Lessons

Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)

Disproving a universal ("for all") claim with a single example where it fails — shown with the arithmetic worked out. Why one counter-example is enough to disprove, why no number of supporting examples can prove, and where to hunt for the case that breaks a statement.

Direct proof — starting from things known to be true and reasoning in valid algebraic steps to the conclusion. Representing the general case (2n, 2n+1, consecutive integers), "show that" manipulations, completing the square for positivity, and always finishing with a clear conclusion.

Proving a statement by splitting it into a finite, complete set of cases and checking every one — either a short list of values, or all integers split by parity or remainder. Why the cases must leave no gaps, and why exhaustion only works when the cases really are finite.

Simplifying, multiplying, dividing, adding and subtracting fractions with polynomials — by factorising and cancelling only common factors.

How composite functions chain together (and why order matters), and how to find an inverse — what it does, how to build it, and the link to its graph.

What a function is, the meaning of domain and range, how to find a range (and why a quadratic's is bounded), and one-to-one vs many-to-one mappings.

Sketching standard curves from their key features — intercepts, repeated-root behaviour, and the asymptotes of reciprocals — and finding where curves meet.

Reading an inequality as a region of the plane — which side of a curve to shade, when the boundary is dashed or solid, and how overlapping inequalities define a region.

What |x| means, how to sketch y = |f(x)|, and how to solve modulus equations and inequalities — including why some "solutions" must be thrown out.

Splitting a single algebraic fraction back into a sum of simpler ones — including the extra term a repeated linear factor needs.

The factor theorem — why f(a) = 0 means (x − a) is a factor — and how to use it with algebraic division to factorise and solve cubics.

Three ways to solve a quadratic — factorising, completing the square, the formula — where the formula comes from, and what the discriminant tells you before you solve.

Solving two equations at once (including a line meeting a curve), and solving linear and quadratic inequalities — including the sign-flip and the inside/outside rule for quadratics.

The laws of indices — and why negative and fractional powers have to mean what they mean — plus how to simplify surds and rationalise a denominator.

The four graph transformations and the two reflections — why a change "inside" the function behaves backwards — and how to combine them.

Why (x − a)² + (y − b)² = r² is just the distance formula squared, reading the centre and radius off it, and completing the square to recover them from the expanded form x² + y² + 2gx + 2fy + c = 0.

Where a line meets a circle (substitute, then read the discriminant), and the three circle facts that solve coordinate problems: tangent ⊥ radius, the perpendicular bisector of a chord passes through the centre, and the angle in a semicircle is 90°.

The distance between two points as Pythagoras on the horizontal and vertical gaps (left in exact surd form), and the midpoint as the average of the coordinates — plus working backwards from a known midpoint.

Parallel lines share a gradient; perpendicular gradients multiply to −1 (the negative reciprocal). Using both — and the perpendicular bisector of a segment — to find equations of lines.

Defining a curve by parametric equations x = f(t), y = g(t), finding the coordinates of points on the curve, finding the parameter value at a given point, and reading off the range of x- and y-values from the domain of t.

Finding where a parametric curve meets the coordinate axes, a given line, or another curve — by solving for the parameter — and using a repeated root to detect a tangent.

Using parametric equations to model real situations where x and y each depend on time — projectiles and rotating points — reading positions, maxima and ranges from the model, and finding the Cartesian equation of the path.

Eliminating the parameter to find the Cartesian equation of a parametric curve, by substitution or rearrangement, and carrying the domain restriction from the parameter through to the Cartesian equation.

Converting trigonometric parametric equations to Cartesian form using identities — sin²+cos²≡1 for circles and ellipses, sec²−tan²≡1, and the double-angle formulae — and recognising the curve that results.

The gradient of a line as a constant rate of change, the three forms of a line's equation — y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 — and how to move between them.

Sequences with a constant common difference — the nth-term rule and the sum formula, and where each comes from.

Expanding (a + b)^n with binomial coefficients — Pascal's triangle and nCr — and picking out a single term or coefficient without expanding everything.

Expanding (1 + x)^n and (a + bx)^n when n is negative or fractional — the series form, when it's valid, and combining it with partial fractions.

Sequences with a constant common ratio — the nth term, the sum of n terms, and the sum to infinity (and exactly when it exists).

