Edexcel A level Maths: grade by grade
Every skill, from the first marks to the top grade. For each grade you also need the skills for the grades below it. Tick them off in the app's Notes section.
Grade E
- Disprove a statement with a counter-example Find one value that makes the statement false, show the working and say that it disproves the statement. Proof by deduction, exhaustion and counter-example
- Use the multiplication, division and power laws \(a^m \times a^n = a^{m+n}\), \(a^m \div a^n = a^{m-n}\) and \((a^m)^n = a^{mn}\). Laws of indices
- Evaluate zero and negative powers \(a^0 = 1\) and \(a^{-n} = \frac{1}{a^n}\), e.g. \(2^{-3} = \frac{1}{8}\). Laws of indices
- Simplify a surd using a square factor \(\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}\). Surds
- Add and subtract like surds \(\sqrt{50} + \sqrt{8} = 5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}\). Surds
- Solve a quadratic by factorising or formula Factorise when you can; otherwise use \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Quadratics and the discriminant
- Solve linear simultaneous equations by elimination Make the coefficients of one unknown match, then add or subtract the equations. Simultaneous equations
- Solve a linear inequality Solve it like an equation, but reverse the sign if you multiply or divide by a negative number. Inequalities
- Expand brackets and collect like terms Multiply every term in one bracket by every term in the other, e.g. \((x + 2)(x^2 - 3x + 1) = x^3 - x^2 - 5x + 2\). Polynomials, algebraic division and the factor theorem
- Sketch a cubic from its factorised form Mark the roots and the \(y\)-intercept; a positive \(x^3\) coefficient goes from bottom left to top right. Curve sketching and the modulus function
- Evaluate a composite function at a number \(fg(3)\) means find \(g(3)\) first, then put the answer into \(f\). Composite and inverse functions
- Translate a graph vertically and horizontally \(f(x) + a\) moves the graph up by \(a\); \(f(x + a)\) moves it left by \(a\). Transformations of graphs
- Use a model to make a prediction Substitute into the formula and give the answer in context, with units. Modelling with functions
- Find the gradient between two points \(m = \frac{y_2 - y_1}{x_2 - x_1}\): the change in \(y\) divided by the change in \(x\). Straight lines
- Find a line from a point and gradient Use \(y - y_1 = m(x - x_1)\), then rearrange if a particular form is asked for. Straight lines
- State the centre and radius from the equation \((x - a)^2 + (y - b)^2 = r^2\) has centre \((a, b)\) and radius \(r\). Circles
- Find a point from a parameter value Substitute the value of \(t\) into both \(x\) and \(y\). Parametric equations
- Expand \((a+b)^n\) for positive integer n Use Pascal's triangle or \(^nC_r\) to write down all n + 1 terms, e.g. \((1+2x)^4 = 1 + 8x + 24x^2 + 32x^3 + 16x^4\). Binomial expansion
- Find terms from an nth term formula Substitute n = 1, 2, 3, … into a formula such as \(u_n = 3n^2 - 1\). Sequences and recurrence relations
- Write out the terms of a sigma sum Substitute each value of r from the bottom limit to the top limit, e.g. \(\sum_{r=1}^{4} r^2 = 1 + 4 + 9 + 16\). Sigma notation
- Find the nth term of an arithmetic sequence Use \(u_n = a + (n-1)d\), e.g. the 20th term of 7, 11, 15, … is \(7 + 19 \times 4 = 83\). Arithmetic sequences and series
- Find the nth term of a geometric sequence Use \(u_n = ar^{n-1}\), e.g. the 6th term of 3, 6, 12, … is \(3 \times 2^5 = 96\). Geometric sequences and series
- Choose an arithmetic or geometric model A fixed amount added each time is arithmetic; a fixed percentage change each time is geometric. Modelling with sequences and series
- Use the sine and cosine rules Choose the rule from what you know: a side and its opposite angle means the sine rule; two sides and the included angle, or three sides, means the cosine rule. Sine and cosine rules, radians, arcs and sectors
- Convert between degrees and radians Use π radians = 180°, e.g. \(60^\circ = \frac{\pi}{3}\) and \(\frac{3\pi}{4} = 135^\circ\). Sine and cosine rules, radians, arcs and sectors
- Sketch the sin, cos and tan graphs Mark the key points: where they cross the axes, the maximum and minimum values, and the asymptotes of tan. Graphs of sin, cos and tan
- Solve \(\sin x=k\) in a given interval Find the calculator value, then use symmetry (or CAST) to find every other solution in the interval. Solving trigonometric equations
- Resolve a vector into components A force F at angle θ to the horizontal has components \(F\cos\theta\) (horizontal) and \(F\sin\theta\) (vertical). Trigonometry in context
- Sketch \(y=a^x\) for \(a>1\) An increasing curve through \((0, 1)\) that gets closer and closer to the x-axis as \(x \to -\infty\) but never touches it. Exponential functions and graphs
- State the gradient function of \(\mathrm{e}^x\) If \(y = \mathrm{e}^x\) then \(\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{e}^x\): the gradient at every point equals the y-coordinate. The function eˣ and its gradient
- Convert between index form and log form \(\log_a b = c\) means exactly the same as \(a^c = b\). Logarithms
- Evaluate simple logarithms without a calculator For example \(\log_2 32 = 5\), \(\log_5 1 = 0\) and \(\log_9 3 = \frac{1}{2}\). Logarithms
- Use the addition and subtraction laws \(\log_a x + \log_a y = \log_a xy\) and \(\log_a x - \log_a y = \log_a \frac{x}{y}\). Laws of logarithms
- Solve \(a^x=b\) by taking logarithms \(x\log a = \log b\), so \(x = \frac{\log b}{\log a}\); e.g. \(5^x = 40\) gives \(x = 2.29\) (3 s.f.). Solving exponential equations
- Find the initial value of a model Put \(t = 0\): \(A\mathrm{e}^{0} = A\), so \(A\) is the starting amount. Exponential growth and decay models
- Interpret a derivative as gradient and rate \(f'(x)\) is the gradient of the tangent at \((x, f(x))\) and the rate of change of \(y\) with respect to \(x\). First principles, second derivatives and concavity
- Differentiate polynomials term by term \(\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}\); a constant differentiates to 0. Differentiating standard functions
- Find the gradient at a point Differentiate, then substitute the x-coordinate into \(\frac{\mathrm{d}y}{\mathrm{d}x}\). Tangents, normals and stationary points
- Integrate as the reverse of differentiation If \(\frac{\mathrm{d}y}{\mathrm{d}x} = 6x^2\) then \(y = 2x^3 + c\), because differentiating \(2x^3 + c\) gives back \(6x^2\). The fundamental theorem of calculus
- Add a constant to every indefinite integral The derivative of any constant is zero, so an indefinite integral is only known up to \(+\,c\). The fundamental theorem of calculus
- Integrate powers of x, including fractional powers Rewrite roots and fractions as powers first, e.g. \(\frac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}\), which integrates to \(4x^{\frac{1}{2}} + c\). Integrating standard functions
- Evaluate a definite integral of a polynomial Integrate, substitute the upper limit, then subtract the value at the lower limit. Definite integrals and areas under curves
- Show a root lies in an interval Work out \(\mathrm{f}(a)\) and \(\mathrm{f}(b)\), show they have opposite signs, and conclude using continuity. Locating roots by change of sign
- Calculate iterates from a formula Substitute \(x_0\) to get \(x_1\), then \(x_1\) to get \(x_2\), and so on, using the ANS key. Iteration and staircase/cobweb diagrams
- Complete a table of values Substitute each \(x\) into the function (radians for trigonometric functions) and round as instructed. Numerical integration: the trapezium rule
- Write vectors in i, j, k and column form \(3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}\). Vectors in 2D and 3D
- Add, subtract and scale vectors Work component by component, e.g. \(2(\mathbf{i} - 3\mathbf{j}) + (4\mathbf{i} + \mathbf{j}) = 6\mathbf{i} - 5\mathbf{j}\). Vectors in 2D and 3D
- Find the magnitude of a 2D vector \(|3\mathbf{i} - 4\mathbf{j}| = \sqrt{3^2 + (-4)^2} = 5\). Magnitude and direction
- Add vectors using the triangle law \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\): follow the route from start to finish. Vector arithmetic and geometry
- Write the position vector of a point The point \((3, -1, 2)\) has position vector \(3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\). Position vectors and distances
- Define population, sample, census and sampling frame A census measures every member of the population; a sample is a selection, chosen from a sampling frame (a list of the sampling units). Sampling methods and the large data set
- Describe how to take a simple random sample Number every item in the sampling frame, then use random numbers to choose the required number of different items. Sampling methods and the large data set
- Calculate frequency densities for a histogram Frequency density = frequency ÷ class width, using the class boundaries. Histograms, box plots and cumulative frequency
- Read the median and quartiles from a box plot The box shows \(Q_1\), the median and \(Q_3\); the whiskers show the extent of the data and crosses show outliers. Histograms, box plots and cumulative frequency
- Describe the correlation shown in a scatter diagram State positive, negative or no correlation, and whether it is strong or weak. Correlation and regression
- Identify the explanatory and response variables The explanatory (independent) variable is set or controlled, or affects the other; the response (dependent) variable responds to it. Correlation and regression
- Find the mean, median and mode of data Mean \(= \dfrac{\Sigma x}{n}\); the median is the middle value of the ordered data; the mode is the most common value. Measures of location and spread
- Find quartiles and the interquartile range Use the rule for positions in an ordered list, then IQR \(= Q_3 - Q_1\). Measures of location and spread
- State what an outlier is An outlier is a value that is very different from (much larger or smaller than) the rest of the data. Outliers and cleaning data
- Use set notation for events Read and write \(A \cap B\) (A and B), \(A \cup B\) (A or B or both) and \(A'\) (not A). Mutually exclusive and independent events; Venn and tree diagrams
- Complete and use a Venn diagram Fill it in from the intersection outwards, using probabilities or frequencies, and read off probabilities. Mutually exclusive and independent events; Venn and tree diagrams
- Find conditional probabilities from a two-way table Divide by the total of the row or column that you are given. Conditional probability
- Calculate expected frequencies from probabilities Expected frequency = number of trials × probability. Modelling with probability
- Use the fact that probabilities sum to 1 Solve \(\Sigma P(X = x) = 1\) to find an unknown such as \(k\). Discrete distributions, including the discrete uniform and the binomial
- Recognise a discrete uniform distribution Each of the \(n\) possible values has probability \(\tfrac{1}{n}\), e.g. the score on a fair die. Discrete distributions, including the discrete uniform and the binomial
- Describe the shape and properties of the normal distribution Bell-shaped and symmetrical about the mean \(\mu\), with mean = median = mode, and total area 1. The normal distribution
- Find normal probabilities using a calculator Use normal CD with \(\mu\) and \(\sigma\) (not \(\sigma^2\)). The normal distribution
- Recognise when a binomial model is suitable The variable counts successes in a fixed number of independent trials with a constant probability of success. The normal approximation to the binomial; choosing a distribution
- Write null and alternative hypotheses \(H_0\) gives the parameter a single value; \(H_1\) says it is greater, less or different. The language of hypothesis testing
- Decide whether a test is one- or two-tailed ‘Increased’ or ‘decreased’ gives a one-tailed test; ‘changed’ or ‘different’ gives a two-tailed test. The language of hypothesis testing
- Write hypotheses for a binomial test \(H_0: p = p_0\), with \(H_1\) from the wording and \(p\) defined in context. Binomial hypothesis tests
- Write hypotheses for a correlation test \(H_0: \rho = 0\) and \(H_1: \rho \gt 0\), \(\rho \lt 0\) or \(\rho \ne 0\), with \(\rho\) defined. Tests for correlation and for the mean of a normal distribution
- State the SI units of basic quantities Length is measured in metres (m), mass in kilograms (kg) and time in seconds (s). SI units and modelling assumptions
- Convert units before substituting Divide km h−1 by 3.6 to get m s−1, divide grams by 1000 to get kilograms, and multiply minutes by 60 to get seconds. SI units and modelling assumptions
- Distinguish distance from displacement Displacement is the vector from the start to the finish; distance is the total length of the path. Displacement, velocity and acceleration
- Find average speed and average velocity Divide the total distance (for speed) or the total displacement (for velocity) by the total time. Displacement, velocity and acceleration
- Find velocity from a displacement–time graph The gradient of a displacement–time graph is the velocity. Kinematics graphs
- Sketch a velocity–time graph from a description Use straight lines for constant acceleration and label the key speeds and times. Kinematics graphs
