the value of money over time, indices used to measure key financial information and tables and diagrams of financial information.

### The value of money over time

 The value of money varies over time. Imagine you were asked if you would like to be given a £1000 now or in ten years time. What would be your response? Even if you didn’t spend the money for ten years it would be better if you had the money now: you could invest it and it would be worth more at the end of the ten years. If, for example, you were able to invest it at 4% interest per year, after 10 years it would be worth £1480. Of course, in that period due to inflation, depending on what you spend the £1000 on it is likely to cost you more. However, some goods come down in price over time: this is often true, for example, for computer equipment. A question you need to consider then is, what is the cost of what you might want to buy likely to be at the end of the ten year period relative to what it costs now? Understanding how money varies over time is, therefore, a very important idea to consider when making all manner of financial decisions

#### What you need to learn

 Financial aspect Mathematical understanding This includes The key idea of present and future values present value, future value, Interest rates: AER calculating the annual effective interest, r, rate given a nominal interest rate, i $r = \Big( 1 + \frac{i}{n} \Big ) ^n - 1$ where $\space n \space$ is the number of compounding periods per year Calculating the future value of a present sum (using ideas of compound interest) $FV = PV(1+r)^n$ where $\space r \space$ is the interest rate expressed as a decimal and $\space n \space$ is the number of time periods understanding as a geometric series $a, \space ar^{2}, \space ar^{3}, \space ...ar^{\mbox{n}-1}, \space ar^{\mbox{n}}$ Use of recurrence relations* eg $P_{n + 1} = P_n(1 + r)$ Calculating the present value of a future sum $PV = \frac{FV}{(1+r)^n}$
*You should understand and be able to use recurrence relations in a range of financial situations, such as iteratively calculating the balance on a credit card, the balance remaining on an outstanding mortgage loan, the accumulating amount in a savings account when you make regular savings and so on.
 Continuous compounding understanding that the idea of continuous compounding leads to exponential functions ie considering the case where $P = P_0 \Big( 1 + \frac{r}{n} \Big ) ^{nt}$ is the amount after $\space t \space$ years for an initial investment of $\space P_0 \space$ when the interest is compounded $n$ times per year, and $\displaystyle \space n \rightarrow \infty \space$ giving $\space P = P_0e^{rt}$ APR (annual percentage rate) Assume no arrangement or exit fees. Use of the simplified version formula for APR in straightforward cases. ie $\mbox{C} = \displaystyle \sum_{k=1}^{m} \left(\frac{A_k}{(1+i)^{(t_k)}}\right)$ where $\space i \space$ is the APR expressed as a decimal, $\space k \space$ is the number identifying a particular instalment, $\space A_k \space$ is the amount of the instalment $\space k \mathrm{,}\space t_k \space$ is the interval in years between the payment of the instalment and the start of the loan. For simple cases only: for example, (i) for a single loan repaid in full after a fixed period in which case $\space \mbox{C} = \frac{A}{(1 + i)^n}$ where $\space n \space$ is the number of years between the advance of the loan and its repayment. (ii) for a loan repaid in a small number of instalments (eg 2,3 or 4). ie working with an equation of the form $\mbox{C} = \frac{A_1}{1 + i} + \frac{A_2}{(1+i)^2} + \frac{A_3}{(1+i)^3}+\frac{A_4}{(1+i)^4}$ In this case you will be expected to either substitute values into the resulting equation for confirmation, or solve for $\space i \space$ using the interval bisection method Applications to financial areas such as: loans credit cards motgages savings Personal Taxation Complex calculations involving multiple rates To include income tax, national insurance and value added tax. Capital gains tax, including the effect of indexation on the taxable gain.

### Indices used to measure key financial information

 When you make a financial decision you need to have measures available that allow you to make sense of data. For example, as you found in section 1 when considering how the value of money varies over time, it is useful, if you are considering borrowing money and investigating which loan you should take that you make sure you have details of the APR (annual percentage rate) for each possibility so that you can compare like with like. In this section you will learn how indices such as the retail price index and the FTSE 100 share index are developed so that you can quickly understand financial information such as how the cost of living is varying or how share prices are increasing or decreasing