Reading and evaluating sums written with Σ, and generating sequences from a recurrence relation — including spotting periodic behaviour.

The compound-angle formulae for sin(A±B), cos(A±B) and tan(A±B) — the sign patterns to get right, and using them for exact values of non-standard angles, expanding, proving identities and solving equations.

The radian formulas for arc length, sector area and segment area, where they come from, and using them in reverse — finding a radius or angle from a given perimeter or area, and the perimeters of sectors and segments.

The double-angle formulae for sin 2A, cos 2A (three forms) and tan 2A, where they come from, choosing the right form of cos 2A, and using them to solve equations and find exact values.

The full method for solving trig equations in a range — quadratics in sin/cos, using sin²+cos²≡1 to convert a mixed equation, and transformed arguments like sin 2x and cos(x − 30°) where the interval must change.

The exact surd values of sin, cos and tan at 0, 30, 45, 60 and 90 degrees, where they come from (two special triangles plus the unit circle), and how to use them to answer "find the exact value" questions without a calculator.

The shapes of y = sin x, cos x and tan x from the unit circle — period, amplitude, symmetry and asymptotes — and why reading them off the graph is how you find ALL the solutions to a trig equation.

The two AS identities — tan ≡ sin/cos and sin²+cos² ≡ 1 — where they come from, and how to use them to simplify expressions and prove "show that" results.

The functions arcsin, arccos and arctan — why the domains of sin, cos and tan must be restricted to invert them, their domains, ranges and graphs, and evaluating exact values and composite expressions.

A reliable method for proving trig identities, drawing on the whole toolkit — Pythagorean, reciprocal, addition and double-angle identities — with the common tactics (one side at a time, convert to sin/cos, combine fractions, factorise).

Writing a sinθ ± b cosθ as a single R sin(θ ± α) or R cos(θ ∓ α), finding R and α by matching coefficients, and using the form to read off maxima/minima, solve equations and answer modelling questions.

What a radian is, converting between degrees and radians, the exact values at the special angles in radians, and solving trigonometric equations over an interval given in radians.

The three reciprocal functions secant, cosecant and cotangent — their definitions, the shape and key features of their graphs, exact values, and solving equations that involve them.

The two new Pythagorean identities 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ — deriving them from sin²+cos²≡1, and using them to prove identities, simplify expressions and turn equations into a solvable quadratic.

Solving any (non-right-angled) triangle — the sine rule, the cosine rule and the area formula ½ab·sinC — and how to choose the right one from what you're given.

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

Functions with the variable in the power, y = a^x — their graph shape, the asymptote and intercept — and the special exponential e^x whose gradient equals itself.

Real-world models of the form N = N0 e^{kt} (or A = a + b e^{kt}): finding the starting value, fitting the rate constant to data with logs, interpreting parameters, rates and limiting values.

The product, quotient and power laws — turning sums and differences of logs into a single log (and back) — plus log_a a = 1, log_a 1 = 0 and the reciprocal rule.

A logarithm is the inverse of an exponential — it answers "what power?". Converting between log and index form, evaluating logs, the special values, and natural logs (ln).

Turning curved laws into straight lines with logs — y = ax^n becomes log-log linear, y = ab^x becomes log-linear — then reading the constants off the gradient and intercept.

Taking logs to solve a^x = b, equations with e^x and ln, hidden quadratics in e^x or a^x, and log equations solved with the laws — always checking which roots are valid.

Differentiating a function of a function — y = f(g(x)) — by multiplying the outside derivative by the inside derivative, with the dy/dx = (dy/du)(du/dx) substitution method and the reciprocal connection dx/dy = 1 / (dy/dx).

Linking the rate of change of one quantity to another through a shared variable using the chain rule — dV/dt = (dV/dr)(dr/dt) — for problems like an expanding balloon, a draining cone or a sliding ladder.

Using the second derivative to decide where a curve is convex (f'' > 0) or concave (f'' < 0), and locating points of inflection where the concavity changes — distinct from a stationary point.

The standard results d/dx eˣ = eˣ, d/dx eᵏˣ = k eᵏˣ, d/dx aˣ = aˣ ln a and d/dx ln x = 1/x, why e is the special base, and combining them with the chain, product and quotient rules.

The definition every differentiation rule is built on — the gradient as the limit of a chord — and the "cancel the h, then let h → 0" technique for using it directly.

