Maths Extension 1 syllabus, dot point by dot point

Model projectile motion in two dimensions using parametric equations and find range, maximum height, time of flight and trajectory equation

How do we model projectile motion in two dimensions, and what quantities (range, maximum height, time of flight) can we extract?

Describe motion in the plane using a position vector r(t), differentiate to get velocity v(t)=r'(t) and acceleration a(t)=r''(t), and use the dot product to analyse the motion (speed, perpendicular velocities, angle between v and a)

How do we describe the motion of a particle in the plane with a position vector, and what do its derivatives tell us about velocity and acceleration?

Differentiate inverse trigonometric functions and integrate functions that involve them

What are the derivatives and antiderivatives of the inverse trigonometric functions?

Differentiate an inverse function using (f^-1)'(x) = 1/f'(f^-1(x)) (equivalently dy/dx = 1/(dx/dy)), including the second derivative

How do we differentiate the inverse of a function from the function itself, and what is the gradient of the inverse at a point?

Model unrestricted growth and decay with

How do we model and solve problems involving exponential growth and decay using $\frac{dN}{dt} = k N$ ?

\frac{dN}{dt} = k N

and solve the resulting separable differential equation

Apply integration by substitution to evaluate definite and indefinite integrals, including reverse chain rule cases

How do we use the substitution method to evaluate integrals that arise from the reverse chain rule?

How do we link the rate of change of one quantity to the rate of change of another using the chain rule?

Solve separable first-order differential equations of the form

How do we solve a first-order differential equation when the variables separate?

\frac{dy}{dx} = f(x) g(y)

by separating variables and integrating both sides

Calculate volumes of revolution about the x-axis and y-axis using the disc method

How do we compute the volume of a solid generated by rotating a region around an axis?

Find the derivative of a function defined parametrically using the chain rule

How do we find the gradient of a curve when x and y are both given as functions of a parameter, and use it for tangents, normals and turning points?

Integrate functions whose antiderivative involves

Which integrals lead to inverse trigonometric antiderivatives, and how do we recognise them?

\arcsin

\arccos

\arctan

Combinatorics (ME-A1)

State and use the binomial theorem, identify general and specific terms, and relate it to Pascal's triangle

How do we expand $(a + b)^n$ , and what does Pascal's triangle reveal about the coefficients?

State and apply the pigeonhole principle in counting and existence problems

When can we guarantee that some box contains more than one object, and how do we apply this to counting and existence arguments?

10 min read →

How do we count the number of unordered selections (combinations) of objects from a larger set?

Use the combination formula

\binom{n}{r}

to count unordered selections, including with restrictions and complementary counting

Use the multiplication principle and the permutation formula to count ordered arrangements, including restrictions and repeated elements

How do we count the number of ordered arrangements of a set of objects?

Functions (ME-F1, ME-F2)

Sketch polynomial functions using leading-term behaviour, intercepts and the multiplicity of each root

How does the multiplicity of a root affect the graph of a polynomial near that root?

Sketch curves given parametrically, eliminate the parameter to obtain Cartesian equations, and use parametric form for circles, parabolas and lines

How do parametric equations describe a curve, and how do we convert between parametric and Cartesian forms?

Solve polynomial and rational inequalities by factoring and analysing the sign of each factor across critical values

How do we solve polynomial and rational inequalities using sign analysis?

Apply the division algorithm for polynomials and use the remainder and factor theorems to identify and verify factors and roots

How do we divide one polynomial by another and use the remainder and factor theorems to find roots?

Use the relationships between roots and coefficients (Vieta's formulas) for polynomials of degree two, three and four

How do the coefficients of a polynomial relate to the sum and product of its roots?

Proof (ME-P1)

Apply mathematical induction to prove general statements about a recursive sequence, a property of a formula, or a recursive procedure

How do we use induction to prove general statements involving a recursion, a formula or a property?

Prove identities for sums of series using the principle of mathematical induction

How do we use mathematical induction to prove identities involving sums of a sequence?