- Choose and use the correct suvat formula List \(s, u, v, a, t\), then pick the formula that links the three you know with the one you want. Constant acceleration (suvat)
- Draw a clear, labelled force diagram Show the weight, normal reaction, tension or thrust, friction and any applied forces, each with an arrow. Forces, resultants and equilibrium of a particle
- Add forces given in vector form Add the \(\mathbf{i}\) components and the \(\mathbf{j}\) components separately to get the resultant. Forces, resultants and equilibrium of a particle
- Apply \(F = ma\) in a straight line The resultant force in the direction of motion equals mass × acceleration. Newton's second law and vectors
- Calculate weight using \(W = mg\) Mass in kg, \(g = 9.8\) m s−2, weight in newtons. Weight and motion under gravity
- Use suvat for vertical motion under gravity Take the acceleration as \(g\) downwards and choose up or down as positive. Weight and motion under gravity
- State Newton's third law The forces between two bodies are equal in size, opposite in direction and act on different bodies. Newton's third law, connected particles and pulleys
- Resolve a force into two components A force \(F\) at \(\theta\) to a direction has components \(F\cos\theta\) along it and \(F\sin\theta\) perpendicular to it. Resolving forces and inclined planes
- Know the direction in which friction acts Along the surface, opposing the motion, or the motion that would happen without friction. Friction
- Calculate the moment of a force Moment = force × perpendicular distance from the point, stating clockwise or anticlockwise. Moments, beams and rods in equilibrium
Grade D
- Write general even, odd and consecutive integers Use \(2n\), \(2n + 1\), and \(n\), \(n + 1\), where \(n\) is an integer, so the argument covers every case. Proof by deduction, exhaustion and counter-example
- Evaluate fractional powers without a calculator \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\), e.g. \(27^{\frac{2}{3}} = 3^2 = 9\). Laws of indices
- Expand brackets containing surds \((3 + \sqrt{5})(2 - \sqrt{5}) = 6 - 3\sqrt{5} + 2\sqrt{5} - 5 = 1 - \sqrt{5}\). Surds
- Complete the square \(x^2 + 6x + 2 = (x + 3)^2 - 7\); when \(a \ne 1\), take \(a\) out of the \(x\) terms first. Quadratics and the discriminant
- Sketch a quadratic with intercepts and vertex Show the shape, the \(y\)-intercept, any roots and the turning point from the completed square. Quadratics and the discriminant
- Find constants from two conditions Substitute two given points into a formula to get two linear equations, then solve them. Simultaneous equations
- Solve one linear and one quadratic equation Rearrange the linear equation for \(x\) or \(y\) and substitute into the quadratic. Simultaneous equations
- Solve a quadratic inequality using a sketch Find the critical values, sketch the parabola and choose the part above or below the \(x\)-axis. Inequalities
- Show a factor using the factor theorem Show \(f(a) = 0\) and conclude that \((x - a)\) is a factor of \(f(x)\). Polynomials, algebraic division and the factor theorem
- Show repeated roots correctly on a sketch A squared factor such as \((x - 2)^2\) means the curve touches the \(x\)-axis at \(x = 2\) without crossing. Curve sketching and the modulus function
- Sketch reciprocal graphs and their asymptotes \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\) both have asymptotes \(x = 0\) and \(y = 0\). Curve sketching and the modulus function
- Use proportional relationships and their graphs \(y \propto x^n\) means \(y = kx^n\); find \(k\) from one pair of values. Curve sketching and the modulus function
- Find a composite function algebraically Replace \(x\) in \(f\) with the whole expression for \(g(x)\): \(fg(x) = f(g(x))\). Composite and inverse functions
- Find the inverse of a function Write \(y = f(x)\), rearrange to make \(x\) the subject, then write the result in terms of \(x\). Composite and inverse functions
- Stretch a graph parallel to either axis \(af(x)\): stretch parallel to the \(y\)-axis, scale factor \(a\); \(f(ax)\): parallel to the \(x\)-axis, scale factor \(\frac{1}{a}\). Transformations of graphs
- Reflect a graph in either axis \(-f(x)\) reflects the graph in the \(x\)-axis; \(f(-x)\) reflects it in the \(y\)-axis. Transformations of graphs
- Split a fraction with two linear factors \(\frac{px + q}{(x - a)(x - b)} \equiv \frac{A}{x - a} + \frac{B}{x - b}\). Partial fractions
- Find constants in a model from data Substitute given pairs of values to form equations and solve for the unknown constants. Modelling with functions
- Give a line's equation with integer coefficients Clear any fractions and collect all the terms on one side, with \(a\), \(b\) and \(c\) integers. Straight lines
- Use the parallel and perpendicular gradient rules Parallel lines have equal gradients; perpendicular gradients multiply to give \(-1\). Straight lines
- Write the equation from centre and radius Substitute into \((x - a)^2 + (y - b)^2 = r^2\); find \(r\) with the distance formula if needed. Circles
- Convert to Cartesian form by substitution Rearrange one equation for \(t\) and substitute: \(x = t + 1\), \(y = t^2\) gives \(y = (x - 1)^2\). Parametric equations
- Find positions at given times Substitute the value of \(t\) into both equations and give the answers in context with units. Parametric equations in modelling
- Find a given term or coefficient The term in \(x^r\) in \((a+bx)^n\) is \(\binom{n}{r}a^{n-r}(bx)^r\), so you can find it without writing out the whole expansion. Binomial expansion
- Generate terms from a recurrence relation Use a rule such as \(u_{n+1} = 2u_n - 3\) with the first term to find \(u_2, u_3, \dots\) in turn. Sequences and recurrence relations
- Count the terms and evaluate short sums From r = a to r = b there are b − a + 1 terms; for a few terms, just add them. Sigma notation
- Find the sum of an arithmetic series Use \(S_n = \frac{n}{2}[2a + (n-1)d]\) or \(S_n = \frac{n}{2}(a + l)\). Arithmetic sequences and series
- Find the sum of a finite geometric series Use \(S_n = \frac{a(1 - r^n)}{1 - r}\) with the correct values of a, r and n. Geometric sequences and series
- Find a value in a given year Use the nth term, taking care over what n = 1 represents in the context. Modelling with sequences and series
- Find the area of a triangle Use \(\frac{1}{2}ab\sin C\), where C is the angle between the sides a and b. Sine and cosine rules, radians, arcs and sectors
- Find arc length and sector area Use \(s = r\theta\) and \(A = \frac{1}{2}r^2\theta\), with θ in radians. Sine and cosine rules, radians, arcs and sectors
- State the period of each graph sin and cos repeat every 360° (2π); tan repeats every 180° (π). Graphs of sin, cos and tan
- Use \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) to simplify e.g. \(\sin\theta = 2\cos\theta\) becomes \(\tan\theta = 2\). Pythagorean identities
- Use \(\tan x=\frac{\sin x}{\cos x}\) to solve equations Turn \(3\sin x = 2\cos x\) into \(\tan x = \frac{2}{3}\) and solve. Solving trigonometric equations
- Find the magnitude and direction of a vector Use \(\sqrt{a^2 + b^2}\) and \(\tan\theta = \frac{b}{a}\), with a sketch to get the direction right. Trigonometry in context
- Sketch \(y=a^x\) for \(0\lt a\lt 1\) A decreasing curve through \((0, 1)\); it is the reflection of \(y = \left(\frac{1}{a}\right)^x\) in the y-axis. Exponential functions and graphs
- Differentiate \(\mathrm{e}^{kx}\) and \(A\mathrm{e}^{kx}\) Multiply by \(k\): the derivative of \(5\mathrm{e}^{-2x}\) is \(-10\mathrm{e}^{-2x}\). The function eˣ and its gradient
- Use ln and e as inverse functions \(\ln(\mathrm{e}^x) = x\) for all \(x\), and \(\mathrm{e}^{\ln x} = x\) for \(x > 0\). Logarithms
- Use the power law, including negative powers \(3\log_a x = \log_a x^3\), \(-\log_a x = \log_a \frac{1}{x}\) and \(-\frac{1}{2}\log_a x = \log_a \frac{1}{\sqrt{x}}\). Laws of logarithms
- Solve \(\mathrm{e}^{kx}=b\) using ln E.g. \(\mathrm{e}^{0.4x} = 9\) gives \(x = \frac{\ln 9}{0.4} = 5.49\) (3 s.f.). Solving exponential equations
- Take logs of \(y=ax^n\) \(\log y = n\log x + \log a\): a straight line when \(\log y\) is plotted against \(\log x\). Using log graphs to estimate parameters
- Take logs of \(y=kb^x\) \(\log y = x\log b + \log k\): a straight line when \(\log y\) is plotted against \(x\). Using log graphs to estimate parameters
- Use a model to predict a value Substitute the given time into the model and include units. Exponential growth and decay models
- Find a second derivative Differentiate \(\frac{\mathrm{d}y}{\mathrm{d}x}\) again to get \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\), the rate of change of the gradient. First principles, second derivatives and concavity
- Differentiate roots and reciprocals of \(x\) Rewrite as powers first: \(\sqrt{x} = x^{\frac{1}{2}}\), \(\frac{5}{x^2} = 5x^{-2}\). Differentiating standard functions
- Find the equation of a tangent Use \(y - y_1 = m(x - x_1)\) with \(m = f'(x_1)\). Tangents, normals and stationary points
- Find the equation of a normal The normal is perpendicular to the tangent, so its gradient is \(-\frac{1}{m}\). Tangents, normals and stationary points
- Use the chain rule on powers of functions E.g. \(\frac{\mathrm{d}}{\mathrm{d}x}(3x - 1)^5 = 15(3x - 1)^4\). Product, quotient and chain rules
- Find a curve through a given point Integrate the gradient function, then substitute the coordinates of the point to find \(c\). The fundamental theorem of calculus
- Integrate exponentials and 1/x \(\int \mathrm{e}^{kx}\,\mathrm{d}x = \frac{1}{k}\mathrm{e}^{kx} + c\) and \(\int \frac{1}{x}\,\mathrm{d}x = \ln|x| + c\). Integrating standard functions
- Find the area under a curve For a curve above the \(x\)-axis, the area between the curve, the axis and the lines \(x = a\) and \(x = b\) is \(\int_a^b y\,\mathrm{d}x\). Definite integrals and areas under curves
- Find where two graphs intersect Set the two equations equal and solve to get the limits of the region. Areas between curves and parametric areas
- Rearrange an equation into f(x) = 0 To find where \(\mathrm{e}^x = 4 - x\), use \(\mathrm{f}(x) = \mathrm{e}^x + x - 4\). Locating roots by change of sign
- Rearrange f(x) = 0 into x = g(x) Use algebra to isolate an \(x\) on one side, e.g. \(x^3 - 4x - 2 = 0 \Rightarrow x = \sqrt{4 + \frac{2}{x}}\). Iteration and staircase/cobweb diagrams
- Differentiate f(x) accurately A correct \(\mathrm{f}'(x)\) is the first mark; use the chain, product or quotient rule where needed. The Newton–Raphson method
- Apply the trapezium rule formula \(\frac{1}{2}h\left[(y_0 + y_n) + 2(y_1 + \cdots + y_{n-1})\right]\) with the values from the table. Numerical integration: the trapezium rule
- Form an equation from a context Turn a condition such as 'the concentration falls to 2 mg per litre' into an equation \(\mathrm{f}(t) = 0\). Numerical methods in context
- Use 3D coordinates and axes The point \((2, -1, 4)\) has position vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) relative to the origin. Vectors in 2D and 3D
- Find the magnitude of a 3D vector \(|x\mathbf{i} + y\mathbf{j} + z\mathbf{k}| = \sqrt{x^2 + y^2 + z^2}\). Magnitude and direction
- Express a route in terms of given vectors For example \(\overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} = -\mathbf{a} + \mathbf{b}\). Vector arithmetic and geometry
- Find the vector between two points \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\): subtract the start from the end. Position vectors and distances
- Find a resultant force Add the force vectors: \(\mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2 + \cdots\). Vector problems
- Describe systematic, quota and opportunity sampling Say how each method is carried out and whether it needs a sampling frame. Sampling methods and the large data set
- Estimate quartiles from a cumulative frequency graph Read across from \(\tfrac{n}{4}\), \(\tfrac{n}{2}\) and \(\tfrac{3n}{4}\) on the cumulative frequency axis. Histograms, box plots and cumulative frequency
- Draw a histogram with unequal class widths Plot frequency density on the vertical axis, with the bars touching at the class boundaries. Histograms, box plots and cumulative frequency
- Interpret the gradient of a regression line Say how much the response variable changes, with units, for each increase of 1 in the explanatory variable. Correlation and regression
- Estimate the mean from grouped data Use the class midpoints: \(\bar{x} \approx \dfrac{\Sigma fx}{\Sigma f}\). Measures of location and spread
- Find outlier limits using quartiles and the IQR Use the rule given, e.g. below \(Q_1 - 1.5 \times \text{IQR}\) or above \(Q_3 + 1.5 \times \text{IQR}\). Outliers and cleaning data
- Use tree diagrams for successive events Multiply along the branches and add the end results, including for selections without replacement. Mutually exclusive and independent events; Venn and tree diagrams
- Use conditional probabilities on tree diagrams The second-stage branches are conditional probabilities, e.g. \(P(B \mid A)\). Conditional probability
- State an assumption made by a model E.g. the die is fair, outcomes are equally likely, or trials are independent, in context. Modelling with probability