### What you need to learn

 Financial aspect Mathematical understanding This includes Understanding of an index as a ratio that describes the relative change in a variable (eg price) compared to a certain base period (eg one specific year). As applied in particular to measures of inflation such as the Retail Price Index (RPI), Consumer Price Index (CPI) and Average Earning Index. The index at any time tells you what percentage the variable is of its respective value at the base time. The value of the index at the base time is 100. Calculations using measures of inflation, including annual changes to pensions and tax allowances. Calculating contributions made by individual items to indices, eg calculating contributions made by the prices of commodities in different shops and regions to a consumer price index. Weighted averages for example, carrying out calculations such as finding the effective costs of a commodity which varies in price between shops. Eg the commodity costs £5 in shop A and £6 in shop B. 0.4 of customers buy the commodity from shop A whereas 0.6 buy it from shop B. The effective cost of the commodity to be used in calculating an index is given by 0.4 × £5 + 0.6 × £6 = £5.60 Understanding the idea of calculating a composite index by combining indices using weighting eg in calculating a price index the index of each commodity multiplied by its weighting is totalled and this sum is divided by the sum of all the commodities weights. Calculating and using different index formulae For the indices below the following apply: is the price of commodity at time is the quantity of commodity at time represents the base period so for example represents the quantity of commodity at the base period Laspeyres index formula (weighted by quantities in the base period) Paasche index formula (weighted by quantities in the calculation period) Fischer index formula The geometric mean of the Laspeyres index formula and the Paasche index formula using indices to understand change understanding ideas of fixed base indices and chain indices fixed base index chain index understanding that for a fixed-base index quantities at time t are compared with the base period ( t = 0 ) understanding that in a chain index comparisons are always made between subsequent points and therefore take account of changes between the start and end points. percentage change calculating average changes using the arithmetic mean eg quarterly change figures from monthly figures understanding and working with basic principles as well as applying to areas of finance such as: FTSE 100 share index
inflation including: consumer price index, retail price index

### Making sense of data over time

 Data you may want to use to make financial decisions is often presented as time-series data, that is a particular measure is given every month, quarter or annually. Sometimes, particularly when the data fluctuates a lot, this may have been processed so that you can identify trends over time. For example, share prices can fluctuate from day to day, as can the FTSE 100 share index. To understand the underlying trend over time it useful to average the data before considering this. Other issues you may need to consider include seasonal variation and cyclical patterns.

#### What you need to learn

 Financial aspect Mathematical understanding This includes Considering data, either primary or secondary (such as financial indices), over time. Interpreting trends. The types of data considered should relate to data or indicators likely to be met when making personal financial decisions: for example, prices of stocks and shares (including 100 share index), interest rates, exchange rates and so on. Time series data in unprocessed form and understanding variability and how this may be random, seasonal or cyclical in nature. Representation graphically and identifying linear trends Inspection of data tables and graphs. Data over different time intervals, for example daily, weekly, quarterly etc. Finding linear equations to model data using gradient and intercept and algebraic substitution. To smooth short-term fluctuations, time-series data can be averaged so that longer-term trends can be identified. For example, indices such as the 100 share index can fluctuate from day-to-day, but over a month or two there may be a distinct trend. Seasonal and cyclical variations may also be more easily detected by such smoothing. Moving averages: for data points $\space p_1, \space p_2, \space ... \space$ the simple moving average, $\space x_m \space$ at interval m takes account of n data points $\displaystyle x_m = \frac{p_m \space + \space p_{m-1} \space + \space p_{m-2}\space + .... \space p_{m(n-1)}}{n}$ Calculating successive values of the simple moving average using $\displaystyle \bar{x}_{m+1} = \bar{x}_m - \frac{P_{m-(n - 1)}}{n} + \frac{P_{m + 1}}{n}$ Interpretation of situations which may include seasonal and cyclical variations. Understanding that the simple weighted average based on a relatively large number of data points can be considered to lag behind the trend of the data The problem of lag in moving averages can be addressed using weighting. The linear weighted moving average (over n intervals) weights the current data with weight n, the previous day with weight (n – 1) and so on. $\displaystyle x_m = \frac{np_m + (n-1) \space p_{m-1} + (n-2) \space p_{m-2}\space + .... \space p_{m - (n-1)}}{ n + (n-1) + (n-2) + .... + 2 + 1 }$ recognising the denominator as a triangular number with sum $\displaystyle \frac{n(n+1)}{2}$

### Tables and diagrams of financial information

Â
 Much basic financial information is presented in as simple a form as possible, for example using indices such as the Retail Price Index and FTSE 100 index which you learned about in section 2. Other information is often quoted in tabular or diagrammatic form giving simplified data and measures so that you can quickly compare like with like. You will have met such ideas in other walks of life; for example, you are probably aware of school performance tables and how attempts have been made to look for measures of "value added" in pupil performance rather than taking raw scores that don't allow for the ability of pupils at entry to different schools. In this section you will learn how to make sense of a range of information presented in tables and diagram relating to personal finance. For example, you will learn to interpret information about how an investment might perform or how to compare financial products.It is not the intention that you should learn specific financial measures other than those highlighted in previous sections but that you should be able to work with and interpret financial information presented in tables and diagrams when basic terms are defined.