Prove inequalities involving an integer parameter

How do we use induction to prove an inequality holds for every positive integer?

n

using mathematical induction

Prove divisibility statements involving an integer parameter

How do we use mathematical induction to prove divisibility statements?

n

using mathematical induction

Statistical Analysis (ME-S1)

Define a Bernoulli random variable, compute its mean and variance, and recognise scenarios that fit the model

What is a Bernoulli trial, and what are its mean and variance?

10 min read →

What is the binomial distribution, and what are its mean and variance?

Define the binomial distribution

B(n, p)

, state its probability mass function, and find its mean

n p

and variance

n p (1 - p)

Use the normal approximation

When and how do we use the normal distribution to approximate binomial probabilities?

X \sim N(n p, n p (1 - p))

to approximate binomial probabilities for large

n

Use the sample proportion

How does the sample proportion behave as a random variable, and how do we use its normal approximation?

\hat{p} = X/n

as a random variable, with mean

p

, variance

pq/n

and standard deviation

\sqrt{pq/n}

, and apply its normal approximation

\hat{p} \sim N(p,\, pq/n)

Apply the central limit theorem to estimate the probability that the sample mean lies within given bounds

How much do sample means vary from sample to sample, and how does the central limit theorem let us find the probability that a sample mean lies within given bounds?

Compute exact probabilities for the binomial distribution including

How do we calculate probabilities involving the binomial distribution, including ranges and complements?

P(X = k)

P(X \le k)

P(X \ge k)

, and use complementary counting

Trigonometric Functions (ME-T1, T2, T3)

Define and sketch the inverse trigonometric functions

What are the inverse trigonometric functions, and what are their domains, ranges and graphs?

\arcsin

\arccos

, including their domains and ranges

\arctan

Use the product-to-sum and sum-to-product identities to simplify trigonometric expressions and integrals

How do we convert between products and sums of trigonometric functions?

Use the sum and difference identities for sine, cosine and tangent to expand or simplify trigonometric expressions

How do we expand $\sin(A \pm B)$ , $\cos(A \pm B)$ and $\tan(A \pm B)$ , and what are these identities used for?

Use the t-formula (Weierstrass substitution) to express

How does the t-substitution $t = \tan \frac{\theta}{2}$ help simplify and solve trigonometric equations?

\sin \theta

\cos \theta

\tan \theta

as rational functions of

t = \tan \frac{\theta}{2}

and use this to solve equations and find extreme values

How do we write $a \sin x + b \cos x$ as a single sinusoid, and what is this used for?

Express

a \sin x + b \cos x

in the form

R \sin(x + \alpha)

R \cos(x - \alpha)

Write general solutions to trigonometric equations using the period and the symmetries of

How do we find every solution to a trigonometric equation, not just the principal one?

\sin

\cos

\tan

Vectors (ME-V1)

Solve relative velocity problems involving constant crosswind/cross-current using vector diagrams, and describe the direction of a vector where required

How do we combine a velocity relative to the air or water with a constant wind or current, and find the resulting speed, direction or required heading?

Perform vector arithmetic with vectors in the plane, including component and column-vector notation, and find the magnitude and unit vector

How do we add, subtract and scale two-dimensional vectors, and how do we find their magnitudes?

Compute the scalar product of two vectors in component or geometric form and use it to find the angle between vectors and test orthogonality

What is the scalar (dot) product of two vectors, and what does it measure?

Compute the scalar and vector projection of one vector onto another and interpret it geometrically

How do we find the projection of one vector onto another, and what does it mean geometrically?

Use vector methods to prove geometric properties, including parallelism, perpendicularity, midpoint and ratio division

How do we use vectors to prove standard geometric results in the plane?

Write parametric vector equations of lines and convert between vector and Cartesian forms

How do we represent a line in the plane using vector equations?

Combinatorics (ME-A1)

Define and evaluate factorials, use the recursive relation n! = n(n-1)!, and simplify factorial expressions and fractions by unrolling and cancelling

What does the factorial n! count, and how do you simplify expressions built from factorials?