- Calculate a binomial probability for one value Use \(P(X = r) = \binom{n}{r}p^r(1 - p)^{n - r}\) or your calculator's binomial PD. Discrete distributions, including the discrete uniform and the binomial
- Use the approximate 68%, 95% and 99.7% rules About 68% (roughly two-thirds) of values lie within \(\mu \pm \sigma\), 95% within \(\mu \pm 2\sigma\) and almost all within \(\mu \pm 3\sigma\). The normal distribution
- State the conditions for the normal approximation \(n\) is large and \(p\) is close to 0.5. The normal approximation to the binomial; choosing a distribution
- Explain what the significance level means It sets how unlikely a result must be to reject \(H_0\), and is the probability of rejecting \(H_0\) when it is true. The language of hypothesis testing
- Carry out a one-tailed test with a p-value Find \(P(X \ge x)\) or \(P(X \le x)\) under \(H_0\) and compare it with the significance level. Binomial hypothesis tests
- Compare \(r\) with a critical value Read the table for \(n\) and the significance level, then decide whether \(r\) is beyond the critical value. Tests for correlation and for the mean of a normal distribution
- Give the units of derived quantities Velocity m s−1, acceleration m s−2, force and weight N (1 N = 1 kg m s−2), moment N m. SI units and modelling assumptions
- Use signs to show direction in one dimension Choose a positive direction; a negative velocity means moving the opposite way at that speed. Displacement, velocity and acceleration
- Find acceleration and distance from a velocity–time graph The gradient gives the acceleration and the area under the graph gives the displacement. Kinematics graphs
- Handle deceleration and direction with signs Choose a positive direction; a deceleration has the opposite sign to the velocity. Constant acceleration (suvat)
- Differentiate displacement to find velocity and acceleration \(v = \frac{ds}{dt}\) and \(a = \frac{dv}{dt}\). Variable acceleration using calculus
- Resolve the initial velocity into components The horizontal component is \(U\cos\alpha\) and the vertical component is \(U\sin\alpha\). Projectiles
- Find the magnitude and direction of a force Use Pythagoras for the magnitude and \(\tan\theta\) for the angle with \(\mathbf{i}\). Forces, resultants and equilibrium of a particle
- Combine \(F = ma\) with suvat Find \(a\) from the forces and then use suvat, or find \(a\) from suvat and then find a force. Newton's second law and vectors
- Find the greatest height and time of flight At the top \(v = 0\); when it lands the displacement is 0 or a known negative value. Weight and motion under gravity
- Solve car and trailer problems Use the whole system to find the acceleration, then one part to find the tension or thrust. Newton's third law, connected particles and pulleys
- Resolve weight on an inclined plane On a slope at angle \(\theta\): \(mg\sin\theta\) down the slope and \(mg\cos\theta\) into the slope. Resolving forces and inclined planes
- Use \(F = \mu R\) on rough level ground Find \(R\) by resolving vertically; when the object slides, the friction is \(\mu R\). Friction
- Find the reactions on a beam on two supports Take moments about one support, then resolve vertically for the other reaction. Moments, beams and rods in equilibrium
- Take moments for forces on an inclined rod For a rod at \(\theta\) to the horizontal, a vertical force at distance \(d\) along it has moment arm \(d\cos\theta\); a horizontal force has \(d\sin\theta\). Ladders, tilting and limiting equilibrium
Grade C
- Prove a statement by exhaustion Split all possibilities into a finite set of cases (e.g. \(n\) even and \(n\) odd) and prove each one. Proof by deduction, exhaustion and counter-example
- Prove an algebraic result by deduction Expand and factorise to show, for example, that an expression equals \(8 \times\) an integer. Proof by deduction, exhaustion and counter-example
- Write the opening assumption correctly Assume the negation of the statement, e.g. for 'there is no greatest odd number' assume there is a greatest odd number, \(N\). Proof by contradiction
- Write expressions as powers of x e.g. \(\frac{3}{\sqrt{x}} = 3x^{-\frac{1}{2}}\), ready to differentiate or integrate. Laws of indices
- Split a fraction into separate powers e.g. \(\frac{x^2 + 4}{2\sqrt{x}} = \frac{1}{2}x^{\frac{3}{2}} + 2x^{-\frac{1}{2}}\). Laws of indices
- Rationalise a single-surd denominator Multiply top and bottom by the surd: \(\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\). Surds
- Use the discriminant to count real roots \(b^2 - 4ac \gt 0\): two distinct real roots; \(= 0\): one repeated root; \(\lt 0\): no real roots. Quadratics and the discriminant
- Find the coordinates of intersection points Solve for one variable, then substitute each value into the linear equation to get the pairs. Simultaneous equations
- Interpret solutions as intersections of graphs Two solutions: the line crosses the curve twice; one: it is a tangent; none: they do not meet. Simultaneous equations
- Write solutions using set notation e.g. \(\{x : x \lt -2\} \cup \{x : x \gt 5\}\) for two regions, \(\{x : -2 \lt x \lt 5\}\) for one. Inequalities
- Find values satisfying two inequalities together Show both solution sets on a number line and find where they overlap. Inequalities
- Define a region using inequalities Decide which side of each line or curve the region is on, and whether the boundary is included. Inequalities
- Divide a cubic by a linear factor Use long division or compare coefficients to find the quadratic factor. Polynomials, algebraic division and the factor theorem
- Factorise a cubic completely Find one factor with the factor theorem, divide, then factorise the quadratic quotient. Polynomials, algebraic division and the factor theorem
- Simplify algebraic fractions by factorising Factorise the top and bottom fully, then cancel common factors (whole brackets only). Polynomials, algebraic division and the factor theorem
- Sketch the modulus graph of a linear function Reflect the part of \(y = ax + b\) below the \(x\)-axis to make a V shape, vertex at \(x = -\frac{b}{a}\). Curve sketching and the modulus function
- Use intersections to count solutions The number of points where \(y = f(x)\) and \(y = g(x)\) meet is the number of real solutions of \(f(x) = g(x)\). Curve sketching and the modulus function
- State the range of a function Use a sketch over the given domain; write the range in terms of \(f(x)\), e.g. \(f(x) \ge 2\). Composite and inverse functions
- Sketch a function and its inverse The graph of \(y = f^{-1}(x)\) is the reflection of \(y = f(x)\) in the line \(y = x\). Composite and inverse functions
- Find where points and asymptotes move Apply the transformation to coordinates, e.g. under \(y = 2f(x)\), \((3, 4)\) moves to \((3, 8)\). Transformations of graphs
- Describe a single transformation fully Name it and give every detail: the vector, or the direction and scale factor, or the mirror line. Transformations of graphs
- Find the constants by substitution Multiply through by the denominator and substitute the roots, e.g. \(x = a\), to find \(A\) and \(B\) quickly. Partial fractions
- Handle three distinct linear factors Use \(\frac{A}{x - a} + \frac{B}{x - b} + \frac{C}{x - c}\) and substitute each root in turn. Partial fractions
- Interpret constants and features in context e.g. the value at \(t = 0\) is the starting value; the vertex of a quadratic is a greatest height. Modelling with functions
- State a sensible domain for a model Only values that make sense, e.g. from \(t = 0\) until the object hits the ground. Modelling with functions
- Find midpoints, lengths and intersections Use the midpoint and distance formulae, and solve the equations of two lines simultaneously. Straight lines
- Use straight-line models in context Interpret the gradient as a rate of change and the intercept as a starting or fixed value. Straight lines
- Complete the square to find centre and radius e.g. \(x^2 + y^2 - 6x + 4y - 12 = 0\) becomes \((x - 3)^2 + (y + 2)^2 = 25\): centre \((3, -2)\), radius 5. Circles
- Find the equation of a tangent The tangent is perpendicular to the radius at that point, so use the negative reciprocal of the radius gradient. Circles
- Find where a curve meets an axis or line Set \(y = 0\) or \(x = 0\), or substitute \(x(t)\) and \(y(t)\) into the line's equation, then solve for \(t\). Parametric equations
- Convert trigonometric parametric equations Use identities such as \(\sin^2 t + \cos^2 t = 1\) or \(\cos 2t = 1 - 2\sin^2 t\) to eliminate \(t\). Parametric equations
- Find when a condition is met e.g. set \(y = 0\) to find when a ball lands, and choose the root that fits the context. Parametric equations in modelling
- Interpret constants and the parameter e.g. the coefficient of \(t\) in \(x = 18t\) is a constant horizontal speed of 18 m s−1. Parametric equations in modelling
- Find an unknown constant from a coefficient Write the coefficient in terms of the constant, set it equal to the given value and solve, e.g. \(28k^2 = 252\). Binomial expansion
- Expand \((1+x)^n\) for negative or fractional n Use \(1 + nx + \frac{n(n-1)}{2!}x^2 + \dots\) to give the first few terms, valid for \(|x| \lt 1\). Binomial expansion
- Find unknown constants in a recurrence relation Write the terms in terms of the constant, form an equation from a given term and solve it. Sequences and recurrence relations
- Evaluate an arithmetic series in sigma notation A linear rule such as 3r + 2 gives an arithmetic series: find the first term, last term and number of terms, then use \(S_n\). Sigma notation
- Find a and d from given information Turn each fact into an equation in a and d, then solve the equations simultaneously. Arithmetic sequences and series
- Use consecutive terms to find an unknown If p, q and r are consecutive terms, then \(q - p = r - q\). Arithmetic sequences and series
- Find a and r from two terms Divide one term equation by the other to eliminate a, e.g. \(\frac{ar^4}{ar} = r^3\). Geometric sequences and series
- Find a total over several periods Use \(S_n\) when the question asks for a total, e.g. the total distance run over 12 weeks. Modelling with sequences and series
- Find both angles in the ambiguous case If the sine rule gives \(\sin B = k\), the angle could be B or 180° − B; check whether both fit in the triangle. Sine and cosine rules, radians, arcs and sectors
- Recall the three small angle approximations \(\sin\theta \approx \theta\), \(\tan\theta \approx \theta\) and \(\cos\theta \approx 1 - \frac{\theta^2}{2}\), for small θ in radians. Small angle approximations
- Use symmetry to find related angles e.g. \(\sin(180^\circ - x) = \sin x\), \(\cos(-x) = \cos x\) and \(\tan(x + 180^\circ) = \tan x\). Graphs of sin, cos and tan
- Sketch translated and stretched trig graphs e.g. \(y = \cos(x + 30^\circ)\) is \(y = \cos x\) translated 30° to the left. Graphs of sin, cos and tan
- Define sec, cosec and cot \(\sec x = \frac{1}{\cos x}\), \(\operatorname{cosec} x = \frac{1}{\sin x}\) and \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\). sec, cosec, cot and inverse trig functions
- Find exact values of reciprocal and inverse functions e.g. \(\sec\frac{\pi}{3} = 2\) and \(\arccos\left(-\frac{1}{2}\right) = \frac{2\pi}{3}\). sec, cosec, cot and inverse trig functions
- Find exact values using \(\sin^2\theta+\cos^2\theta=1\) Given one ratio and the quadrant, find the other ratios with the correct signs. Pythagorean identities
- Find exact values using compound angles e.g. \(\sin 75^\circ = \sin(45^\circ + 30^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}\). Compound and double angles; R cos(θ ± α)
- Use the double angle formulae Know \(\sin 2A\), the three forms of \(\cos 2A\) and \(\tan 2A\), and choose the most useful form. Compound and double angles; R cos(θ ± α)
- Solve equations with multiple angles For \(\cos 2x = k\), solve for 2x in the doubled interval, then halve. Solving trigonometric equations
- Solve equations with a shifted angle For \(\cos(x - \alpha) = k\), shift the interval by α, solve, then add α back on. Solving trigonometric equations
- Prove simple identities using \(\sin^2x+\cos^2x=1\) e.g. show that \((1 - \cos x)(1 + \cos x) \equiv \sin^2 x\). Proving trigonometric identities
- Solve bearings problems using sine and cosine rules Draw north lines, find the angle inside the triangle, then use the right rule. Trigonometry in context
- Know the graphs of \(y=\mathrm{e}^x\) and \(y=\mathrm{e}^{-x}\) \(\mathrm{e} \approx 2.718\), so \(y = \mathrm{e}^x\) lies between \(y = 2^x\) and \(y = 3^x\); \(y = \mathrm{e}^{-x}\) is its reflection in the y-axis. Exponential functions and graphs
- Find the gradient at a given point Differentiate first, then substitute, giving exact answers such as \(2\mathrm{e}\). The function eˣ and its gradient
- Solve equations using ln and e Take ln of both sides of \(\mathrm{e}^{3x+1} = 10\), or write \(2y - 1 = \mathrm{e}^4\) when \(\ln(2y - 1) = 4\). Logarithms
- Sketch \(y=\ln x\) Through \((1, 0)\), increasing, defined only for \(x > 0\), with the y-axis as a vertical asymptote. Logarithms
- Write an expression as a single logarithm Use the power law first, then combine, writing plain numbers as logs, e.g. \(2 = \log_3 9\). Laws of logarithms