#### You need to learn:

• to be able to extract and understand data from tables and diagrams
• to work with the data carrying out calculations using basic mathematics, such as calculating with percentages
• to interpret the original data and results of your calculations in terms of the financial situation
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## 3.5 FSMQ Hypothesis Testing (9994)

### Hypothesis Testing (9994)

 You should learn: Including: Binomial Distribution Multiplication law for probabilities of independent events Addition law for probabilities of exclusive events Tree Diagram For independent events only Binomial probabilities notation Normal approximations and the standard normal distribution. Use of normal distribution tables Sampling Sampling from a parent population Precision and sample size The knowledge that improving accuracy by removing bias and increasing sample size can cost both time and money Distribution of the sample mean and sample proportion from a normal population Confidence intervals Hypothesis Testing The null and alternative hypotheses One-tail and two-tail tests With reference to the binomial and normal distributions only. Significance level and critical region Specific tests to include Opinion Polls Food Tasting Triangle test and pairs-difference test Clinical Trials Placebo and double-blind trials Non-Parametric Tests The sign test The Mann-Whitney U test

## 3.6 FSMQ Decision Mathematics (9997)

### Decision Mathematics (9997)

 What you need to learn Throughout your work you need to develop a critical and questioning approach to your own use of decision mathematics diagrams and techniques and also learn how these can be used to draw conclusions and summarise findings. You will carry out work that involves you in:selecting appropriate data to usedrawing appropriate network(s)carrying out an analysis using an algorithmic approachdrawing conclusions and summarising findings.The key ideas that you will meet and some specific techniques that you need to be able to use are set out below. Using networks to model real world situations You should be able to represent a situation so that some of the relationships are clarified by the use of appropriate networks.In drawing networks you should consider and understand:terminology such as vertices, edges, edge weights, paths and cyclesconnectednessdirected and undirected edges and graphsYou should:be able to store graphs as matrices e.g. adjacency/distance matricesunderstand the degree of a vertex and be aware of odd and even vertices
Trees and spanning treesYou should understand that a tree is a connected graph with no cycles and that every connected graph contains at least one tree connecting all the vertices of the original graph.

 In your study of trees you should: This includes: understand the idea of a minimum connector (a spanning tree of minimum length) finding minimum connectors using Prim's and Kruskal's algorithms You will be expected to apply these algorithms in graphical and, for Prim's algorithms, also in tabular form understand when a situation requires a minimum spanning tree to be found commenting on the appropriateness of a solution in its context appreciate the relative advantages of Prim's and Kruskal's algorithms
Shortest PathsIn developing ideas about shortest paths you will need to appreciate that problems of finding paths of minimum time and cost can both be considered to be shortest path problems

 In developing ideas about shortest paths you should: This includes: be able to apply Dijkstra's algorithm using a labelling technique to identify the shortest pathcommenting on the appropriateness of a solution in its context
Route Inspection ProblemIn developing ideas about route inspection you will need to appreciate the connection with the classical problem of finding an Eulerian trail.

 In developing ideas about route inspection you should: This includes: understand the significance of odd vertices problems with 0, 2 or 4 odd vertices be able to apply the Chinese Postman algorithm commenting on the appropriateness of a solution in its context
Travelling Salesperson ProblemIn developing ideas about the Travelling Salesperson Problem you will need to appreciate the connection with the classical problem of finding a Hamiltonian cycle.

 In developing ideas about the Travelling Salesperson Problem you should: This includes: be able to determine upper bounds by using the nearest neighbour algorithm converting a practical problem into the classical problem be able to determine lower bounds finding the length of a minimum spanning tree for a network formed by deleting a given node and then adding the two shortest distances to the given node. appreciate when a solution is sufficiently good realising that a solution is not necessarily the bestcommenting on the appropriateness of a solution in its context
Critical Path AnalysisIn developing ideas about Critical Path Analysis you will need to understand both how to construct and how to interpret activity networks with vertices representing activities.

 In developing ideas about Critical Path Analysis you should: This includes: be able to find earliest and latest times using forward and reverse passes be able to identify critical activities and find a critical path the calculation of floats know how to construct and interpret cascade diagrams
 Mathematical modelling You should be able to apply mathematical modelling to situation relating to the topics covered in this module. You will need to interpret results in contexts.
 Using calculators and computers The use of a standard scientific calculator is sufficient for this unit. However, software for the construction of networks or for the carrying out of algorithms is available commercially.

## 3.7 FSMQ Calculus (9998)

### Calculus (9998)

This qualification has been developed to allow you to demonstrate your ability to use

• differentiation
• integration
• differential equations

to analyse, make sense of and describe real world situations and to solve problems. You will also investigate the use of numerical methods to find gradients and evaluate integrals and compare these with analytic methods.