Count arrangements of n distinct objects around a circle as (n-1)! by treating rotations as identical, extend this to groups and blocks around a circle, alternating patterns and 'not together' restrictions handled by the complement, and recognise that for a necklace or bracelet, where a reflection also coincides, the count is (n-1)!/2

When objects are arranged in a closed ring rather than a line, which arrangements should count as the same, and how does that change the count from n! to (n-1)!?

Use the multiplication principle to count ordered selections across stages, count ordered selections with repetition as n^r and without repetition as nPr = n!/(n-r)!, and apply restriction techniques: deal with the difficulty first, fix a position, keep items together as a block, and count 'at least one' by the complement

How do you count the number of ordered ways to make a sequence of choices, and how does allowing or forbidding repetition change the count?

Count the distinct arrangements of n objects when some are identical: divide n! by the factorial of each repeat count to get n! over r1! r2! ... rk!, and handle the two-type special case where every object is one of two kinds (2^n arrangements in all, or choose the positions of one kind with a combination)

When some of the objects being arranged are identical, why does plain n! overcount, and how do you divide out the repeats to count only the genuinely different arrangements?

Count unordered selections (combinations) using ^{n}C_{r} = ^{n}P_{r}/r! = n!/(r!(n-r)!), recognise that an n-element set has 2^n subsets and that the r-element subsets number ^{n}C_{r} with the symmetry ^{n}C_{r} = ^{n}C_{n-r} and row sum 2^n, and apply the at-least, complement, conditional and geometric counting techniques to selection problems

When the order of a selection does not matter, how do you count the choices, and how do the restriction moves you learned for arrangements carry across to unordered selections?

Apply three reusable counting principles to ordered arrangements with restrictions: keep tied items together by grouping them into a block and ordering inside it, count an unwanted condition and subtract it from the total (the complement), and split an 'or' count into non-overlapping cases, subtracting the overlap by inclusion-exclusion when the cases meet

When an ordered arrangement carries a restriction, how do you choose between grouping items into a block, counting the complement, and splitting into cases with an overlap correction?

17 min read →

When every outcome is equally likely, how do you turn a probability into a ratio of two counts, and how do the counting moves (multiplication principle, permutations, combinations, complement, cases) supply both the favourable count and the total?

Compute probabilities of equally likely outcomes as P = (favourable outcomes)/(total outcomes) where both counts come from the counting methods (multiplication principle, ^{n}P_{r} and ^{n}C_{r}), and apply the complement, separate-pool multiplication, cases and inclusion-exclusion to find the favourable count for selection, card-hand, digit-string and with/without-replacement problems

19 min read →

Functions (ME-F1)

Define a curve parametrically by

How does a pair of equations $x = f(t)$ and $y = g(t)$ describe a curve, how do we eliminate the parameter $t$ to recover the Cartesian equation, and how do we read off the direction of travel and any excluded points?

x = f(t)

, eliminate the parameter

y = g(t)

t

by algebra or by a Pythagorean identity to obtain the Cartesian equation, parametrise standard curves (lines, parabolas, circles, ellipses and the rectangular hyperbola), and determine the direction of travel (orientation) and any excluded points

21 min read →

Given the graph of $y = f(x)$ , how do we sketch the graph of its reciprocal $y = 1/f(x)$ without first finding a formula?

Sketch the graph of

y = \dfrac{1}{f(x)}

from the graph of

y = f(x)

: turn zeroes into vertical asymptotes, send large values to small ones, fix the points where

f = \pm 1

, and flip a maximum to a minimum where

f

keeps its sign

by adding or subtracting ordinates: at a zero of one function the sum meets the other, opposite ordinates give a zero, equal ordinates double, and an oblique asymptote can emerge

Given the graphs of $y = f(x)$ and $y = g(x)$ , how do we sketch their sum and difference without first finding a formula?

Sketch the graph of

y = f(x) \pm g(x)

by reflecting the part of

Given the graph of $y = f(x)$ , how do we sketch $y = |f(x)|$ and $y = f(|x|)$ , and why are they two completely different transformations?