- Express logarithms in terms of given logs Given \(\log_a 2 = p\) and \(\log_a 3 = q\), write \(\log_a 12 = 2p + q\). Laws of logarithms
- Solve \(a^{px+q}=b\) Bring the whole bracket down: \((px + q)\ln a = \ln b\), then rearrange. Solving exponential equations
- Choose the right axes for a straight line Power law: \(\log y\) against \(\log x\). Exponential: \(\log y\) against \(x\). Using log graphs to estimate parameters
- Find when a quantity reaches a value Rearrange to \(\mathrm{e}^{kt} = \ldots\), then take ln of both sides. Exponential growth and decay models
- Give a limitation of a model E.g. a growth model predicts that a population increases without limit, which is unrealistic. Exponential growth and decay models
- Sketch the gradient function of a curve Where the curve has a stationary point, \(y = f'(x)\) meets the x-axis; where the curve increases, \(f'(x) > 0\). First principles, second derivatives and concavity
- Differentiate \(\mathrm{e}^{kx}\), \(\sin kx\) and \(\cos kx\) \(k\mathrm{e}^{kx}\), \(k\cos kx\) and \(-k\sin kx\), with \(x\) in radians. Differentiating standard functions
- Differentiate \(\ln x\) and \(\ln kx\) Both give \(\frac{1}{x}\), because \(\ln kx = \ln k + \ln x\). Differentiating standard functions
- Find stationary points Solve \(\frac{\mathrm{d}y}{\mathrm{d}x} = 0\), then find each y-coordinate from the curve's equation. Tangents, normals and stationary points
- Classify stationary points \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} > 0\): minimum; \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \lt 0\): maximum; if it is 0, test the gradient on each side. Tangents, normals and stationary points
- Differentiate composite exponential, log and trig functions \(\mathrm{e}^{f(x)} \to f'(x)\mathrm{e}^{f(x)}\), \(\ln f(x) \to \frac{f'(x)}{f(x)}\), \(\sin f(x) \to f'(x)\cos f(x)\). Product, quotient and chain rules
- Use the product rule \(\frac{\mathrm{d}}{\mathrm{d}x}(uv) = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}\). Product, quotient and chain rules
- Write a rate using derivative notation 'Volume increasing at 3 cm3 s−1' means \(\frac{\mathrm{d}V}{\mathrm{d}t} = 3\); a decrease gives a negative derivative. Connected rates of change
- Connect rates with the chain rule E.g. \(\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\mathrm{d}A}{\mathrm{d}r} \times \frac{\mathrm{d}r}{\mathrm{d}t}\). Connected rates of change
- Differentiate parametric equations \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}\), giving an answer in terms of \(t\). Implicit and parametric differentiation
- Differentiate terms in \(y\) implicitly Differentiate with respect to \(y\) and multiply by \(\frac{\mathrm{d}y}{\mathrm{d}x}\): \(\frac{\mathrm{d}}{\mathrm{d}x}(y^3) = 3y^2\frac{\mathrm{d}y}{\mathrm{d}x}\). Implicit and parametric differentiation
- Translate 'rate of change' into derivative notation 'The rate of change of \(P\) with respect to time' is \(\frac{\mathrm{d}P}{\mathrm{d}t}\). Constructing differential equations
- Write 'proportional to' using a constant 'Proportional to \(P\)' becomes \(kP\); 'inversely proportional to \(\sqrt{h}\)' becomes \(\frac{k}{\sqrt{h}}\). Constructing differential equations
- Use a negative sign for a decrease A decreasing quantity gives, for example, \(\frac{\mathrm{d}m}{\mathrm{d}t} = -km\), where \(k > 0\). Constructing differential equations
- Evaluate a definite integral as F(b) − F(a) Find an integral \(\mathrm{F}\), substitute the upper limit and subtract its value at the lower limit; no constant is needed. The fundamental theorem of calculus
- Integrate sin kx and cos kx in radians \(\int \cos kx\,\mathrm{d}x = \frac{1}{k}\sin kx + c\) and \(\int \sin kx\,\mathrm{d}x = -\frac{1}{k}\cos kx + c\). Integrating standard functions
- Integrate functions of a linear expression If \(\int \mathrm{f}(x)\,\mathrm{d}x = \mathrm{F}(x) + c\), then \(\int \mathrm{f}(ax + b)\,\mathrm{d}x = \frac{1}{a}\mathrm{F}(ax + b) + c\), e.g. \(\int \frac{1}{3x - 2}\,\mathrm{d}x = \frac{1}{3}\ln|3x - 2| + c\). Integrating standard functions
- Find the limits where a curve meets the axis Solve \(y = 0\) to find where a region bounded by the curve and the \(x\)-axis starts and ends. Definite integrals and areas under curves
- Find the area between a curve and a line Integrate (top − bottom), or subtract the area of a triangle or trapezium from the area under the curve. Areas between curves and parametric areas
- Interpret f(x)δx as a rectangle's area Each term is a thin strip of height \(\mathrm{f}(x)\) and width \(\delta x\). Integration as the limit of a sum
- Write a limit of a sum as an integral Keep the same function and limits, and replace \(\sum\) and \(\delta x\) by \(\int\) and \(\mathrm{d}x\). Integration as the limit of a sum
- Integrate f′(x)/f(x) by inspection When the numerator is a multiple of the derivative of the denominator, the answer is a multiple of \(\ln|\mathrm{f}(x)|\). Integration by substitution
- Integrate f′(x)[f(x)]ⁿ by inspection \(\int \mathrm{f}'(x)[\mathrm{f}(x)]^n\,\mathrm{d}x = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + c\) for \(n \neq -1\); adjust the constant if only a multiple of the derivative is there. Integration by substitution
- Choose u and dv/dx, then apply the formula Pick \(u\) so that \(\frac{\mathrm{d}u}{\mathrm{d}x}\) is simpler, and \(\frac{\mathrm{d}v}{\mathrm{d}x}\) so that it is easy to integrate. Integration by parts
- Integrate x times an exponential or trig function Let \(u = x\), so \(\frac{\mathrm{d}u}{\mathrm{d}x} = 1\) and the new integral is a standard one. Integration by parts
- Split into partial fractions with distinct factors Write \(\frac{px + q}{(ax + b)(cx + d)} = \frac{A}{ax + b} + \frac{B}{cx + d}\) and find \(A\) and \(B\) by substituting values or comparing coefficients. Integration using partial fractions
- Integrate A/(ax + b) to a logarithm \(\int \frac{A}{ax + b}\,\mathrm{d}x = \frac{A}{a}\ln|ax + b| + c\). Integration using partial fractions
- Separate the variables correctly Get every \(y\) on the side with \(\mathrm{d}y\) and every \(x\) on the side with \(\mathrm{d}x\). Separable differential equations
- Find a general solution with one constant Integrate both sides and add a single constant, usually on the \(x\) side. Separable differential equations
- Solve exponential growth and decay models \(\frac{\mathrm{d}N}{\mathrm{d}t} = kN\) gives \(N = N_0\mathrm{e}^{kt}\); \(k \gt 0\) means growth and \(k \lt 0\) decay. Differential equations in context: interpreting solutions
- Use given data to find the constants An initial value gives the constant of integration; a second data point gives \(k\), usually by taking logs. Differential equations in context: interpreting solutions
- Show a root is correct to given accuracy Test the bounds of the rounding interval, e.g. 1.2515 and 1.2525 for a root of 1.252 to 3 d.p. Locating roots by change of sign
- Draw staircase and cobweb diagrams Go vertically to the curve \(y = \mathrm{g}(x)\), then horizontally to the line \(y = x\), and repeat. Iteration and staircase/cobweb diagrams
- Confirm a root with a sign change After iterating, use a change of sign in \(\mathrm{f}(x)\) to show the root to the accuracy required. Iteration and staircase/cobweb diagrams
- Apply one Newton–Raphson step Substitute \(x_0\) into \(x_0 - \frac{\mathrm{f}(x_0)}{\mathrm{f}'(x_0)}\), showing the values of \(\mathrm{f}(x_0)\) and \(\mathrm{f}'(x_0)\). The Newton–Raphson method
- Find the strip width h \(h = \frac{b - a}{n}\) for \(n\) strips; there are \(n + 1\) values of \(y\). Numerical integration: the trapezium rule
- Estimate totals from data with the trapezium rule Distance \(= \int v\,\mathrm{d}t\), volume \(= \int (\text{flow rate})\,\mathrm{d}t\), cross-section area \(= \int (\text{depth})\,\mathrm{d}x\). Numerical methods in context
- Solve contextual equations numerically Use a given iteration or Newton–Raphson, and a change of sign to locate or confirm the root. Numerical methods in context
- Find unknowns by equating components If two vectors are equal, each pair of components is equal, giving equations to solve. Vectors in 2D and 3D
- Recognise parallel vectors Parallel vectors are scalar multiples of each other: every component is multiplied by the same number. Vectors in 2D and 3D
- Find a unit vector in a given direction Divide the vector by its magnitude: \(\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}\). Magnitude and direction
- Convert magnitude and angle to components A vector of magnitude \(r\) at angle \(\theta\) anticlockwise from \(\mathbf{i}\) is \(r\cos\theta\,\mathbf{i} + r\sin\theta\,\mathbf{j}\). Magnitude and direction
- Show that two vectors are parallel Show that one is a scalar multiple of the other, e.g. \(4\mathbf{a} - 6\mathbf{b} = 2(2\mathbf{a} - 3\mathbf{b})\). Vector arithmetic and geometry
- Use a ratio to locate a point If \(AP : PB = m : n\), then \(\overrightarrow{AP} = \frac{m}{m + n}\overrightarrow{AB}\). Vector arithmetic and geometry
- Find the distance between two points \(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). Position vectors and distances
- Find a midpoint or dividing point \(\overrightarrow{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})\); for \(AP : PB = m : n\), \(\overrightarrow{OP} = \mathbf{a} + \frac{m}{m + n}(\mathbf{b} - \mathbf{a})\). Position vectors and distances
- Use equilibrium: resultant force is zero Set the sum of all the forces equal to \(\mathbf{0}\) and solve for the unknown force. Vector problems
- Find position with constant velocity \(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\), where \(\mathbf{r}_0\) is the position when \(t = 0\). Vector problems
- Calculate the numbers for a stratified sample Sample each stratum in proportion to its size, then choose randomly within each stratum. Sampling methods and the large data set
- Estimate frequencies from histogram bar areas Use area to find frequency, assuming values are spread evenly within a class for part of a bar. Histograms, box plots and cumulative frequency
- Judge reliability: interpolation or extrapolation Estimates within the range of the data are reasonably reliable; estimates outside it are not. Correlation and regression
- Interpret \(r\), knowing correlation is not causation Relate \(r\) to how close the points are to a straight line, and recognise that a third variable may explain the link. Correlation and regression
- Calculate standard deviation from summary statistics Use \(\sigma = \sqrt{\dfrac{\Sigma x^2}{n} - \bar{x}^2}\), or \(\sqrt{\dfrac{S_{xx}}{n}}\). Measures of location and spread
- Interpolate medians and percentiles from grouped data Find the position, locate its class, then go the right fraction of the way through that class. Measures of location and spread
- Find outlier limits using mean and standard deviation Use the rule given, e.g. outside \(\bar{x} \pm 2\sigma\). Outliers and cleaning data
- Draw a box plot showing outliers Mark outliers with crosses and end the whisker at the most extreme value that is not an outlier. Outliers and cleaning data
- Use the addition rule for probabilities \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Mutually exclusive and independent events; Venn and tree diagrams
- Show whether two events are independent Compare \(P(A \cap B)\) with \(P(A) \times P(B)\). Mutually exclusive and independent events; Venn and tree diagrams
- Apply the conditional probability formula \(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\), using a Venn diagram or given values. Conditional probability
- Show independence using conditional probability Check whether \(P(A \mid B) = P(A)\) to decide whether A and B are independent. Conditional probability
- Compare a model's predictions with observed data Decide whether observed frequencies are close to the expected ones, and what that says about the model. Modelling with probability
- Use cumulative binomial probabilities with inequalities Rewrite ‘more than’, ‘at least’ and ‘between’ in terms of \(P(X \le r)\). Discrete distributions, including the discrete uniform and the binomial
- State binomial conditions and assumptions in context Fixed number of trials, two outcomes, independent trials, constant probability of success. Discrete distributions, including the discrete uniform and the binomial
- Find a value from a given probability Use inverse normal, remembering that most calculators work with the area to the left. The normal distribution
- Use the points of inflection at \(\mu \pm \sigma\) The mean is halfway between the points of inflection and \(\sigma\) is the distance from the mean to either one. The normal distribution
- Find the parameters of the approximating normal \(\mu = np\) and \(\sigma^2 = np(1 - p)\). The normal approximation to the binomial; choosing a distribution
- Apply a continuity correction correctly Extend half a unit beyond the end integers you want, e.g. \(P(X \le 40) \approx P(Y \lt 40.5)\). The normal approximation to the binomial; choosing a distribution
- Define critical region and critical value The critical region is the set of values that lead to rejecting \(H_0\); the critical value is its boundary. The language of hypothesis testing