 Before you start this qualification You must: This includes: be able to use algebraic methods to rearrange and solve linear and quadratic equations Solution of a quadratic equation by at least one of the following methods: use of a graphics calculatoruse of formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$(which must be memorised)completing the squareSolution by factorisation will be acceptable where the quadratic factorises. have knowledge of basic functions and how geometric transformations can be applied to them using transformations by the vector $\left [ \begin{matrix} a \\ 0 \end{matrix} \right ] \space$ and by the vector $\space \left [ \begin{matrix} 0 \\ a \end{matrix} \right ]$stretches of scale factor $a$ with the invariant line $\space x = 0 \space$ and with the invariant line $\space y = 0$ being familiar with graphs and functions of: powers of $\space x$ , eg $\space y = kx^{-2}; \space y = kx^{-1};$$\space y = kx^{\frac{1}{2}}; \space y = kx^{3}; \space y = kx^4$quadratics: $y = ax^2 + bx + c$$y = (ax - b)(x - c)$trigonometric functions: $y = A \sin (mx + c)$$y = A \cos (mx + c)$exponential functions: $y = k\mbox {e}^{mx}$( $m \space$ positive or negative)logarithmic functions: $y = a \ln (bx)$

### Using calculators and computers

Using calculators and computers

When carrying out calculations, you may find the use of a standard scientific calculator sufficient.

You should learn to use your calculator effectively and efficiently. This will include learning to use:

• memory facilities
• function facilities (e.g., $\mbox {e}^x, \space \sin(x), \space ...$ )

It is important that you are also able to carry out certain calculations without using a calculator, using both written methods and 'mental' techniques.

Whenever you use a calculator you should record your working as well as the result.

### Understanding and using differentiation

 Understandingandusingdifferentiation You should learn to: This includes: understand and calculate gradient at a point, $\space a \space$ , on a function $\space y = \mbox {f}(x) \space$ using the numerical approximation: $\text{gradient} \approx \frac{\mbox {f}(a + h) - \mbox {f}(a)}{h}$where $\space h$ is small understanding how to improve the calculation of gradient at a point by using a smaller interval, $\space h.$ understand and interpret gradients in terms of their physical significance understand and use the correct units with which to measure gradients /rates of change sketch graphs of gradient functions curves that you do not know as functionscurves defined as functions identify the key features of gradient functions in terms of the gradient of the original function zeros of gradient functions linking to local turning points understand how $\space \frac{\mbox {f}(x+h) - \mbox {f}(x)}{h} \space$ can be used to generate a gradient function differentiate functions using notations $\space \frac{\mbox {d}y}{\mbox {d}x} \space$ and $\space \mbox {f}'(x) \space$polynomialstrigonometric functions using radiansexponential functions Differentiate sums and differences of functionsfunctions multiplied by a constantproducts of functions find the second derivatives of functions using notations $\space \frac{\mbox {d}^2y}{\mbox {d}x^2} \space$ and $\space \mbox {f}''(x)$ identify the key features of a second derivative linking positive values to increasing gradientlinking negative values to decreasing gradientlinking zero values to points of inflexion Applications of differentiation to gradients, maxima and minima and stationary points, increasing and decreasing functions Application to determining maxima and minimaunderstanding the importance of the second derivative and its value at such pointsunderstanding that zero values of the second derivative can occur at maximum and minimum points as well as points of inflexion

### Understanding and using integration

 Understandingandusingintegration You should learn to: This includes: estimate areas under graphs of functions using numerical methods the trapezium ruleunderstanding how to improve your calculation of the area under a graph by using a smaller interval. understand and find areas under curves, between $\space x = a \space$ and $\space x = b \space$ using $\space \int_a^b \mbox {f}(x)\mbox {d}x \space$ , $\space (\mbox {f}(x) \geq 0) \space$ understand integration as the reverse process of differentiation understand and determine indefinite integrals of functions $x^n \space$ (including $\space n = -1 \space$ and fractional)$A \sin (mx+c) \space$$A \cos (mx+c) \space$$k\mbox {e}^{mx} \space$ ( $m$ positive or negative)sums, differences and constant multiples of these using a constant of integration Integration by inspection and by one use of integration by parts eg: $\int e^{-5x}dx$, $\int \mbox {sin}6xdx$  eg: $\int xe^{-5x}dx$,$\int x\mbox {cos}4xdx$, $\int x\mbox {ln}xdx$ understand the idea of constant of integration and be able to calculate this in known situations be able to determine definite integrals for functions those functions defined above

### Understanding and using differential equations

 Understandingandusingdifferentialequations You should learn to: This includes: find families of solutions to first order differential equations with separable variables find particular solutions when boundary conditions are given