Sketch

y = |f(x)|

y = f(x)

below the

x

-axis upward, and sketch

y = f(|x|)

by keeping the part right of the

y

-axis and mirroring it across the

y

-axis, recognising that

y = f(|x|)

is always even and that the two transformations coincide only in special cases

20 min read →

How do we solve an inequation that contains an absolute value, or that has the unknown in the denominator, without losing or inventing solutions?

Solve absolute value inequations of the form |ax + b| < k, and inequations with the unknown in the denominator, by multiplying through by the square of the denominator

Form the inverse relation by reflecting

How do we form the inverse of a relation by swapping $x$ and $y$ , when is that inverse itself a function, and how do we find and verify a rule for $f^{-1}(x)$ ?

y = f(x)

in the line

y = x

, use the horizontal line test to decide whether the inverse is a function, find the rule for

f^{-1}(x)

and verify it by showing

f(f^{-1}(x)) = x

, swap the domain and range, and restrict a domain so that a many-to-one function gains an inverse

22 min read →

Where can a function change sign, and how do we use that to solve inequations and to begin a sketch?

Examine the sign of a function by building a table of test values that dodge around its zeroes and discontinuities, and use the resulting sign pattern to solve inequations and to start a curve sketch

17 min read →

Year 11: Polynomials

Define a polynomial and use the language of degree, leading term, leading coefficient, monic and the zero polynomial, including the degree of sums and products and equating coefficients of identically equal polynomials

What exactly is a polynomial, and how do its degree and coefficients control its behaviour?

Relate the coefficients of a polynomial to the elementary symmetric functions of its zeroes (with alternating signs) for quadratics, cubics and quartics, and use these relations to find a missing zero, to evaluate symmetric expressions of the zeroes such as the sum of squares and the sum of reciprocals, to find unknown coefficients from a condition on the zeroes, and to handle zeroes in arithmetic or geometric progression or of a special form

How do the coefficients of a polynomial already know the sum, the product and every symmetric combination of its roots, so that you can answer questions about the roots without ever finding them?

19 min read →

How do the leading term and the multiplicity of each zero control the shape of a polynomial graph?

Sketch the graph of a polynomial in factored form using the behaviour of the leading term for large x and the multiplicity of each zero, deciding where the curve crosses, touches or has a horizontal inflection, and locate a zero between integers from a table of values

Use the remainder theorem (the remainder on division by x - a is P(a)) and the factor theorem ((x - a) is a factor if and only if P(a) = 0), test divisors of the constant term to locate integer zeroes, and combine these with division to find unknown coefficients and to fully factor a polynomial

How can the remainder and factor theorems tell us the result of a division, and even fully factor a polynomial, without carrying out the long division?

17 min read →

How does long division split one polynomial by another into a quotient and a remainder, and what does the identity P(x) = D(x)Q(x) + R(x) tell us?

Divide one polynomial by another using long division, expressing the result in the form P(x) = D(x)Q(x) + R(x) where the remainder has degree less than the divisor, handling missing terms, and writing the result in the rational form P/D = Q + R/D

Apply the factor theorem, multiplicity and the sum and product of roots to the geometry of curves: a line is tangent to a curve exactly when the equation formed by solving them simultaneously has a double root; the x-coordinate of the midpoint of a chord is the average of the roots; and the number and nature of the intersections of two curves are read from the roots of the difference of the two polynomials

When a line meets a curve, what does it mean for the line to be a tangent rather than a secant, and how can the sum and product of the roots of one equation locate the point of contact, the midpoint of a chord, and a common tangent without any calculus?

20 min read →

Once the factor theorem links each zero to a factor, what does that force a polynomial to look like: how many zeroes can it have, when is it pinned down by its graph, and how many times can two curves meet?

Develop the structural consequences of the factor theorem: distinct zeroes give distinct linear factors and a degree-n polynomial has at most n zeroes; a polynomial agreeing with another at n + 1 points is identical, so a degree-n graph is fixed by n + 1 points; the number of intersections of two curves is bounded by degree via F(x) = P(x) - Q(x); and re-express a polynomial in powers of (x - a)