One handout per topic, in plain English. Read the handout before the textbook, not after it — each one takes about five minutes and is designed to make the idea land first, so the formal version has somewhere to stick.
14 handoutsCambridge O LevelPrintableFree to copy and share
A function is a machine: put a number in, get exactly one number out — and the whole topic is about controlling that machine.
Picture itA vending machine takes a code and gives you one item. Press B4 and you always get the same thing. If pressing B4 sometimes gave crisps and sometimes chocolate, it would be broken — and that is precisely why a function must give one output per input.
Domain and range are the input and output sets
The domain is every value you are allowed to put in; the range is every value that can come out. For f(x) = √x the domain is x ≥ 0, because you cannot square-root a negative, and the range is f(x) ≥ 0.
Composite functions run one machine into another
fg(x) means do g first, then f. The order matters and is the commonest slip in the topic — fg(x) and gf(x) are usually different. Work from the inside out.
Inverse functions undo
f⁻¹ reverses f. Find it by writing y = f(x), swapping x and y, then rearranging for y. A function only has an inverse if it is one-to-one — each output comes from just one input — which is why the domain is sometimes restricted.
Modulus makes everything positive
|x| is the size of x ignoring its sign. Graphs of |f(x)| reflect any part below the x-axis up above it. Solving |x − 3| = 5 means considering both x − 3 = 5 and x − 3 = −5, giving two answers.
The bit that catches people outThe graph of f⁻¹ is the reflection of f in the line y = x, not in the x-axis or y-axis. That reflection is why swapping x and y is the algebraic method — the two are the same operation.
The grown-up words
What it means
What it is called
Note
Set of allowed inputs
domain
Watch for square roots and division by zero
Set of possible outputs
range
One function inside another
composite function
fg means g first
The function that undoes f
inverse function
f⁻¹, reflect in y = x
Each output from only one input
one-to-one
Required for an inverse to exist
Size ignoring sign
modulus
|x| — gives two cases when solving
Check you have got it
If f(x) = 2x + 1 and g(x) = x², find fg(3).
g first: g(3) = 9. Then f(9) = 19.
Why does f(x) = x² have no inverse over all real numbers?
It is not one-to-one — both 3 and −3 give 9, so the inverse would not know which to return. Restricting the domain to x ≥ 0 fixes it.
Edvia Free Resources · Additional Mathematics 4037 · Topic 1 — free to copy and share
Topic 2
Quadratic functions
Every quadratic is the same U-shaped curve, moved and stretched — and completing the square tells you exactly where it has been moved to.
Picture itThrow a ball and its path is a quadratic. Ask where it peaks and you are asking for the vertex. Complete the square and the peak's coordinates fall out of the algebra without any calculus at all.
Completing the square reveals the vertex
Rewriting ax² + bx + c as a(x + p)² + q tells you immediately that the vertex is at (−p, q). It also gives you the maximum or minimum value for free — q — and whether it is a maximum or minimum depends on the sign of a.
The discriminant counts the roots
b² − 4ac. If it is positive, two distinct real roots. If zero, one repeated root — which geometrically means the curve just touches the axis, so it is also the condition for a tangent. If negative, no real roots and the curve misses the axis entirely.
Three ways to solve, and when to use each
Factorise when the numbers are friendly. Complete the square when you want the vertex or an exact surd answer. Use the formula when nothing factorises. All three give the same roots.
Quadratic inequalities need the sketch
Find the roots, sketch the parabola, then read off where it is above or below the axis. For x² − 5x + 6 > 0 the curve is above the axis outside the roots, so x < 2 or x > 3. Trying to do this algebraically without the sketch is how people get the inequality backwards.
The bit that catches people out'Discriminant = 0' is the condition for a tangent, and questions rarely say the word 'discriminant' — they say 'the line is a tangent to the curve' and expect you to set the two equal, form a quadratic, and set b² − 4ac to zero.