Differentiating a relation that mixes x and y — like x² + y² = 25 — without solving for y, by differentiating both sides with respect to x and treating y as a function of x (so d/dx of y² is 2y dy/dx).

Finding dy/dx for a curve given parametrically as dy/dx = (dy/dt) / (dx/dt), using it for tangents and normals, and the second derivative of a parametric curve.

Bring the power down to the front and knock it down by one — and why that rule is exactly what the gradient-of-a-chord limit always produces.

Differentiating a product of two functions, y = uv, as u'v + uv' — why the naive "differentiate each and multiply" is wrong, and how to combine the rule with the chain rule.

Differentiating a quotient y = u/v as (u'v - uv') / v² — getting the order in the numerator and the squared denominator right, and when to prefer rewriting as a product instead.

Where a curve is momentarily flat — found by setting dy/dx = 0 — and how the second derivative tells a maximum from a minimum, plus reading off where a curve is increasing or decreasing.

How to find the equation of the tangent and the normal to a curve at a point: get the gradient from the derivative, the point from the curve, then write a straight line — and why the normal's gradient is the negative reciprocal.

The six standard trig derivatives — sin→cos, cos→−sin, tan→sec², sec→sec tan, cosec→−cosec cot, cot→−cosec² — why x must be in radians, and combining them with the chain, product and quotient rules.

Using definite integrals for areas — the trap when a curve dips below the axis, and finding the area between two curves (or a curve and a line) as the integral of top minus bottom between their intersections.

Reversing the product rule to integrate a product — ∫u(dv/dx)dx = uv − ∫v(du/dx)dx — choosing u to differentiate to something simpler, the ∫ln x trick, and applying it twice.

Why every indefinite integral carries a "+C", what that constant actually represents, and how extra information (a point on the curve, an initial condition) pins it down.

Why evaluating F(b) − F(a) gives the area under a curve, what a negative integral actually means, and why "integral" and "area" are not always the same number.

Solving first-order separable differential equations by separating the variables and integrating both sides, finding a particular solution from a boundary condition, and modelling rates of change.

What integration actually is — running differentiation backwards — and why the power rule for integration ("raise the power, divide by the new power") is the exact undo of the differentiation rule.

Finding the area under a curve given parametrically as ∫ y (dx/dt) dt with the limits in t — and combining it with tangents/normals to find regions bounded by the curve, a line and an axis.

Splitting a rational function into partial fractions so each piece integrates to a logarithm — turning an impossible-looking fraction into a sum of ln terms.

Spotting integrals of the form ∫f'(x)[f(x)]ⁿ, ∫f'(x)/f(x) → ln|f(x)| and ∫f'(x)e^{f(x)} — "integration by recognition", where the derivative of the inside appears as a factor.

The integrals that are just the derivative rules in reverse — e^x, 1/x → ln|x|, sin, cos and sec² — together with the f(ax+b) rule for a linear inside function (integrate as normal, then divide by a).

The formal version of the reverse chain rule — substitute u = g(x), replace dx using du = g'(x) dx, integrate in u, then convert back (or change the limits for a definite integral).

Integrals like sin²x, cos²x and tan²x that have no direct rule — rewritten with the double-angle and Pythagorean identities into things you can integrate term by term.

Solving f(x) = 0 by rearranging it into x = g(x) and iterating x_(n+1) = g(x_n) from a starting value — why some rearrangements converge and others diverge, and reading staircase and cobweb diagrams.

Trapping a root of f(x) = 0 in an interval using the change-of-sign rule — why continuity is essential, what a sign change does and doesn't guarantee, and refining a root to a given number of decimal places.

A fast iteration that uses the tangent — x_(n+1) = x_n − f(x_n)/f'(x_n) — its geometric meaning, why it converges so quickly, and the cases where it fails (a horizontal tangent or a poor starting value).

Estimating a definite integral by slicing the area into trapezium-topped strips: the formula (h/2)[y₀ + yₙ + 2(y₁ + … + y_(n−1))], choosing h, and deciding whether the estimate is an over- or under-estimate from the curve's shape.

Using vectors to prove geometry: parallel vectors as scalar multiples, collinear points as parallel vectors through a shared point, dividing a line in a ratio, expressing figures in terms of base vectors, and comparing areas.

