AS Maths (Edexcel): grade by grade

Edexcel 8MA0

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
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
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
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
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 and the area of a triangle
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 simple trigonometric equations
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 powers of x
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
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
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
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
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
Use the fact that probabilities sum to 1 Solve \(\Sigma P(X = x) = 1\) to find an unknown such as \(k\). Discrete distributions and the binomial 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
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
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

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 Multiply out, then collect like terms: \((3 + \sqrt{5})(2 - \sqrt{5}) = 1 - \sqrt{5}\). 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
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
Use proportional relationships and their graphs \(y \propto x^n\) means \(y = kx^n\); find \(k\) from one pair of values. Curve sketching
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
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
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
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 and the area of a triangle
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 simple trigonometric equations
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 powers of x
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
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
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
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
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
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 and the binomial 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
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
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

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 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
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
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 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 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
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 and the area of a triangle
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
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
Solve equations with multiple angles For \(\cos 2x = k\), solve for 2x in the doubled interval, then halve. Solving simple 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 simple trigonometric equations
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
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
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
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
Find unknowns by equating components If two vectors are equal, each pair of components is equal, giving equations to solve. Vectors in 2D
Recognise parallel vectors Parallel vectors are scalar multiples of each other: every component is multiplied by the same number. Vectors in 2D
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
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
Use cumulative binomial probabilities with inequalities Rewrite ‘more than’, ‘at least’ and ‘between’ in terms of \(P(X \le r)\). Discrete distributions and the binomial distribution
State binomial conditions and assumptions in context Fixed number of trials, two outcomes, independent trials, constant probability of success. Discrete distributions and the binomial 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
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
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

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
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
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
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
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
Solve quadratic equations in sin, cos or tan Factorise, solve each factor, and reject any impossible value. Solving simple trigonometric equations
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 function is increasing or decreasing Solve \(f'(x) \geq 0\) (increasing) or \(f'(x) \leq 0\) (decreasing). Tangents, normals and stationary points
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
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
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
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
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
Criticise a binomial model in context Explain why independence or a constant probability may not hold. Discrete distributions and the binomial 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
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 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

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
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
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
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
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 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
Use identities before solving Use a Pythagorean or double angle identity to get an equation in one trig function. Solving simple trigonometric equations
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
Simplify an expression before differentiating Split fractions, expand brackets and use log laws, e.g. \(\ln x^3 = 3\ln x\). Differentiating powers of x
Solve optimisation problems Form an expression in one variable, differentiate, solve, then justify the maximum or minimum. Tangents, normals and stationary points
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
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
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
Find \(n\) or \(p\) from a probability condition E.g. solve \(1 - (1 - p)^n \gt 0.9\) using logarithms. Discrete distributions and the binomial 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
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
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

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
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 simple trigonometric equations
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
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 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
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

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