The grown-up words
What it means
What it is called
Note
a(x + p)² + q form
completed square
Vertex at (−p, q)
b² − 4ac
discriminant
Counts the real roots
Turning point of the parabola
vertex
Maximum if a < 0, minimum if a > 0
Touches but does not cross
tangent
Discriminant = 0
Where the curve meets the x-axis
roots
Solutions of the equation
Check you have got it
For what value of k is y = kx − 2 a tangent to y = x² − 3x + 2?
Set equal: x² − (3+k)x + 4 = 0. Tangent means discriminant zero: (3+k)² − 16 = 0, so k = 1 or k = −7.
Express x² + 6x + 11 in completed square form and state the minimum value.
(x + 3)² + 2. The minimum is 2, at x = −3.
Edvia Free Resources · Additional Mathematics 4037 · Topic 2 — free to copy and share
Topic 3
Factors of polynomials
If substituting a number into a polynomial gives zero, you have found a factor — and that turns a hard cubic into an easy quadratic.
Picture itA cubic looks intimidating until you spot that x = 1 makes it vanish. Then (x − 1) divides out cleanly and you are left with a quadratic you can solve on sight. One lucky guess collapses the whole problem.
The factor theorem
If f(a) = 0 then (x − a) is a factor. To find that first root, try the factors of the constant term — for x³ − 6x² + 11x − 6, try ±1, ±2, ±3, ±6. One of them almost always works in an exam.
The remainder theorem
f(a) gives the remainder when f(x) is divided by (x − a). So the factor theorem is just the special case where the remainder happens to be zero. Questions about 'the remainder is 5' are asking you to set f(a) = 5.
Then divide and finish
Once you have one factor, use long division or comparing coefficients to get the quadratic, then factorise or use the formula. Three roots for a cubic, if they are all real.
Watch for (ax − b) factors
If the leading coefficient is not 1, the root may be a fraction. For 2x³ + …, try x = ½ as well as whole numbers — and the corresponding factor is (2x − 1), not (x − ½).
The bit that catches people outThe factor theorem uses (x − a) with a minus sign. If (x + 2) is a factor, the root is x = −2, so you substitute −2, not 2. Getting this sign wrong makes every subsequent step wrong while looking perfectly reasonable.
The grown-up words
What it means
What it is called
Note
f(a) = 0 means (x − a) is a factor
factor theorem
Try factors of the constant term
f(a) is the remainder on dividing by (x − a)
remainder theorem
Factor theorem is the zero case
Highest power of x
degree
A cubic has degree 3
Value that makes the polynomial zero
root
Corresponds to a factor
Dividing one polynomial by another
long division
Or compare coefficients
Check you have got it
Show that (x − 2) is a factor of x³ − 3x² + 4.
Substitute x = 2: 8 − 12 + 4 = 0. Since f(2) = 0, (x − 2) is a factor.
f(x) = x³ + 2x − 5. What is the remainder when divided by (x − 1)?
f(1) = 1 + 2 − 5 = −2, so the remainder is −2.
Edvia Free Resources · Additional Mathematics 4037 · Topic 3 — free to copy and share
Topic 4
Equations, inequalities and graphs
Sketching first turns most equation and inequality questions from algebra into reading a picture.
Picture itAsked to solve |2x − 1| < 5, most people start manipulating symbols and lose a sign. Sketch the V-shaped graph and a horizontal line at 5, and the answer is simply the bit of the V below the line — no sign errors possible.
Modulus equations split into two cases
|f(x)| = k means f(x) = k or f(x) = −k. Solve both, then check both answers in the original — squaring or splitting can produce solutions that do not actually work.
Modulus graphs bounce off the axis
The graph of y = |f(x)| is the graph of f with everything below the x-axis reflected upwards. Sketch f first, lightly, then flip the negative part. The corners are where f crossed the axis.
Inequalities need a sketch or a sign check
Find the critical values where the expression is zero or undefined, mark them on a number line, then test a point in each interval. Never multiply an inequality by something that might be negative without splitting into cases — it flips the sign.
Cubic and reciprocal graphs have recognisable shapes
Know what y = x³, y = 1/x and y = |x| look like before you plot anything. Most graph questions are transformations of a shape you already know, and recognising the parent shape saves the whole question.