- Use a p-value to reach a decision Reject \(H_0\) if the p-value is less than the significance level. The language of hypothesis testing
- Find a one-tailed critical region Find the boundary where the tail probability is no more than the significance level. Binomial hypothesis tests
- Carry out a two-tailed correlation test Use the one-tailed critical value for half the significance level and compare \(|r|\) with it. Tests for correlation and for the mean of a normal distribution
- State the distribution of the sample mean If \(X \sim N(\mu, \sigma^2)\), then \(\bar{X} \sim N\left(\mu, \dfrac{\sigma^2}{n}\right)\). Tests for correlation and for the mean of a normal distribution
- State what each modelling term means For example, light means the mass is ignored and inextensible means the string does not stretch. SI units and modelling assumptions
- Find speed and direction from a velocity vector Speed is \(\sqrt{a^2 + b^2}\) for \(a\mathbf{i} + b\mathbf{j}\); find the direction using \(\tan\theta\) and a sketch. Displacement, velocity and acceleration
- Use area to find an unknown time Write the total area in terms of the unknown and set it equal to the given distance. Kinematics graphs
- Solve two-stage and two-object problems The final velocity of one stage is the initial velocity of the next; objects meet when their displacements are equal. Constant acceleration (suvat)
- Integrate to find velocity or displacement Integrate, then use the initial conditions to find the constant of integration. Variable acceleration using calculus
- Find when a particle is at rest Solve \(v = 0\); for the greatest or least velocity, solve \(a = 0\). Variable acceleration using calculus
- Solve horizontal projection problems Vertically \(u = 0\) and the acceleration is \(g\); horizontally the velocity is constant; the time links them. Projectiles
- Find time of flight, range and greatest height At the greatest height the vertical velocity is zero; on landing the vertical displacement is known. Projectiles
- Use equilibrium to find unknown forces In equilibrium the resultant force is zero, so the components in each direction add to zero. Forces, resultants and equilibrium of a particle
- Use \(F = ma\) with forces as vectors Add the force vectors and divide by the mass to get the acceleration vector. Newton's second law and vectors
- Apply \(F = ma\) to vertical motion For an object lifted by a cable, \(T - mg = ma\). Weight and motion under gravity
- Find the reaction force in a lift Apply \(F = ma\) to the person; the force on the floor is equal and opposite (Newton's third law). Newton's third law, connected particles and pulleys
- Solve problems with particles over a pulley Apply \(F = ma\) to each particle, with the same acceleration and the same tension. Newton's third law, connected particles and pulleys
- Find the acceleration on a smooth slope Use \(F = ma\) parallel to the slope; the forces perpendicular to the slope balance. Resolving forces and inclined planes
- Handle a pulling force at an angle The horizontal component drives the motion; the vertical component changes the normal reaction. Resolving forces and inclined planes
- Understand when friction is limiting Friction equals \(\mu R\) only when the object is moving or about to move; otherwise \(F \leqslant \mu R\). Friction
- Include an angled force with friction A force with a vertical component changes \(R\), and so changes the limiting friction \(\mu R\). Friction
- State the conditions for equilibrium of a rigid body The resultant force is zero and the resultant moment about any point is zero. Moments, beams and rods in equilibrium
- Find the centre of mass of a non-uniform rod Put the weight at an unknown distance \(x\) from one end and take moments. Moments, beams and rods in equilibrium
- Recognise when a beam is about to tilt On the point of tilting about one support, the reaction at the other support is zero. Ladders, tilting and limiting equilibrium
- Find limits on loads without tilting Take moments about the support it would tilt about, with the other reaction set to zero. Ladders, tilting and limiting equilibrium
Grade B
- Prove an expression is always positive Complete the square to get \((x - a)^2 + k\) with \(k \gt 0\), then use \((x - a)^2 \ge 0\). Proof by deduction, exhaustion and counter-example
- Prove simple number results by contradiction e.g. to prove 'if \(n^2\) is odd then \(n\) is odd', assume \(n\) is even and show \(n^2\) is even. Proof by contradiction
- Solve equations using a common base e.g. \(8^x = 4^{x + 1}\) gives \(2^{3x} = 2^{2x + 2}\), so \(x = 2\). Laws of indices
- Solve equations with fractional or negative powers e.g. \(x^{-\frac{3}{2}} = \frac{1}{8}\) gives \(x^{\frac{3}{2}} = 8\), so \(x = 8^{\frac{2}{3}} = 4\). Laws of indices
- Rationalise denominators using the conjugate For \(a + \sqrt{b}\), multiply top and bottom by \(a - \sqrt{b}\); the denominator becomes \(a^2 - b\). Surds
- Find the range of k for given roots Form an inequality in \(k\) from the discriminant and solve it, often as a quadratic inequality. Quadratics and the discriminant
- Solve a quadratic in a function of x Put \(u = x^2\), \(u = \sqrt{x}\) or \(u = 2^x\), solve for \(u\), then find \(x\). Quadratics and the discriminant
- Use the discriminant to decide if graphs meet Substitute the line into the curve, form a quadratic and test the sign of \(b^2 - 4ac\). Simultaneous equations
- Solve inequalities using graphs \(f(x) \gt g(x)\) where the graph of \(f\) is above the graph of \(g\); find the intersections first. Inequalities
- Use factors of the form ax − b \((ax - b)\) is a factor of \(f(x)\) if and only if \(f\left(\frac{b}{a}\right) = 0\). Polynomials, algebraic division and the factor theorem
- Solve modulus equations and inequalities Solve both \(ax + b = g(x)\) and \(-(ax + b) = g(x)\), then check each answer with a sketch. Curve sketching and the modulus function
- State the domain and range of an inverse The domain of \(f^{-1}\) is the range of \(f\); the range of \(f^{-1}\) is the domain of \(f\). Composite and inverse functions
- Solve equations involving composite functions Form \(fg(x)\) algebraically, set it equal to the value and solve. Composite and inverse functions
- Sketch |f(x)| and f(|x|) from y = f(x) \(|f(x)|\) reflects parts below the \(x\)-axis upwards; \(f(|x|)\) reflects the \(x \ge 0\) part in the \(y\)-axis. Transformations of graphs
- Handle a repeated linear factor For \((x - b)^2\) you need both \(\frac{B}{x - b}\) and \(\frac{C}{(x - b)^2}\). Partial fractions
- Compare coefficients to find a constant Equate the coefficients of \(x^2\) (or the constants) when substitution does not give every constant. Partial fractions
- Find a maximum using completed square form Write the model as \(a - b(t - c)^2\) to read off the maximum \(a\) when \(t = c\). Modelling with functions
- Criticise the limitations of a model Give a reason in context, e.g. it predicts negative values or unlimited growth, or relies on extrapolation. Modelling with functions
- Find the perpendicular bisector of a segment Use the midpoint of the segment and the negative reciprocal of its gradient. Straight lines
- Use chord and semicircle properties The perpendicular bisector of a chord passes through the centre; an angle in a semicircle is \(90^\circ\). Circles
- Find where a line meets a circle Substitute the line into the circle's equation and solve the resulting quadratic. Circles
- State the domain and range of the curve Use the range of the parameter: \(x = 2\cos t\) gives \(-2 \le x \le 2\). Parametric equations
- Sketch a curve given parametrically Recognise the Cartesian form (e.g. a circle) or plot points, and draw only the part given by the parameter. Parametric equations
- Find greatest heights and distances Find the vertex of \(y(t)\) by completing the square or by symmetry, then find the position. Parametric equations in modelling
- Convert a model to a Cartesian path Eliminate \(t\) to get \(y\) in terms of \(x\), with a domain that matches the context. Parametric equations in modelling
- Expand \((a+bx)^n\) and state its validity Take out \(a^n\) to get \(a^n\left(1+\frac{b}{a}x\right)^n\), expand, and give the valid range \(|x| \lt \left|\frac{a}{b}\right|\). Binomial expansion
- Describe a sequence as increasing, decreasing or periodic Use the definitions: \(u_{n+1} \gt u_n\) for all n, \(u_{n+1} \lt u_n\) for all n, or terms repeating in a cycle. Sequences and recurrence relations
- Evaluate a geometric series in sigma notation A rule such as \(3 \times 2^r\) gives a geometric series: find the first term by substituting, then use \(S_n\) or \(S_\infty\). Sigma notation
- Find the number of terms from a sum Solve the quadratic in n from \(S_n\), then choose the correct positive integer. Arithmetic sequences and series
- Find and use the sum to infinity Check that \(|r| \lt 1\), then use \(S_\infty = \frac{a}{1-r}\). Geometric sequences and series
- Find when a model reaches a target Form an equation or inequality in n and solve it with a quadratic or with logarithms. Modelling with sequences and series
- Find the area of a segment Segment = sector − triangle \(= \frac{1}{2}r^2(\theta - \sin\theta)\). Sine and cosine rules, radians, arcs and sectors
- Approximate an expression for small θ Replace each trig function with its approximation and simplify to an expression in θ. Small angle approximations
- Replace θ by a multiple such as 3θ \(\sin 3\theta \approx 3\theta\) and \(\cos 3\theta \approx 1 - \frac{(3\theta)^2}{2} = 1 - \frac{9\theta^2}{2}\). Small angle approximations
- Estimate a value and its percentage error Substitute a small angle into the approximation and compare with the calculator value in radian mode. Small angle approximations
- Find maximum, minimum and period after transformations For \(y = a\sin(bx + c) + d\) with b > 0: maximum \(d + |a|\), minimum \(d - |a|\), period \(\frac{360^\circ}{b}\). Graphs of sin, cos and tan
- Sketch the graphs of sec, cosec and cot The asymptotes are where the function you take the reciprocal of is zero; where cos or sin is 1 or −1, so is its reciprocal. sec, cosec, cot and inverse trig functions
- State domains and ranges of inverse functions arcsin: domain \([-1, 1]\), range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\); arccos: domain \([-1, 1]\), range \([0, \pi]\); arctan: domain all real numbers, range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). sec, cosec, cot and inverse trig functions
- Solve equations in sec, cosec and cot Turn them into cos, sin or tan, e.g. \(\sec 2x = -2\) becomes \(\cos 2x = -\frac{1}{2}\). sec, cosec, cot and inverse trig functions
- Derive the sec and cosec identities Divide \(\sin^2\theta + \cos^2\theta = 1\) by \(\cos^2\theta\), or by \(\sin^2\theta\). Pythagorean identities
- Solve quadratic equations using an identity Replace \(\cos^2 x\) by \(1 - \sin^2 x\) (or similar) to get a quadratic in one function. Pythagorean identities
- Use given ratios to find compound values e.g. given tan A and sin B, find \(\cos(A + B)\) or \(\sin 2A\) exactly, taking care with signs. Compound and double angles; R cos(θ ± α)
- Write \(a\cos\theta + b\sin\theta\) in R form Expand \(R\cos(\theta - \alpha)\) and compare coefficients: \(R = \sqrt{a^2 + b^2}\) and \(\tan\alpha = \frac{b}{a}\). Compound and double angles; R cos(θ ± α)
- Solve quadratic equations in sin, cos or tan Factorise, solve each factor, and reject any impossible value. Solving trigonometric equations
- Prove identities involving sec, cosec and cot Write everything in terms of sin and cos, then simplify. Proving trigonometric identities
- Use double angle formulae in proofs Replace \(\sin 2\theta\) and \(\cos 2\theta\) with expressions in θ, choosing the form of \(\cos 2\theta\) that simplifies best. Proving trigonometric identities
- Interpret the parameters of a trig model In \(y = a + b\sin(ct)\) with b > 0, a is the mean level, b is the amplitude and \(\frac{360^\circ}{c}\) is the period. Trigonometry in context
- Sketch transformed exponential graphs For \(y = A\mathrm{e}^{kx} + c\), show the y-intercept \((0, A + c)\) and the asymptote \(y = c\). Exponential functions and graphs
- Find exact axis intercepts using logarithms Set \(y = 0\) and solve with ln, e.g. \(4 - \mathrm{e}^{2x} = 0\) gives \(x = \frac{1}{2}\ln 4 = \ln 2\). Exponential functions and graphs
- Find where the gradient has a given value Solve an equation such as \(6\mathrm{e}^{2x} - 5 = 7\) using ln. The function eˣ and its gradient
- Sketch transformed log graphs E.g. \(y = \ln(x + 3)\) has asymptote \(x = -3\) and meets the axes at \((-2, 0)\) and \((0, \ln 3)\). Logarithms
- Solve equations involving logarithms Combine into one log, then change to index form, e.g. \(\log_2 x(x - 2) = 3 \Rightarrow x(x - 2) = 8\). Laws of logarithms
- Solve equations with different bases E.g. \(2^{x+3} = 5^x\): take logs, expand, collect the \(x\) terms and factorise. Solving exponential equations
- Find \(a\) and \(n\) from a line The gradient is \(n\); the intercept \(c\) is \(\log a\), so \(a = 10^{c}\) (or \(a = \mathrm{e}^{c}\) if the axes use ln). Using log graphs to estimate parameters