## 3.8 Mathematical Applications (USE2)

### Mathematical Applications (USE2)

Before you start this qualification You must:
Have completed or be following courses for two of the following units:
 Algebra USE1 Hypothesis Testing 9994 Data Analysis 9993 Decision Mathematics 9997 Dynamics 9995 Calculus 9998 Mathematical Principles for Personal Finance 9996
Portfolio requirements Candidates are required to produce two pieces of work in their Coursework Portfolios.
• Each piece must be one coherent task with a consistent theme.
• The two pieces must be based upon the work of two different units from the above list. There can be some overlap between the two pieces but what is outside the overlap must be significant.
• Work going beyond that of a specific unit is, of course, acceptable but each piece can score full marks without the candidate working beyond the specification.
• A substantial proportion of the work for each piece must be on work beyond Higher Level GCSE Mathematics.
Assessing the Coursework Portfolio The two tasks must be marked separately (using the same grid) and the two marks totalled to produce one final mark for the unit.

For each of the two tasks, the candidate will be given a mark, from 0 to 7, for each of three themes:

• Structuring and presenting work
• Using appropriate mathematics (and technology) and working accurately
• Interpreting mathematics

The marking grid gives a description under each of these themes for work at various marks.

Structuring and presenting work Using appropriate mathematics (and technology) and working accurately Interpreting mathematics

### 1

substantial omissions and is poorly presented.
There is little evidence of using
mathematics accurately at the appropriate level.
There is little evidence of
relating mathematics to the situation(s) investigated or there are substantial errors in interpretation.

### 3

The portfolio task has been completed with only a little advice and is well presented so that it is easy to follow.A significant proportion of the work is beyond GCSE and is substantially correct.The candidate has interpreted the main mathematical findings in terms of the situation(s) investigated.

### 5

The candidate has worked independently and produced a portfolio task that is well-structured and reported with clarity.A significant proportion of the work is beyond Higher Level GCSE and is substantially correct, using relevant mathematical techniques and ICT where appropriateThe candidate has used mathematics to correctly summarise and draw conclusions about the situation(s) investigated.

### 7

The candidate has shown initiative in developing their portfolio task and has structured it logically and has reported their work fluently.The candidate has used appropriate, efficient and concise methods of working.The candidate has considered, how their initial data, and assumptions where appropriate, affect their findings.
 Evidence to Support the Award of Marks Portfolios must be presented in a clear and helpful way for the moderator. An indication must also be given at the appropriate point in the work, or in accompanying information, of any further guidance given by the teacher (or other person) which has significant assessment implications. When the assessment of the portfolio is complete, the mark awarded for each theme and the total mark awarded must be entered on the Candidate Record Form, with supporting information given in the spaces provided.

## 3.9 Mathematical Comprehension (USE3)

### Mathematical Comprehension (USE3)

In their study of this unit candidates will build on the mathematical knowledge, skills and understanding they develop in working towards the compulsory units Algebra (studied as part of AS Use of Mathematics) and FSMQ Calculus. This unit can be studied alongside these other units.

In particular, this assessment will concentrate on:

• reading and making sense of the mathematics of other people
• the processes involved when mathematics is used to solve problems developing clarity in the communication of mathematics
 Written Examination The comprehension paper will be one and a half hours in duration. It encourages candidates to communicate their results, working and reasoning clearly by allowing time for them to present their solutions with care. The paper will have 45 marks allocated in the ratio 2:1 between sections A and B. The questions of Section A will be based in contexts using data from a Data Sheet that will be made available to candidates up to 14 days, and at least 7 days, prior to the examination. Candidates will be expected to familiarise themselves with the contexts outlined in the Data Sheet and should discuss these with others, including teachers.Section B will require candidates to read a brief unseen article, describing the application of mathematical principles in context, upon which a number of questions will be based.Both sections will include questions that test the mathematical principles developed in the AS unit Algebra and the FSMQ unit Calculus together with the mathematical processes and principles set out later in section 3.9.Candidates will be required to answer all questions.The use of a graphics calculator will be expected.Candidates will be expected to remember all appropriate formulae as no formulae sheet will be provided.
Prior learningThis unit extends and develops the mathematics of the Algebra and Calculus units. You are therefore expected to be familiar with the content as outlined in their specifications. It is possible to study for this unit either alongside one or both of these units or to complete your studies for them prior to starting study of this unit.

 You should be able to use a graphics calculator to: This includes: plot graphs of paired variable data plot graphs of functions use function facilities $\mathrm{e}^x, \space \sin x, \space$ etc. trace graphs finding intersections of functions with other functions and axes use zoom facilities (if possible) finding significant features of functions such as turning points
You should be able to sketch graphs you have plotted with your graphics calculator showing clearly all significant features such as intercepts with axes, turning points and asymptotes.

### What you need to learn

 Whatyou need to learn This unit emphasises 'process' skills - the skills you need to develop to understand mathematical models of real situations.These skills are identified and described in the five learning objectives below.