The bit that catches people outWhen you multiply or divide an inequality by a negative number, the inequality sign reverses. −2x > 6 becomes x < −3, not x > −3. This is the single most common error in the topic and it is entirely avoidable by pausing whenever a negative appears.
The grown-up words
What it means
What it is called
Note
Size ignoring sign
modulus
Splits into two cases
Values where the expression is zero or undefined
critical values
Mark them on a number line
Sign flips on multiplying by a negative
inequality rule
The classic trap
Rough drawing showing shape and key points
sketch
Not a plotted graph
The basic shape before transformations
parent graph
x², x³, 1/x, |x|
Check you have got it
Solve |x − 4| = 3.
x − 4 = 3 gives x = 7; x − 4 = −3 gives x = 1. Both check out.
Solve −3x ≥ 12.
Divide by −3 and flip the sign: x ≤ −4.
Edvia Free Resources · Additional Mathematics 4037 · Topic 4 — free to copy and share
Topic 5
Simultaneous equations
Two equations, two unknowns — and when one of them is a curve, the answers tell you where a line meets it.
Picture itSolving a linear and a quadratic simultaneously is the same question as asking where a straight line crosses a parabola. Two solutions means it cuts twice, one means it is a tangent, none means it misses entirely.
Linear pairs: eliminate or substitute
Elimination when the coefficients match up neatly — multiply one equation through and add or subtract. Substitution when one equation is already arranged for a single letter. Both work; pick the one with less arithmetic.
One linear, one quadratic: always substitute
Rearrange the linear equation for one letter and put it into the quadratic. You get a quadratic in one variable, which you solve normally. Then substitute back to find the partner values — and remember each solution comes as a pair.
The number of solutions has a meaning
Two solutions: the line cuts the curve twice. One repeated solution: the line is a tangent. No real solutions: they never meet. Questions asking 'for what value of k is the line a tangent' are discriminant questions in disguise.
Always give answers in pairs
If x = 2 and x = 5, you must find the matching y for each. Writing four separate numbers with no pairing loses marks even when all four are correct.
The bit that catches people outSubstitute back into the linear equation, not the quadratic. It is simpler, and substituting into the quadratic can introduce a spurious partner value that does not satisfy both equations.
The grown-up words
What it means
What it is called
Note
Adding or subtracting to remove a letter
elimination
Match coefficients first
Replacing one letter with an expression
substitution
Best with a curve involved
Line touching a curve once
tangent
Repeated root, discriminant zero
Where two graphs meet
point of intersection
The simultaneous solution
Answers that belong together
solution pair
Each x with its own y
Check you have got it
Solve y = x + 1 and y = x² − 1 simultaneously.
x + 1 = x² − 1 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 or −1, giving (2, 3) and (−1, 0).
A line meets a curve at exactly one point. What does that tell you about the resulting quadratic?
Its discriminant is zero — there is a repeated root, so the line is a tangent.
Edvia Free Resources · Additional Mathematics 4037 · Topic 5 — free to copy and share
Topic 6
Logarithmic and exponential functions
A logarithm answers the question 'what power do I need?' — and it is the tool that turns curves into straight lines.
Picture itEarthquake magnitude, sound in decibels and pH are all logarithmic scales. A magnitude 7 quake is not slightly worse than a 6 — it releases about thirty times more energy. Logs exist because some quantities span ranges too vast for ordinary numbers.
Logs and powers are the same statement
log_a b = c means exactly a^c = b. Every log question is a power question rewritten. If you get stuck, convert to power form and the way forward usually becomes obvious.
Three rules do all the manipulation
log a + log b = log(ab) · log a − log b = log(a/b) · log(aⁿ) = n log a. That third one is the workhorse: it brings a power down to the front, which is how you solve equations with the unknown in the exponent.
Solving 2ˣ = 20
Take logs of both sides: x log 2 = log 20, so x = log 20 ÷ log 2 ≈ 4.32. Any equation with x in the power is solved this way.
e and ln are just a particular base
e ≈ 2.718 turns up naturally in growth and decay. ln means log to base e. All the same rules apply — there is nothing special to learn beyond the notation.