- Find \(k\) and \(b\) from a line The intercept \(c\) is \(\log k\) and the gradient \(m\) is \(\log b\), so \(k = 10^{c}\) and \(b = 10^{m}\). Using log graphs to estimate parameters
- Find the constants from given data Use the starting value for \(A\), then a second value to find \(k\) using ln. Exponential growth and decay models
- Differentiate \(x^2\) or \(x^3\) from first principles Expand \(f(x + h)\), simplify \(\frac{f(x+h) - f(x)}{h}\), then let \(h \to 0\). First principles, second derivatives and concavity
- Find where a curve is convex or concave Convex where \(f''(x) \geq 0\); concave where \(f''(x) \leq 0\). First principles, second derivatives and concavity
- Differentiate \(a^{kx}\) and \(\tan kx\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{kx}) = ka^{kx}\ln a\) and \(\frac{\mathrm{d}}{\mathrm{d}x}(\tan kx) = k\sec^2 kx\). Differentiating standard functions
- Find where a function is increasing or decreasing Solve \(f'(x) \geq 0\) (increasing) or \(f'(x) \leq 0\) (decreasing). Tangents, normals and stationary points
- Use the quotient rule \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{u}{v}\right) = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}\) (given in the formula booklet). Product, quotient and chain rules
- Use \(\frac{\mathrm{d}y}{\mathrm{d}x}=1\div\frac{\mathrm{d}x}{\mathrm{d}y}\) When \(x\) is given in terms of \(y\), differentiate with respect to \(y\) and take the reciprocal. Product, quotient and chain rules
- Find an unknown rate at an instant Differentiate the formula first, then substitute the value at that instant. Connected rates of change
- Differentiate products such as \(xy\) implicitly Use the product rule: \(\frac{\mathrm{d}}{\mathrm{d}x}(xy) = x\frac{\mathrm{d}y}{\mathrm{d}x} + y\). Implicit and parametric differentiation
- Find tangents and normals to parametric curves Find the value of \(t\), the point \((x, y)\) and the gradient, then use \(y - y_1 = m(x - x_1)\). Implicit and parametric differentiation
- Find the constant of proportionality Substitute a known rate and value at the same moment, e.g. \(-5 = -k\sqrt{400}\) gives \(k = 0.25\). Constructing differential equations
- Integrate twice from a second derivative Each integration brings a new constant, so you need two conditions, such as a point on the curve and a stationary point. The fundamental theorem of calculus
- Reverse standard derivatives such as tan kx Because \(\frac{\mathrm{d}}{\mathrm{d}x}(\tan kx) = k\sec^2 kx\), \(\int \sec^2 kx\,\mathrm{d}x = \frac{1}{k}\tan kx + c\). Integrating standard functions
- Deal with regions below the x-axis The integral is negative there, so split the integral at each root and add the sizes of the separate answers. Definite integrals and areas under curves
- Find the area between two curves Integrate the difference of the two functions between the intersection points. Areas between curves and parametric areas
- Handle regions bounded by tangents or normals Find the line's equation and where it meets the axis, then add or subtract areas. Areas between curves and parametric areas
- Evaluate the limit by integrating Integrate and substitute the limits to give an exact value. Integration as the limit of a sum
- Use a given substitution to integrate Replace every \(x\) and \(\mathrm{d}x\) so that the integral is entirely in \(u\), then integrate. Integration by substitution
- Change the limits for a definite integral Convert each \(x\)-limit to a \(u\)-limit so that you never need to go back to \(x\). Integration by substitution
- Integrate products involving ln x Let \(u = \ln x\); for \(\int \ln x\,\mathrm{d}x\), write it as \(\int 1 \times \ln x\,\mathrm{d}x\). Integration by parts
- Evaluate definite integrals by parts exactly Apply the limits to both \(uv\) and the remaining integral. Integration by parts
- Integrate a repeated-factor term \(\int \frac{B}{(ax + b)^2}\,\mathrm{d}x = -\frac{B}{a(ax + b)} + c\): a power, not a logarithm. Integration using partial fractions
- Combine logarithms into an exact answer Use the laws of logarithms to write the answer in the form the question asks for. Integration using partial fractions
- Find a particular solution from a condition Substitute the given values of \(x\) and \(y\) to find the constant. Separable differential equations
- Rearrange a logarithmic solution into y = … From \(\ln|y| = \mathrm{f}(x) + c\), write \(y = A\mathrm{e}^{\mathrm{f}(x)}\), where \(A\) is a constant. Separable differential equations
- Find a time or value from a solution Substitute into the solution and rearrange, using logs to solve for \(t\). Differential equations in context: interpreting solutions
- Interpret terms and constants in context Say what each term represents, with units, e.g. a constant inflow rate or a rate proportional to the amount present. Differential equations in context: interpreting solutions
- Explain how a sign change can mislead A vertical asymptote (a break in the graph) can give a sign change with no root. Locating roots by change of sign
- Explain how roots can be missed With no sign change there may still be two roots, or a root where the curve just touches the axis. Locating roots by change of sign
- Explain convergence or divergence from iterates Say whether the values approach a limit, and whether it is the root you want. Iteration and staircase/cobweb diagrams
- Iterate to the accuracy required Repeat using ANS until the values agree to the accuracy asked. The Newton–Raphson method
- Explain the method using tangents \(x_{n+1}\) is where the tangent to the curve at \(x = x_n\) crosses the \(x\)-axis. The Newton–Raphson method
- State over- or underestimate with a reason Convex curve (bending upwards): overestimate. Concave curve (bending downwards): underestimate. Numerical integration: the trapezium rule
- Explain how to improve the estimate Use more strips (a smaller \(h\)), so the trapezia fit the curve more closely. Numerical integration: the trapezium rule
- Interpret numerical answers in context Give units and convert where sensible, e.g. 7.154 hours is about 7 hours 9 minutes. Numerical methods in context
- Describe edges and diagonals of cuboids Write edges and diagonals of a cuboid in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) from the coordinates of its vertices. Vectors in 2D and 3D
- Find the direction of a 2D vector Use \(\tan\) with a sketch to find the angle with the positive \(x\)-axis in the correct quadrant. Magnitude and direction
- Prove that three points are collinear Show that two of the vectors joining them are parallel and share a point. Vector arithmetic and geometry
- Find a missing vertex of a parallelogram Use equal opposite sides: in \(ABCD\), \(\overrightarrow{AD} = \overrightarrow{BC}\), so \(\mathbf{d} = \mathbf{a} + \mathbf{c} - \mathbf{b}\). Position vectors and distances
- Find speed and direction from velocity Speed is \(|\mathbf{v}|\); the direction is an angle or bearing found with trigonometry. Vector problems
- Prove the type of a quadrilateral Compare the sides as vectors (parallel, equal) and as lengths. Vector problems
- Give advantages and disadvantages in context Link each point to the situation, e.g. no list of shoppers exists, so a quota sample is practical. Sampling methods and the large data set
- Find a histogram's scale from a known bar When area is not equal to frequency, use one bar to find the frequency per unit of area. Histograms, box plots and cumulative frequency
- Recognise distinct groups in a scatter diagram Spot separate clusters (e.g. two species) and explain why one line or one \(r\) for all the points may mislead. Correlation and regression
- Use coding to find mean and standard deviation Reverse the coding for the mean; for the standard deviation only the scaling matters. Measures of location and spread
- Choose a suitable diagram for the data Match the diagram to the data and the purpose, e.g. box plots to compare distributions. Outliers and cleaning data
- Decide whether an outlier should be removed Remove it only if it is an error; keep a genuine extreme value. Outliers and cleaning data
- Use mutually exclusive and independent conditions Use \(P(A \cap B) = 0\) or \(P(A \cap B) = P(A)P(B)\) to find missing probabilities. Mutually exclusive and independent events; Venn and tree diagrams
- Reverse a tree diagram Find the probability of a first-stage event given a second-stage event: one path divided by the total of all paths ending in that event. Conditional probability
- Criticise an assumption in context Give a specific reason why an assumption may not hold in the situation described. Modelling with probability
- Criticise a binomial model in context Explain why independence or a constant probability may not hold. Discrete distributions, including the discrete uniform and the binomial
- Find an unknown mean or standard deviation Standardise: \(\dfrac{x - \mu}{\sigma} = z\), with \(z\) from the inverse normal for \(Z \sim N(0, 1)\). The normal distribution
- Approximate binomial probabilities using a normal distribution Combine the parameters, the continuity correction and normal CD. The normal approximation to the binomial; choosing a distribution
- Find the actual significance level of a test Calculate the probability that the test statistic falls in the critical region when \(H_0\) is true. The language of hypothesis testing
- Find critical regions for a two-tailed test Allow at most half the significance level in each tail. Binomial hypothesis tests
- Find the actual significance level Find the probability of the critical region under \(H_0\) (adding both tails for a two-tailed test). Binomial hypothesis tests
- Test a normal mean using a p-value Find \(P(\bar{X} \ge \bar{x})\) or \(P(\bar{X} \le \bar{x})\) under \(H_0\) and compare it with the significance level. Tests for correlation and for the mean of a normal distribution
- Explain how an assumption was used Link the assumption to a step in your working, e.g. a smooth pulley means the tension is the same on both sides. SI units and modelling assumptions
- Use \(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\) for constant velocity Write the position vector at time \(t\) and use it to answer questions about where the object is. Displacement, velocity and acceleration
- Compare two objects on one graph Equal areas from the same start mean the objects are level; lines crossing only means equal velocities. Kinematics graphs
- Derive the constant acceleration formulae Use the gradient and area of a straight-line velocity–time graph, then eliminate variables. Constant acceleration (suvat)
- Interpret both roots of a quadratic in \(t\) Two positive roots mean the object is at that displacement twice, for example on the way out and on the way back. Constant acceleration (suvat)
- Differentiate and integrate vectors in terms of \(t\) Work on the \(\mathbf{i}\) and \(\mathbf{j}\) components separately; the constant of integration is a vector. Variable acceleration using calculus
- Find the speed and direction at a given time Combine the horizontal and vertical velocity components using Pythagoras and \(\tan\theta\). Projectiles
- Use the vector form of projectile motion With \(\mathbf{j}\) vertically up, \(\mathbf{a} = -g\mathbf{j}\), so \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) and \(\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\). Projectiles
- Resolve forces given by magnitude and direction A force \(F\) at angle \(\theta\) to a direction has components \(F\cos\theta\) along it and \(F\sin\theta\) perpendicular to it. Forces, resultants and equilibrium of a particle
- Find an unknown force from vector motion Find \(\mathbf{a}\) from the motion, work out the resultant force \(m\mathbf{a}\), then subtract the known forces. Newton's second law and vectors
- Solve problems with two objects under gravity Write both heights in terms of \(t\), measured from the same point, and equate them. Weight and motion under gravity
- Find the force exerted on a pulley It is the resultant of the two tensions that act on the pulley. Newton's third law, connected particles and pulleys
- Solve equilibrium on a slope with a horizontal force Resolve the horizontal force and the weight parallel and perpendicular to the slope. Resolving forces and inclined planes
- Solve rough inclined plane problems Resolve perpendicular to the slope for \(R\), then use \(F = ma\) or equilibrium along the slope. Friction
- Find the moment of a force at an angle Use the perpendicular component \(F\sin\theta\), or the perpendicular distance \(d\sin\theta\). Moments, beams and rods in equilibrium
- Draw the forces on a ladder correctly Smooth wall: horizontal reaction only; rough ground: normal reaction up and friction towards the wall. Ladders, tilting and limiting equilibrium
- Find \(\mu\) for a ladder in limiting equilibrium Resolve both ways, take moments about the foot, then use \(F = \mu R\). Ladders, tilting and limiting equilibrium
Grade A
- Use remainders to cover every integer Write \(n\) as \(3k\), \(3k + 1\) or \(3k + 2\) (or similar) and prove the result for each form. Proof by deduction, exhaustion and counter-example
- Prove that the square root of 2 is irrational Assume \(\sqrt{2} = \frac{a}{b}\) in lowest terms, then show \(a\) and \(b\) are both even. Proof by contradiction