The magnitude of a vector as Pythagoras on its components (left as an exact surd), unit vectors as a vector scaled to length one, direction as an angle or bearing, and finding the angle in a vector triangle with the cosine rule.

Vectors as physical quantities — displacement, velocity and force — with constant-velocity motion r = r₀ + tv, speed as the magnitude of velocity, bearings from north, resultant forces and equilibrium, and "passes through O".

What a vector is (a size and a direction), the ways we write one (bold letter, arrow, column, i–j components), and how to add, subtract and scale vectors — both geometrically (nose-to-tail) and component by component.

Pinning points to the origin with position vectors, the "destination minus start" rule AB = b − a, the distance between two points as |b − a|, the midpoint as ½(a + b), and finding a point that divides a segment in a ratio.

The two sampling methods that don't use chance: quota sampling (fill preset quotas for each group, choosing respondents non-randomly) and opportunity (or convenience) sampling (use whoever is available). How they work, when they're used, and why quota is not the same as stratified sampling.

What a population, a census and a sample actually are; the sampling units and the sampling frame; why you would sample rather than take a census (and when a census is impossible); and how a sample can be unrepresentative — including the trap of combining overlapping samples.

The three sampling methods that use chance: simple random sampling (every sample equally likely), systematic sampling (every kth unit after a random start), and stratified sampling (proportional allocation across groups), with how to carry each out and the advantages and drawbacks that examiners ask for.

Bivariate data and scatter diagrams; describing correlation by its direction (positive/negative) and strength; why correlation does not imply causation; and using a given regression line y = a + bx to make and interpret predictions — including why prediction inside the data range (interpolation) is safe but outside it (extrapolation) is not. Interpreting the PMCC is flagged as A-level.

Building a cumulative frequency diagram and reading the median, quartiles and percentiles off it by plotting against the UPPER class boundary; testing for outliers with the 1.5×IQR rule or the mean ± 2 standard deviations rule; drawing a box plot whose whiskers stop at the most extreme value that is NOT an outlier; and comparing two distributions with a measure of location and a measure of spread.

Fitting y = ab^x and y = ax^n to data by taking logs to get a straight line, then reading the constants off a regression line of log y on x (or on log x), and using the model to predict.

Why a histogram for continuous data plots frequency DENSITY, not frequency, so that the AREA of each bar is proportional to the frequency — and how that one rule lets you find a missing frequency, a missing class width, or read a frequency off a drawn bar. Plus the frequency polygon, which joins class midpoints.

The three averages (mean, median, mode) and the other measures of location (quartiles and percentiles); how to find them from a list and from a frequency table; and how to estimate the median and quartiles of grouped continuous data by linear interpolation.

Range, interquartile and interpercentile range, and — the big one — variance and standard deviation from sums Σx and Σx² (and Σfx, Σfx²). Plus coding: how a linear change of variable rescales the mean and standard deviation, and how to decode back to the original data.

The product moment correlation coefficient r as a number between −1 and 1 measuring linear correlation — computing it from the summary statistics S_xy, S_xx, S_yy, interpreting it in context, and why coding leaves it unchanged.

Qualitative vs quantitative data, and within quantitative the split between discrete and continuous; how continuous data is grouped into classes, and how to read off the class boundaries, class width and midpoint you need before you can draw a histogram or estimate an average.

The probability of A given B, P(A|B) = P(A ∩ B) / P(B), as restricting the sample space to B — reading it off a Venn diagram, the multiplication formula P(A ∩ B) = P(B)P(A|B), and testing independence with P(A|B) = P(A).

Using tree diagrams where the second-stage branches are conditional probabilities — multiplying along branches for P(A ∩ B), adding the paths to an event, and reversing the condition to find P(first | second).

The two key relationships between events: mutually exclusive (they cannot both happen, so P(A∩B) = 0 and P(A∪B) = P(A) + P(B)) and independent (one does not affect the other, so P(A∩B) = P(A)×P(B)). How to use each rule, how to TEST whether two events are independent, and why mutually exclusive and independent are not the same thing.

Probability as the proportion of equally likely outcomes that are favourable; listing the sample space (including for two dice and two-way tables); the fact that probabilities of all outcomes sum to 1, and the complement rule P(A′) = 1 − P(A); and the difference between theoretical probability and experimental (relative frequency) probability.