Logs straighten exponential graphs
If y = kxⁿ, then log y = n log x + log k, so plotting log y against log x gives a straight line of gradient n. If y = kaˣ, plotting log y against x gives a straight line. This is the main reason logs appear in practical work.
The bit that catches people outYou cannot take the log of a negative number or of zero. When you solve a log equation, always check your answers back in the original — solutions that make any log's argument zero or negative must be rejected, and questions expect you to say so explicitly.
The grown-up words
What it means
What it is called
Note
The power you need to raise the base to
logarithm
log_a b = c means a^c = b
log a + log b
log(ab)
Adding logs multiplies
log(aⁿ) = n log a
power rule
Brings the power down
Log to base e
natural logarithm (ln)
e ≈ 2.718
Turning a curve into a straight line
linearising
Plot log y against log x or x
Answers that must be discarded
rejected solutions
Log of a negative or zero
Check you have got it
Solve 3ˣ = 50, to 3 significant figures.
x = log 50 ÷ log 3 = 3.56.
You suspect y = kxⁿ. What do you plot to test it, and what does the gradient give you?
Plot log y against log x. A straight line confirms it, and the gradient is n.
Edvia Free Resources · Additional Mathematics 4037 · Topic 6 — free to copy and share
Topic 7
Straight-line graphs
Any relationship can be tested by rearranging it into y = mx + c — and then a straight line proves it.
Picture itSuppose you think two quantities follow a square law. Plot them raw and you get a curve, which could be almost anything. Plot one against the square of the other and if you get a straight line, you have proved it — and the gradient hands you the constant.
Everything reduces to y = mx + c
Whatever the relationship, the exam wants you to manipulate it until it looks like this. Then whatever multiplies the horizontal variable is the gradient, and whatever is left over is the intercept.
Choosing what to plot is the real question
For y = ax² + b, plot y against x² — gradient a, intercept b. For y = a/x + b, plot y against 1/x. For y = axⁿ, take logs and plot log y against log x. Identify the two things that go on the axes before doing any arithmetic.
Reading the constants off
Once you have a straight line, gradient and intercept give you the unknown constants directly. Draw a gradient triangle spanning at least half the line — small triangles magnify reading errors.
Why this matters beyond the exam
A straight line is the only graph the human eye can judge reliably. We are poor at telling a curve from a slightly different curve, but we can spot a bend in a straight line immediately. That is the entire reason this technique exists.
The bit that catches people outLabel the axes with what you are actually plotting, not the original variables. If you are plotting y against x², the horizontal axis is labelled x², not x. Mislabelling it makes the gradient meaningless and examiners mark it as such.
The grown-up words
What it means
What it is called
Note
y = mx + c
straight-line form
The target of every rearrangement
What multiplies the horizontal variable
gradient
Gives one unknown constant
Where the line crosses the vertical axis
intercept
Gives the other constant
Rearranging into straight-line form
linearising
Choose the axes carefully
Triangle used to measure steepness
gradient triangle
Span at least half the line
Check you have got it
You think y = ax³ + b. What should you plot, and what do the gradient and intercept give?
Plot y against x³. The gradient is a and the vertical intercept is b.
For y = kxⁿ, what do you plot and why?
log y against log x, because taking logs gives log y = n log x + log k — a straight line of gradient n.
Edvia Free Resources · Additional Mathematics 4037 · Topic 7 — free to copy and share
Topic 8
Coordinate geometry of the circle
A circle is every point a fixed distance from a centre — and that one sentence, written algebraically, is the entire equation.
Picture itThe equation (x−3)² + (y+2)² = 25 looks abstract until you read it as Pythagoras: the horizontal distance squared plus the vertical distance squared equals 5². It is the distance formula, rearranged.
The standard equation
(x − a)² + (y − b)² = r², with centre (a, b) and radius r. Watch the signs — a centre at (3, −2) gives (x − 3)² + (y + 2)², because subtracting a negative adds.
Expanded form needs completing the square
If you are given x² + y² + 2gx + 2fy + c = 0, complete the square on the x terms and the y terms separately to get back to standard form and read off the centre and radius.