### Learning Objectives

 Learning Objectives The five Learning Objectives define the skills, techniques and understanding that you need to develop during your study of this unit. You will learn to: LO1 understand how mathematics can be used to model different situations LO2 extend, develop and use a range of knowledge, skills and understanding in the areas of algebra, graphs and calculus when modelling situations LO3 appreciate that general mathematical principles may be applicable in a range of different contexts LO4 make sense of mathematics LO5 work accurately, structure mathematical arguments carefully and communicate mathematics clearly

### LO1 Understanding how mathematics can be used to model different situations

 LO1 Understanding how mathematics can be used to model different situations Throughout the course leading to this qualification you will use mathematical models of a range of real life contexts. It is important that you appreciate how these models relate to the situation and are aware of their limitations. You should: Notes / Examples LO1a appreciate the main stages involved in developing a mathematical model of a real situation you may find a diagram useful in assisting your understanding of the stages of the process for example: LO1b understand that simplifying assumptions will be made when a mathematical model is developed and that these may introduce limitations into the usefulness of the model for example: when modelling the growth of savings in a bank account a simplifying assumption may be that interest is added twice a year. In fact interest is often calculated daily, which can have the effect of increasing or decreasing the amount of interest added to the account over a periodwhen using numerical integration methods, such as the trapezium rule, you effectively assume that the variable follows a mathematical rule between the known data points (for example, a linear rule in the case of the trapezium rule). Consequently this can lead to an under or over estimate of the true value. LO1c be able to interpret the main features of mathematical models in terms of the real situations which they model and use these models to predict what will happen in situations for which you have no data for example: when using direct proportion to model the relationship between two variables understand that when one variable is zero so is the otherwhen using a function to model a situation you may use it to predict long term behaviour - perhaps by using linear extrapolationwhen using a differential equation to model a situation you can find the rate of change of one variable with respect to another without having access to data about this. LO1d consider the validity of mathematical models for example: by considering the predicted behaviour in extreme $\space (t \rightarrow \infty ) \space$ or simple $\space (t = 0 ) \space$ cases LO1e appreciate that a general mathematical model allows you to solve a variety of related problems for example: the model $\space a = Bc^t \space$ gives the rate of radioactive decay of all radioactive substances where the coefficients $B$ and $c$ depend on the substancea situation that can be modelled by a differential equation of the form $\space \frac{\mathrm{d}y}{\mathrm{d}x} = ky \space$ has ageneral solution $\space y = Ae^{kt} \space$ and consequently exhibits features of exponential growth or decay.

### LO2 Extending, developing and using a range of numerical, algebraic, graphical and calculus techniques when modelling situations

 LO2 Extending, developing and using a range of numerical, algebraic, graphical and calculus techniques when modelling situations LO2a You should learn to: Notes/Examples appreciate when algebraic, graphical, numerical or calculus techniques (or combinations of these) are most appropriate, when they are inappropriate and when possibly unsound for example when finding the solutions of $\space t^2 - t -6 = 0 \space$ the algebraic method of factorising, i.e. $t^2-t-6 = (t-3)(t+2)=0$$\Rightarrow \space t = 3 \space$ or $t = -2 \space$ is efficient and accurate giving all the solutionswhen finding distance travelled by computing the area under a linear speed-time graph calculating the area of a trapezium is appropriate and efficient, negating the use of more complicated methods involving calculus. establish links between different ways of understanding mathematical ideas you may use a graphics calculator and its trace facility to determine where $\space t^3 - t + 1 = 0.\space$You will not be able to solve this algebraically and should be aware that any solution of this equation found using a graphics calculator is not exactyou should avoid looking for solutions to equations by substituting numerical valuesyou should be aware that substitution is a valid method of checking the validity of a solutionyou should be aware that finding the gradient at many distinct points on a function allows you to plot a graph of the "gradient function" and you should understand how this is related to the first derivative of the function and what this informs you about the context that the function models.you should be aware that the gradient at a single point on a function can be estimated by calculating the gradient of a line segment between this point and another close to it and how this might inform "differentiation from first principles". LO2b find solutions to equationsshow that particular solutions to equations are validknow when each of these methods is appropriate you should be able to:use algebraic and calculus methods and techniques to find solutionssubstitute numerical values or algebraic expressions into equations to show or verify that solutions are validuse graphs to visualise solutions LO2c use algebra to solve systems of: two linear equationsone linear and one quadratic equation you should: be aware that in general a system of n equations is needed to find n unknownshave a graphical understanding of when systems of equations have: one or more solutionsno unique solutionno solutionunderstand that every equation in a system of equations is satisfied simultaneously by each solutionsubstitute solutions you find into all equations to check their validity LO2d be able to solve linear inequalities you should be able to: substitute numerical values into linear expressions to determine whether or not inequalities holdsolve equations algebraically to assist in the solution of inequalitiespresent your solutions graphically: using dashed lines to indicate boundaries not includedusing full lines to indicate boundaries includedusing shading to indicate regions not included LO2e use algebraic, graphical and calculus techniques to investigate continuous models in a wide variety of contexts such as population modelling, drug absorption, traffic flow and so on. This includes the use of any of the functions specified in the AS unit Algebra and any of the methods and techniques specified in this unit and in the FSMQ Calculus.