Writing events with set notation — intersection ∩, union ∪ and complement ′ — matching each to a region of a Venn diagram, and the probability rules that go with them (P(A′) = 1 − P(A), P(A ∩ B′) = P(A) − P(A ∩ B)).

Using a tree diagram for two-stage experiments: multiply probabilities ALONG the branches of a path (AND), add across the paths that make up an event (OR), and use 1 − P(none) for "at least one". The crucial distinction between sampling WITH replacement (probabilities stay the same) and WITHOUT replacement (the totals, and so the later probabilities, change).

Using a Venn diagram to organise events: the intersection A∩B ("and"), the union A∪B ("or"), and the complement A′ ("not"); filling in the regions from the numbers in a problem (starting with the overlap); reading probabilities off the diagram; and the addition formula P(A∪B) = P(A) + P(B) − P(A∩B).

Finding P(X ≤ r), P(X < r), P(X ≥ r), P(X > r) and P(a ≤ X ≤ b) for a binomial distribution by building everything from the cumulative function P(X ≤ r) (binomcdf), with care over the off-by-one when an inequality is strict or "at least". Includes the complement for "≥" and "two-stage" problems where the success of one binomial becomes the trial of another.

When the count of "successes" in a fixed number of repeated trials follows a binomial distribution X ~ B(n, p): the four conditions (a fixed number of trials, two outcomes per trial, a constant probability of success, and independent trials), how to identify n and p from a context, how to state a condition for a binomial model, and when the binomial is NOT appropriate.

The binomial probability formula P(X = r) = nCr · p^r · (1−p)^(n−r), why it has that shape (choose which r of the n trials succeed, then multiply the success and failure probabilities), how to evaluate it including the special cases r = 0 and r = n, and how it matches the binompdf function on a calculator.

Working backwards from given probabilities to find an unknown mean or standard deviation (or both) by standardising to a z-value and solving — including a pair of simultaneous equations when both are unknown.

When n is large and p not too extreme, approximating B(n, p) by N(np, np(1−p)) — and applying the continuity correction (±0.5) when moving from the discrete binomial to the continuous normal.

The normal model N(μ, σ²) for continuous data — its symmetric bell shape, mean = median = mode = μ, spread set by σ, and standardising any normal variable to the standard normal Z = (X − μ)/σ.

Finding P(X < a), P(X > a) and P(a < X < b) for a normal distribution as areas under the curve, and the inverse problem — finding the value x for a given probability (the inverse normal).

What a discrete random variable is, and how a probability distribution lists each value it can take with the probability of that value — as a table or as a probability function. The key rule that the probabilities must sum to 1 (used to find an unknown), reading cumulative probabilities like P(X ≤ 2) off a distribution, and the special case of a discrete uniform distribution.

Testing whether a population has any linear correlation by comparing the sample PMCC r against a critical value from tables — choosing one- or two-tailed hypotheses on ρ, and concluding in context.

The critical-region method for a one-tailed test: the critical value is the boundary and the critical region is the set of outcomes that would lead to rejecting H0. How to find it (the smallest c with P(X ≥ c) ≤ α for an upper test, the largest c with P(X ≤ c) ≤ α for a lower test), why the discreteness of the binomial means the actual significance level is usually below the nominal α, and how to use the region to conclude.

The language and structure of a binomial hypothesis test: the population proportion p, the null hypothesis H0 : p = p0 and the alternative hypothesis H1, deciding between a one-tailed (p < p0 or p > p0) and a two-tailed (p ≠ p0) test from the wording, the test statistic X ~ B(n, p0) under H0, and what the significance level means.

Testing a claim about the mean μ of a normal population from a sample, using that the sample mean is N(μ, σ²/n) — forming the test statistic z = (x̄ − μ₀)/(σ/√n) and comparing it to the critical z-value.

Carrying out a one-tailed binomial hypothesis test by the probability (p-value) method: assume H0, model X ~ B(n, p0), find the probability of a result at least as extreme as the one observed in the direction of H1 — P(X ≥ x) for an upper test or P(X ≤ x) for a lower test — compare it with the significance level, and write a full conclusion in context.