- Prove that there are infinitely many primes Assume a finite list of all primes, multiply them together and add 1, and show this has a prime factor not in the list. Proof by contradiction
- Simplify expressions with several different bases e.g. \(\frac{2^{x + 1} \times 4^{x}}{8^{x - 1}} = 2^{x + 1 + 2x - 3x + 3} = 2^4 = 16\). Laws of indices
- Solve linear equations with surd coefficients Collect the \(x\) terms, factorise, divide and rationalise to give an exact answer. Surds
- Use the discriminant for tangency conditions Substitute a line into a curve and use \(b^2 - 4ac = 0\) for the line to be a tangent. Quadratics and the discriminant
- Handle curves with xy and y-squared terms Substitute carefully, expanding brackets like \((3 - x)^2\) and \(x(3 - x)\) in full. Simultaneous equations
- Solve inequalities with x in the denominator Multiply both sides by the square of the denominator, which is positive, then solve. Inequalities
- Find unknown coefficients from given factors Each known factor gives an equation, e.g. \(f(-1) = 0\); solve the equations simultaneously. Polynomials, algebraic division and the factor theorem
- Find values of k for a number of intersections Use a sketch (and sometimes the discriminant) to find where a line or curve meets a graph a given number of times. Curve sketching and the modulus function
- Restrict domains and find ranges of composites Restrict a many-to-one function to one side of its turning point so it has an inverse; find the range of \(fg\) by applying \(f\) to the range of \(g\). Composite and inverse functions
- Apply combinations of transformations in order e.g. \(y = 2f(x - 1) + 3\): move right 1, stretch vertically by factor 2, then move up 3. Transformations of graphs
- Split an improper algebraic fraction If the numerator's degree is at least the denominator's, include a polynomial part, e.g. \(A + \frac{B}{x - 1} + \frac{C}{x + 2}\). Partial fractions
- Use partial fractions to integrate or expand Each simple fraction can be integrated (giving a \(\ln\) term) or expanded with the binomial series. Partial fractions
- Suggest and justify refinements to a model e.g. include air resistance, or use a function that levels off instead of growing forever. Modelling with functions
- Solve multi-step problems such as triangle areas Find vertices from intercepts and intersections, then use \(\frac{1}{2} \times\) base \(\times\) height. Straight lines
- Use the discriminant for tangency conditions The line is a tangent when the substituted quadratic has \(b^2 - 4ac = 0\), and misses when \(b^2 - 4ac \lt 0\). Circles
- Find the circle through three points The centre is where the perpendicular bisectors of two chords meet. Circles
- Convert harder parametric equations with domains e.g. using \(1 + \tan^2 t = \sec^2 t\), or with a restricted parameter, and state \(y = f(x)\) with its domain. Parametric equations
- Model circular motion with trigonometric parameters e.g. \(x = r\sin(kt)\), \(y = h - r\cos(kt)\) for a point on a wheel turning at a steady rate. Parametric equations in modelling
- Criticise and refine a parametric model e.g. air resistance, spin and wind are ignored, so the real path is not exactly a parabola. Parametric equations in modelling
- Use an expansion to approximate a value Choose a value of x inside the valid range that turns the expression into the number you want, then substitute it. Binomial expansion
- Find the order and sum of periodic sequences Find the length of the cycle, then add many terms using complete cycles plus the leftover terms. Sequences and recurrence relations
- Prove that a sequence is increasing Show that \(u_{n+1} - u_n \gt 0\) for every positive integer n, not just for a few terms. Sequences and recurrence relations
- Change the limits of a sum Use \(\sum_{r=k}^{n} f(r) = \sum_{r=1}^{n} f(r) - \sum_{r=1}^{k-1} f(r)\), or find the first term, last term and number of terms directly. Sigma notation
- Find an unknown limit from a given sum Write the sum in terms of n, set it equal to the value and solve, keeping only a positive integer n. Sigma notation
- Prove the formula for the sum Write the series forwards and backwards, add the two lines, and divide by 2. Arithmetic sequences and series
- Use logarithms to find the least n Rearrange to get \(r^n\) on its own, take logs, and take care with the direction of the inequality. Geometric sequences and series
- Prove the formula for the sum Write \(S_n\), multiply by r, subtract and rearrange. Geometric sequences and series
- Model regular savings with compound interest Each payment earns interest for a different number of years, so the total is a geometric series. Modelling with sequences and series
- Form equations from sector and triangle facts Combine the formulae to show that θ satisfies an equation such as \(3\theta = 4\sin\theta\). Sine and cosine rules, radians, arcs and sectors
- Find the approximate value of a fraction Approximate the top and bottom separately, then cancel the powers of θ, e.g. to get \(\frac{9}{4}\). Small angle approximations
- Find constants from a graph's features Use the maximum, the minimum and the position of a turning point to find the unknown constants. Graphs of sin, cos and tan
- Simplify expressions involving inverse trig functions e.g. show that \(\tan(\arcsin x) = \frac{x}{\sqrt{1 - x^2}}\), using an identity or a right-angled triangle. sec, cosec, cot and inverse trig functions
- Solve equations with sec and tan together Use \(\tan^2\theta = \sec^2\theta - 1\) to get a quadratic in sec θ, then change to cos θ. Pythagorean identities
- Solve equations using the R form Rewrite as \(R\cos(\theta - \alpha) = c\), adjust the interval for \(\theta - \alpha\), solve and convert back. Compound and double angles; R cos(θ ± α)
- Find maximum and minimum values using R form The maximum of \(R\cos(\theta - \alpha)\) is R and the minimum is −R; use this in fractions and in models. Compound and double angles; R cos(θ ± α)
- Use identities before solving Use a Pythagorean or double angle identity to get an equation in one trig function. Solving trigonometric equations
- Combine fractions to prove an identity Use a common denominator, then simplify using \(\sin^2 x + \cos^2 x = 1\) and factorising. Proving trigonometric identities
- Use a proved identity to solve an equation Replace the complicated expression with the simpler side of the identity, then solve in the interval. Proving trigonometric identities
- Solve a trig model for given values Set the model equal to the value, solve the trig equation in the interval and interpret the answers as times. Trigonometry in context
- Use exponential graphs to count solutions Sketch both sides of an equation such as \(\mathrm{e}^x = 3 - x\) and explain why the graphs meet exactly once. Exponential functions and graphs
- Explain why exponential models suit many situations For \(y = A\mathrm{e}^{kx}\), \(\frac{\mathrm{d}y}{\mathrm{d}x} = ky\): the rate of change is proportional to the amount present. The function eˣ and its gradient
- Find inverses of exponential and log functions E.g. for \(f(x) = 3 + \mathrm{e}^{2x}\), \(f^{-1}(x) = \frac{1}{2}\ln(x - 3)\) with domain \(x > 3\). Logarithms
- Solve log equations leading to quadratics Form and solve a quadratic, then check every root in the original equation, rejecting any that make a log's argument zero or negative. Laws of logarithms
- Solve disguised quadratics in \(a^x\) Write \(9^x = (3^x)^2\), substitute \(u = 3^x\), solve, then reject any \(u \leq 0\). Solving exponential equations
- Solve equations with \(\mathrm{e}^x\) and \(\mathrm{e}^{-x}\) Multiply every term by \(\mathrm{e}^x\) to get a quadratic in \(\mathrm{e}^x\). Solving exponential equations
- Interpret the constants and criticise the model E.g. \(k\) is the value when \(x = 0\) and \(b\) is the multiplier per unit of \(x\); predictions far outside the data are unreliable. Using log graphs to estimate parameters
- Find and interpret a rate of change Differentiate the model and substitute; say whether the quantity is increasing or decreasing, with units. Exponential growth and decay models
- Find and justify points of inflection Solve \(f''(x) = 0\) and show that \(f''(x)\) changes sign there. First principles, second derivatives and concavity
- Simplify an expression before differentiating Split fractions, expand brackets and use log laws, e.g. \(\ln x^3 = 3\ln x\). Differentiating standard functions
- Solve optimisation problems Form an expression in one variable, differentiate, solve, then justify the maximum or minimum. Tangents, normals and stationary points
- Prove the derivatives of sec, cosec and cot Write e.g. \(\sec x = (\cos x)^{-1}\) and use the chain rule or the quotient rule. Product, quotient and chain rules
- Combine rules and factorise to find stationary points Factorise answers like \(2x(1 - x)\mathrm{e}^{-2x}\) so that \(\frac{\mathrm{d}y}{\mathrm{d}x} = 0\) is easy to solve. Product, quotient and chain rules
- Use similar triangles to link variables In a cone of water, write \(r\) in terms of \(h\) before finding \(\frac{\mathrm{d}V}{\mathrm{d}h}\). Connected rates of change
- Use chains of three or more rates E.g. \(\frac{\mathrm{d}V}{\mathrm{d}t} = \frac{\mathrm{d}V}{\mathrm{d}x} \times \frac{\mathrm{d}x}{\mathrm{d}A} \times \frac{\mathrm{d}A}{\mathrm{d}t}\). Connected rates of change
- Find stationary points on implicit curves Set the numerator of \(\frac{\mathrm{d}y}{\mathrm{d}x}\) equal to 0, then solve together with the curve's equation. Implicit and parametric differentiation
- Form equations with inflow and outflow Net rate = rate in − rate out, e.g. \(\frac{\mathrm{d}V}{\mathrm{d}t} = 40 - kV\). Constructing differential equations
- Use the chain rule to change the variable Turn a rate of change of volume into a rate of change of depth with \(\frac{\mathrm{d}h}{\mathrm{d}t} = \frac{\mathrm{d}V}{\mathrm{d}t} \div \frac{\mathrm{d}V}{\mathrm{d}h}\). Constructing differential equations
- Find an unknown limit or constant Set a definite integral equal to its given value, solve the equation and reject any value that breaks a given condition. The fundamental theorem of calculus
- Use trigonometric identities before integrating Replace \(\sin^2 x\), \(\cos^2 x\) or \(\tan^2 x\) using the double angle formulae or \(\tan^2 x = \sec^2 x - 1\), then integrate. Integrating standard functions
- Find exact areas involving e, ln or π Integrate standard functions, then simplify using laws of logarithms and exact trigonometric values. Definite integrals and areas under curves
- Find the area under a parametric curve Use \(\int y\frac{\mathrm{d}x}{\mathrm{d}t}\,\mathrm{d}t\) and change the \(x\)-limits into \(t\)-limits. Areas between curves and parametric areas
- Find upper and lower bounds with rectangles Use the greatest and least heights on each strip; the true area lies between the two totals. Integration as the limit of a sum
- Explain why narrower strips give better bounds As \(\delta x \to 0\) the upper and lower sums both approach the value of the integral. Integration as the limit of a sum
- Choose a suitable substitution yourself Let \(u\) be the awkward part: the inside of a root, a bracket or an exponent, or the denominator. Integration by substitution
- Apply integration by parts twice For \(x^2\mathrm{e}^{x}\) or \(x^2\sin x\), the first application leaves \(x\) times a function, so apply the formula again. Integration by parts
- Divide first for improper fractions If the degree of the numerator is at least that of the denominator, divide to get a polynomial plus a proper fraction. Integration using partial fractions
- Factorise before separating For example \(\frac{\mathrm{d}y}{\mathrm{d}x} = xy^2 + y^2 = y^2(x + 1)\): take out the common factor first. Separable differential equations
- Find the long-term (limiting) value As \(t \to \infty\), \(\mathrm{e}^{-kt} \to 0\) for \(k \gt 0\); the limit is also where the rate of change is zero. Differential equations in context: interpreting solutions
- Evaluate a model and state its limitations Compare predictions with data, say why the model may not hold, and suggest a sensible refinement. Differential equations in context: interpreting solutions
- Count roots using a sketch Sketch \(y = \mathrm{g}(x)\) and \(y = \mathrm{h}(x)\): the number of intersections is the number of roots of \(\mathrm{g}(x) = \mathrm{h}(x)\). Locating roots by change of sign
- Use the gradient of g to predict behaviour Near the root, \(|\mathrm{g}'(x)| \lt 1\) gives convergence: a staircase if \(\mathrm{g}'\) is positive, a cobweb if it is negative. Iteration and staircase/cobweb diagrams
- Explain why a starting value fails If \(\mathrm{f}'(x_0) = 0\) the tangent is horizontal and never meets the axis; near a stationary point, \(x_1\) can land far away. The Newton–Raphson method
- Use the estimate for a related integral For example \(\int_a^b (k + \mathrm{f}(x))\,\mathrm{d}x = k(b - a) + \int_a^b \mathrm{f}(x)\,\mathrm{d}x\). Numerical integration: the trapezium rule
- Compare an estimate with the exact value Percentage error \(= \frac{|\text{estimate} - \text{exact}|}{\text{exact}} \times 100\%\). Numerical integration: the trapezium rule
- Judge the reliability of an estimate Consider over- or underestimates, how often the data were recorded, and whether the model holds. Numerical methods in context