### LO3 Appreciating that general mathematical principles may be applicable in a range of different contexts

 LO3 Appreciating that general mathematical principles may be applicable in a range of different contexts If you are to appreciate the power of mathematics as a tool that can be used to analyse a wide range of different situations, you need to be able to extract and appreciate general mathematical principles that underpin the work that you have been doing.
 You should learn to: Notes/Examples LO3a be able to identify the use of particular ideas in mathematics across a range of situations or contexts you should aim to be able to transfer your use of mathematics into your work in other areas, for example you should: be familiar with using a range of different notations for common mathematical models (eg $\space s = t^2 + 2t + 1,$$\mathrm{g}(c) = (c + 1)^2, \space N = m(m + 1) \space$ are all quadratic functions and have the usual properties of such functions)understand that numerical and analytic methods of integration can be used to find the area under the graph of a function and that in some situations this may have physical significance.able to use algebraic and graphical models of direct proportion in different situations such as problems involving motion at constant speed (uniform motion) and those involving scaling lengths in plans and maps and be able to recognise the principles that underpin and are applicable in both situationsable to identify the use of quadratic models in a range of situations using different notations such as those above and then be able to use mathematical methods to assist your analysis of these situationsaware that differentiating any continuous function (eg quadratic, trigonometric, exponential and combinations of these) gives information about the rate of change of a variableaware of how the nature of the second derivative of a function can be used to inform you of the nature of significant features (such as turning points) of the original function. LO3b understand when mathematical methods will lead to solutions for example: when you have three unknowns that you want to find, you need to have three equations involving these three unknownsbeing aware of when it will be possible to integrate a function using analytical techniques or whether you will need to resort to numerical techniquesappreciating that when you have a general solution to a differential equation you will be able to find a particular solution if you know boundary conditions.understanding the problems that discontinuities in functions pose for calculus methods. LO3c have an appreciation of general features of graphs of functions, including intercepts with axes and asymptotes including understanding: that the values of $\space x \space$ where the graph of $\space y = \mathrm{f}(x) \space$ crosses the x-axis are solutions of the equation $\space \mathrm{f}(x) = 0 \space$the x-coordinates of the points of intersection of the graphs $\space y = \mathrm{f}(x) \space$ and $\space y = \mathrm{g}(x) \space$ are solutions of the equation $\space \mathrm{f}(x) = \mathrm{g}(x) \space$that when $\space \mathrm{f}(t) \space$ is continuous and $\space \mathrm{f}(a) \space$ is of a different sign to $\space \mathrm{f}(b) \space$ there is at least one solution of the equation $\space \mathrm{f}(t) = 0 \space$ between $\space t = a \space$ and $\space t = b \space$of maximum and minimum points and points of inflexion, and being able to interpret these in terms of the situation that the function models.the nature of discontinuities in functions of the form$\space \mathrm{f}(x) = \frac{k}{x}, \space \mathrm{g}(x) = \frac{k}{x - a}\space$limiting values of functions of the form $\space P(t) = A\mathrm{e}^{kt}, \space g(x) = K - A\mathrm{e}^{kx}$the nature of horizontal asymptotesthat vertical asymptotes may be incorrectly displayed by graphics calculatorsof how first and second derivatives of a function can inform you about important characteristics of key features of functions (eg whether a turning point is a maximum or minimum). LO3d develop your understanding of how geometric transformations can be applied to the graphs of basic functions and be able to use these when working with graphs that model real situations including:the transformations of graphs of $\space y = \mathrm{f}(x) \space$ as defined in the AS unit Algebrahaving an appreciation of the symmetries of functions such as quadratic and trigonometric functions and being able to use such properties when working with models based on such functions