Carrying out a two-tailed binomial test (H1 : p ≠ p0), where a change in either direction counts as evidence. Splitting the significance level α into two tails of α/2, finding both critical regions and the actual significance level (the sum of the two tail probabilities), the equivalent p-value method (compare the observed tail with α/2, or 2× the tail with α), and writing a full conclusion in context.

What each standard modelling word actually lets you ignore — particle, light, inextensible, smooth, rigid, uniform — so you can read an assumption off a question and know its consequence, the SI units mechanics is built on, and how to answer "state a limitation" by naming what the model leaves out.

The five suvat equations, why they all fall out of the velocity–time graph (gradient = acceleration, area = displacement), and the one condition that makes them valid — constant acceleration.

Why integrating acceleration gives velocity and integrating velocity gives displacement, where the suvat formulas actually come from, and why they silently break the moment acceleration varies.

What the gradient and the area mean on each kind of motion graph — gradient of displacement–time is velocity, gradient of velocity–time is acceleration, area under velocity–time is displacement — and how to model a journey as a graph and read unknowns off its area.

Why the a → v → r calculus chain works component-by-component in 2D, that speed is the magnitude of the velocity vector, and how "parallel to" and "perpendicular to" a direction become simple component conditions.

Why an object moving freely under gravity is just a constant-acceleration problem with a = ±g, how to choose a positive direction and translate "thrown up", "returns", "hits the ground" into suvat values, and why the up-and-down trip makes distance and displacement differ.

A particle in equilibrium has zero resultant, so resolving in two perpendicular directions gives two equations — used to find unknown forces or angles, including on rough slopes where friction is at its limit (F = μR) on the point of moving.

Why two bodies joined by a light inextensible string share one acceleration and one tension, the two-step method that runs every problem — whole system for the acceleration, a single body for the internal force — and why the connecting force is internal to the system but external to each body.

Two particles joined by a light inextensible string over a smooth pulley — one on a slope, one hanging — share the same acceleration and string tension; write F = ma along each particle's line of motion (including the slope component and friction) and solve simultaneously.

What each force on a free-body diagram is and which way it points (weight always vertically down, normal reaction perpendicular to the surface, tension pulling along a string), why forces add as vectors so a resultant is found component-by-component, and what "in equilibrium" means — the resultant is zero, so the components balance.

Friction as a force opposing motion, with magnitude up to a maximum F ≤ μR (and F = μR at the point of slipping) — on rough horizontal surfaces and slopes, the angle of friction tanα = μ, and F = ma with friction included.

Resolving the weight on a slope into components along (mg sinα) and perpendicular (mg cosα) to the surface, the normal reaction R = mg cosα, and applying F = ma along the slope (acceleration g sinα on a smooth slope).

Newton's three laws and the one equation that runs all of dynamics — the RESULTANT force equals mass times acceleration — applied along a line and to forces given as 2-D vectors, why weight is mg and the normal reaction is whatever balances the rest, and how a lift problem is just F = ma done vertically.

Why a light inextensible string over a smooth pulley gives the two masses the same tension and the same size of acceleration in opposite directions, the two-equations-add method for finding a and T, and the classic twist — when one mass lands the string goes slack and the other flies on freely under gravity.

Splitting a force at an angle into two perpendicular components, F cosθ and F sinθ — to add forces in any directions and to solve equilibrium by resolving in two perpendicular directions.

A rigid body in equilibrium needs both zero resultant force and zero resultant moment — using these two conditions to find the reaction forces on a supported beam, and taking moments about a support to eliminate an unknown.

The moment (turning effect) of a force as force × perpendicular distance from the pivot, measured clockwise or anticlockwise, in newton-metres — including forces acting at an angle to a rod.

Beams whose weight does not act at the centre — finding the weight or the position of the centre of mass by taking moments — and "on the point of tilting" problems, where the reaction at the support about to be left is zero.

A body projected horizontally moves under gravity: horizontal velocity stays constant while the vertical motion is free fall — treat the two directions independently with suvat.

The standard results for a projectile launched at speed u and angle θ over level ground — time of flight 2u sinθ/g, greatest height u²sin²θ/(2g), range u²sin2θ/g — and using them in reverse to find u or θ.

Resolving the launch velocity into u cosθ (horizontal, constant) and u sinθ (vertical, under gravity), then using suvat in each direction to find the time of flight, greatest height and range.