- Find the angle between a vector and an axis In 3D, \(\cos\theta_x = \frac{x}{|\mathbf{a}|}\), and similarly for the \(y\)- and \(z\)-axes. Magnitude and direction
- Find unknowns from a given magnitude Set the sum of the squared components equal to the magnitude squared and solve, keeping both roots unless one is ruled out. Magnitude and direction
- Find where lines meet by comparing routes Write the same vector in two ways with unknown scalars, and compare the coefficients of two non-parallel vectors. Vector arithmetic and geometry
- Find angles in a triangle from coordinates Find all three side lengths, then use the cosine rule. Position vectors and distances
- Find unknown coordinates from a distance Form an equation from the distance squared and solve it, usually a quadratic. Position vectors and distances
- Solve for unknown times or constants For example, 'due north' means the \(\mathbf{i}\)-components of the two position vectors are equal. Vector problems
- Critique a sampling method and suggest improvements Identify who is left out or over-represented, and why, and propose a random method that fixes it. Sampling methods and the large data set
- Compare distributions using location and spread in context Compare a median (or mean) and an IQR (or standard deviation), with values and in the context of the data. Histograms, box plots and cumulative frequency
- Linearise \(y=ax^n\) or \(y=kb^x\) using logarithms Take logs of both sides: \(\log y\) is linear in \(\log x\) for a power model, and in \(x\) for an exponential model. Correlation and regression
- Combine data sets using \(\Sigma x\) and \(\Sigma x^2\) Rebuild \(\Sigma x = n\bar{x}\) and \(\Sigma x^2 = n(\sigma^2 + \bar{x}^2)\) for each set, add them, then recalculate. Measures of location and spread
- Clean data and recalculate summary statistics Take errors and missing values out of \(n\), \(\Sigma x\) and \(\Sigma x^2\), then recalculate. Outliers and cleaning data
- Find unknown probabilities by forming equations Set up and solve equations from the total probability of 1 and the conditions given. Mutually exclusive and independent events; Venn and tree diagrams
- Solve conditional problems with unknown probabilities Use \(P(A \cap B) = P(A)P(B \mid A)\) and the addition rule to form and solve equations. Conditional probability
- Refine a model with more realistic assumptions Use relative frequencies or conditional probabilities instead of equal or constant probabilities. Modelling with probability
- Find \(n\) or \(p\) from a probability condition E.g. solve \(1 - (1 - p)^n \gt 0.9\) using logarithms. Discrete distributions, including the discrete uniform and the binomial
- Find \(\mu\) and \(\sigma\) using simultaneous equations Form two standardised equations and solve them together. The normal distribution
- Choose and justify a distribution for a context Decide between discrete uniform, binomial and normal, and explain when none of them fits. The normal approximation to the binomial; choosing a distribution
- Interpret test results and their limitations Explain that not rejecting \(H_0\) does not prove it, and why the actual and stated significance levels differ. The language of hypothesis testing
- Carry out a complete test in context Hypotheses, distribution, probability or critical region, decision and conclusion in context. Binomial hypothesis tests
- Find a critical region for the sample mean Use \(\mu_0 \pm z\dfrac{\sigma}{\sqrt{n}}\) with the right \(z\) value. Tests for correlation and for the mean of a normal distribution
- Criticise a model and suggest a refinement Name one specific effect the model ignores, such as air resistance, and say how including it would change the answer. SI units and modelling assumptions
- Solve problems with two moving objects Write both position vectors in terms of \(t\), then equate components or use the vector between them. Displacement, velocity and acceleration
- Handle graphs with negative velocity Area below the time axis is negative displacement; for the distance, add the sizes of all the areas. Kinematics graphs
- Use suvat in vector form Use \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) and \(\mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2\) with \(\mathbf{i}\) and \(\mathbf{j}\) components. Constant acceleration (suvat)
- Use calculus with exponential or trigonometric functions Apply the Pure rules, e.g. \(v = 4e^{-0.5t}\) gives \(a = -2e^{-0.5t}\). Variable acceleration using calculus
- Find the distance when the direction changes Split the time interval where \(v = 0\), find each displacement and add their magnitudes. Variable acceleration using calculus
- Derive general projectile formulae Find the time of flight, range and greatest height in terms of \(U\), \(\alpha\) and \(g\). Projectiles
- Solve equilibrium problems with strings at angles Resolve horizontally and vertically and solve the simultaneous equations for the tensions. Forces, resultants and equilibrium of a particle
- Solve problems where the forces change When a force is removed or changes, find the new acceleration and link the stages by the velocity. Newton's second law and vectors
- Handle motion starting above the ground Use a negative displacement for a point below the start, solve the quadratic and reject the negative root. Weight and motion under gravity
- Analyse motion after a string goes slack Once a particle hits the ground the tension is zero, and the other particle continues with the speed it had. Newton's third law, connected particles and pulleys
- Solve multi-stage problems on inclined planes Find the new acceleration when a force is removed and link the stages with suvat. Resolving forces and inclined planes
- Decide whether a particle moves Compare the friction needed for equilibrium with the greatest friction available, \(\mu R\). Friction
- Solve hinged rod problems with non-parallel forces Take moments about the hinge, then resolve to find the components of the force at the hinge. Moments, beams and rods in equilibrium
- Solve ladder problems with a person climbing Put the person at distance \(x\) up the ladder and find the value of \(x\) at which it is about to slip. Ladders, tilting and limiting equilibrium
Grade A*
- Construct a full proof in an unfamiliar context Choose a method, state every assumption, justify each step and finish with a precise conclusion. Proof by deduction, exhaustion and counter-example
- Adapt the method to unfamiliar statements e.g. show there are no integers with \(x^2 - y^2 = 10\), or that \(\sqrt[3]{2}\) is irrational. Proof by contradiction
- Expand a fraction using partial fractions first Split into partial fractions, expand each one, add the series, and give the narrower of the validity ranges. Binomial expansion
- Solve problems combining sums, terms and convergence Form equations from several conditions, solve for r and reject any value with \(|r| \ge 1\) if a sum to infinity is involved. Geometric sequences and series
- Criticise or refine a sequence model Explain in context why the model may fail and how it could be improved. Modelling with sequences and series
- Use approximations in a geometry problem Combine them with the cosine rule or with area formulae when an angle in a diagram is small. Small angle approximations
- Eliminate θ between two equations e.g. from \(x = 2\operatorname{cosec}\theta\) and \(y = 5\cot\theta\), use \(1 + \cot^2\theta = \operatorname{cosec}^2\theta\). Pythagorean identities
- Derive new identities from compound angles e.g. write \(\sin 3x\) in terms of \(\sin x\) by expanding \(\sin(2x + x)\). Compound and double angles; R cos(θ ± α)
- Factorise instead of dividing to keep solutions e.g. \(\sin 2x = \sin x\) gives \(\sin x(2\cos x - 1) = 0\), so the solutions of \(\sin x = 0\) are not lost. Solving trigonometric equations
- Prove identities using compound angles e.g. prove that \(\sin 3x \equiv 3\sin x - 4\sin^3 x\) by expanding \(\sin(2x + x)\). Proving trigonometric identities
- Use the R form to analyse a model Combine \(p\cos t + q\sin t\) into one term to find the maximum, the minimum and when they occur. Trigonometry in context
- Refine a model and describe long-term behaviour E.g. \(\theta = 20 + 60\mathrm{e}^{-0.1t}\) tends to 20 as \(t \to \infty\): the temperature of the room. Exponential growth and decay models
- Differentiate \(\sin x\) or \(\cos x\) from first principles Use the addition formulae with \(\frac{\sin h}{h} \to 1\) and \(\frac{\cos h - 1}{h} \to 0\) as \(h \to 0\). First principles, second derivatives and concavity
- Find where tangents are parallel to an axis Horizontal: \(\frac{\mathrm{d}y}{\mathrm{d}x} = 0\); vertical: the denominator of \(\frac{\mathrm{d}y}{\mathrm{d}x}\) is 0. Implicit and parametric differentiation
- Simplify parametric area integrals with identities Use trigonometric identities to turn \(y\frac{\mathrm{d}x}{\mathrm{d}t}\) into something you can integrate, e.g. \(\sin 2t\sec^2 t = 2\tan t\). Areas between curves and parametric areas
- Use trigonometric substitutions with identities For example \(x = 2\sin\theta\) turns \(\sqrt{4 - x^2}\) into \(2\cos\theta\); then use a double angle formula. Integration by substitution
- Solve integrals where the original reappears For \(\int \mathrm{e}^x\sin x\,\mathrm{d}x\), two applications bring back the original integral \(I\); rearrange to find \(I\). Integration by parts
- Use partial fractions inside longer problems Partial fractions often appear after a substitution or when separating the variables in a differential equation. Integration using partial fractions
- Separate when integrals need further methods The \(y\)-side may need partial fractions, e.g. \(\int \frac{1}{y^2 - 1}\,\mathrm{d}y\), and the \(x\)-side may need parts or substitution. Separable differential equations
- Solve and interpret logistic-type models Use partial fractions to solve \(\frac{\mathrm{d}P}{\mathrm{d}t} = kP(M - P)\), then interpret the limiting value \(M\). Differential equations in context: interpreting solutions
- Derive the formula from the tangent Write the tangent \(y - \mathrm{f}(x_n) = \mathrm{f}'(x_n)(x - x_n)\), put \(y = 0\) and solve for \(x\). The Newton–Raphson method
- Combine calculus with numerical methods For example, set a derivative equal to zero to find a maximum, then solve the resulting equation numerically. Numerical methods in context
- Reason with knowledge of the large data set Use facts about the stations, dates, variables and abbreviations to explain patterns or problems in a sample. Sampling methods and the large data set
- Link a histogram's shape to a probability model Recognise that a roughly symmetrical, bell-shaped histogram suggests a normal model, and treat areas as probabilities. Histograms, box plots and cumulative frequency
- Find model constants from a logarithmic regression line Match the line to \(\log y = \log a + n\log x\) or \(\log y = \log k + x\log b\) and undo the logs. Correlation and regression
- Find unknown data values from summary statistics Form and solve equations (sometimes quadratics) using \(\Sigma x\), \(\Sigma x^2\), the mean or the variance. Measures of location and spread
- Evaluate how cleaning affects the conclusions Explain which statistics change a lot (mean, standard deviation, range, \(r\)) and which barely change (median, IQR). Outliers and cleaning data
- Solve three-event Venn problems with conditions Combine independence, mutual exclusivity and the total of 1 in a three-circle Venn diagram. Mutually exclusive and independent events; Venn and tree diagrams
- Handle multi-step conditional problems in context Work through several conditions in an unfamiliar context, organising them in a tree, table or Venn diagram. Conditional probability
- Explain how changed assumptions affect probabilities Say whether a more realistic model gives a higher or lower probability, and why. Modelling with probability
- Combine normal and binomial models in multi-stage problems E.g. find \(p = P(X \gt a)\) from a normal model, then use \(B(n, p)\). The normal distribution
- Judge a model against evidence E.g. show that a normal model gives impossible values, or compare an approximation with the exact probability. The normal approximation to the binomial; choosing a distribution
- Find a sample size that gives significance E.g. find the smallest \(n\) for which \(X = 0\) is in the critical region. Binomial hypothesis tests
- Find the sample size needed for significance Rearrange the standardised inequality to find the smallest \(n\). Tests for correlation and for the mean of a normal distribution
- Find angles of projection through a point Use the equation of the path with \(\sec^2\alpha = 1 + \tan^2\alpha\) to get a quadratic in \(\tan\alpha\). Projectiles
- Find the range of forces for equilibrium Use both limiting cases: about to slide up (friction down the slope) and about to slide down (friction up the slope). Friction
- Solve rod-on-peg and rough-wall problems A smooth peg's reaction is perpendicular to the rod; at every rough contact that is about to slip, \(F = \mu R\). Ladders, tilting and limiting equilibrium
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