### LO4 Making sense of mathematics

 LO4 Making sense of mathematics Making sense of mathematics is an important skill to develop since at some stage in your studies or future work you can expect that you may need to do this in a range of situations. When you come to explore the mathematics developed by someone else, particularly in workplace situations, you may find that it looks unfamiliar. Although the work will often be based on the same mathematical principles that you know, the context or setting of the mathematics and perhaps its development over time may result in it looking different.
 You should: Notes / Examples LO4a learn to explain steps in mathematical working by developing sub-steps if necessary for example, you should be able to set out the mathematical working that explains the statement. LO4b learn to relate mathematics in new situations to mathematics in situations with which you are familiar for example, when calculating the possible percentage error due to rounding to the nearest year in a population model such as $\space P= A\mathrm{e}^{kt}$ ,   you may relate this to earlier experience of calculating percentage errors in finding the area of a rectangle when you rounded lengths to the nearest centimetre LO4c develop strategies to assist you in making sense of mathematics for example, this could involve you in considering: extremes (eg in modelling a population using $P = 100 - 30$  x  $0.8^t$ ,   considering what happens as   $t \rightarrow \infty )$ boundary conditions (e.g. when   $t = 0$ the initial population predicted by the model $P = 100 - 30$   x   $0.8^t$   is $P = 100- 30 = 70$ ) simple values (e.g. a financial measure 'debtor days' is given by $\frac {outstanding \space debt}{annual \space turnover}$ x 365 ) - you may make sense of this by substituting 'simple' values of 1 million for outstanding debt and 2 million for annual turnover a situation that can be modelled by a function   $y = \mathrm{f}(t)$   for   $t > 0$ ,   such that $\mathrm{f}' (t) = t^2$ is such that the gradient is always positive and consequently the function is always increasing.

### LO5 Working accurately, structuring mathematical arguments carefully and communicating mathematics clearly

 LO5 Working accurately, structuring mathematical arguments carefully and communicating mathematics clearly When applying mathematics to analyse a situation or solve a problem you will need to work accurately and structure your work carefully. You should aim to present your mathematics so that the logic of your thinking and argument can be followed easily by others.
 You should learn to: Notes / Examples LO5a identify possible errors in your working by: For example: questioning whether your solutions are reasonable and / or valid you could consider the magnitude of solutions (e.g. when modelling population growth of a country if you calculate that the population will increase ten-fold in ten years consider whether this is reasonable)you could consider the dimensions of values you are calculating (e.g. $\space k \times l \times h$ has the dimensions of area if $\space k \space$ is a dimensionless constant and both $\space l \space$ and $\space h \space$ are lengths)if you have used definite integration to calculate the area under the graph of a function and the resulting value is negative, but it is clear from inspection of the graph that the area is positive you should be aware that you have made an error.if you have found solutions to simultaneous equations check that these are correct by substituting the values you have found back into the equations using checking techniques wherever possible using differentiation to check your solution to an indefinite integral LO5b decide on the appropriate degree of accuracy of values you calculate for example: when using a calculator to find the area of a circle of radius 10 cm (correct to the nearest cm) your calculator will use $\space \pi = \space$ 3.14159265. You should not quote your final answer to the accuracy of the value on your calculator displaywhen using values read from graphs think carefully about the accuracy of any values that you subsequently calculate LO5c ensure that you write clear and unambiguous mathematical statements this includes correct use of brackets your mathematics should read correctly (as an English sentence), e.g. when solving$10t - 5t^2 = 0 \space$ write:$5t(2 - t) = 0 \Rightarrow t = 0 \space$ or   $t = 2$NOT: $10t - 5t^2 = 5t(2 - t) = 0,2$include the constant of integration, e.g. $s = \int u - gt\space \mathrm{d}t$$= ut - \frac{gt^2}{2} + c$subsequently you may establish the value of $c$ by substituting known values into the solution. use notation correctly notation you use should include: therefore, $\space \therefore$equals, $\space =$approximately equals, $\space \approx$inequalities, $\space <, \space >, \space \leq, \space \geq$implies, $\space \Rightarrow$ LO5d use algebraic, graphical, numerical and calculus techniques effectively to communicate your mathematics for example: when working with a population that may be modelled by the function$P = 100 - 30 \times 0.8^t$ you could:most effectively show how this varies with time by plotting or sketching a graph of $\space P \space$ against $\space t \space$explore the long term behaviour predicted by this population model by substituting a large value for $\space t \space$ and note that $\space P \space$ is approximately 100when exploring how the length of daylight hours varies throughout the year you might differentiate a function used to model this (such as $T = 12 - 4 \cos \Big ( \frac{n \pi}{180} \Big )\space$ , where $T$ is the number of daylight hours, n is the number of days after January 1st) to show how the rate of change will be a maximum when $\space \frac{\mathrm{d}^2 T}{\mathrm{d}n^2} \space$ , that is when$\frac{\pi^2}{45 \times 180} \cos \Big( \frac{n \pi}{180} \Big) = 0 \space$, and therefore n = 90 and n = 270. Interpreting this in terms of the situation implies that 90 and 270 days after January 1st the rate of change of daylight hours is a maximum (sunrise and sunset are getting further apart most quickly).