Tangents meet radii at right angles
That single fact solves most circle questions. To find a tangent at a point: find the gradient of the radius to that point, take the negative reciprocal, and use the point to get the equation.
Intersections tell you the geometry
Substitute a line into the circle's equation to get a quadratic. Two solutions means the line is a chord; one repeated solution means it is a tangent; none means it misses. Same discriminant logic as everywhere else.
The bit that catches people outWhen completing the square to find a circle's centre, remember the constant moves to the other side. From x² − 6x + y² + 4y − 12 = 0 you get (x−3)² + (y+2)² = 12 + 9 + 4 = 25, so r = 5, not √12. Forgetting to add the correction terms is the standard error.
The grown-up words
What it means
What it is called
Note
(x − a)² + (y − b)² = r²
equation of a circle
Centre (a, b), radius r
Line touching the circle at one point
tangent
Perpendicular to the radius
Line cutting the circle at two points
chord
Quadratic has two roots
Chord through the centre
diameter
Twice the radius
Rewriting to find the centre
completing the square
Remember to adjust the constant
Check you have got it
Find the centre and radius of (x + 1)² + (y − 4)² = 9.
Centre (−1, 4), radius 3.
Why is the tangent to a circle perpendicular to the radius at that point?
Because the shortest distance from the centre to the tangent line is the radius, and the shortest distance is always along a perpendicular.
Edvia Free Resources · Additional Mathematics 4037 · Topic 8 — free to copy and share
Topic 9
Circular measure
Measuring angles in radians instead of degrees makes arc length and sector area almost trivial.
Picture itWhy invent a new angle unit? Because in radians, arc length is simply rθ. In degrees you need a fraction of 360 and a π every time. Radians were chosen to make the formulas disappear.
What a radian is
The angle where the arc length equals the radius. Since the whole circumference is 2πr, a full turn is 2π radians. So π radians = 180°, and that single conversion handles everything.
Two formulas, and both are clean
Arc length s = rθ. Sector area A = ½r²θ. Both require θ in radians. Compare that with the degree versions, which need θ/360 fractions, and the appeal is obvious.
Segments need a subtraction
A segment is the region between a chord and the arc. Find it as the sector area minus the triangle area: ½r²θ − ½r² sin θ. Nearly every segment question is this one line.
Common angles worth memorising
π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°. Recognising these instantly saves conversion time and reduces calculator errors.
The bit that catches people outBoth formulas only work in radians. If the question gives degrees, convert first — multiply by π/180. Using s = rθ with θ in degrees gives an answer that is wrong by a factor of about 57, and it is a very common way to lose a whole question.
The grown-up words
What it means
What it is called
Note
Angle where arc = radius
radian
2π in a full circle
s = rθ
arc length
θ must be in radians
A = ½r²θ
sector area
θ must be in radians
Region between a chord and an arc
segment
Sector minus triangle
Multiply by π/180
degrees to radians
And 180/π to go back
Check you have got it
A sector has radius 6 cm and angle 0.5 radians. Find its arc length and area.
s = rθ = 3 cm. A = ½r²θ = ½ × 36 × 0.5 = 9 cm².
Convert 135° to radians.
135 × π/180 = 3π/4.
Edvia Free Resources · Additional Mathematics 4037 · Topic 9 — free to copy and share
Topic 10
Trigonometry
Trigonometric functions repeat forever — and that periodicity is why one equation has infinitely many solutions.
Picture itAsk 'what angle has a sine of 0.5?' and your calculator says 30°. It is not lying, but it is only telling you one of infinitely many answers — 150°, 390°, 510° and so on all work too. The calculator gives you one; the question usually wants all of them in a range.
The three graphs and their shapes
sin and cos wave between −1 and 1 with period 360°, cos being sin shifted 90° left. tan repeats every 180° and shoots off to infinity at 90°, 270° and so on. Sketching the graph is the reliable way to find every solution.
Identities you must know
sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. Nearly every 'prove that' question uses one of these. If you are stuck, try converting everything to sines and cosines.
Finding all solutions in a range
Get the principal value from your calculator, then use the graph's symmetry to find the others in the given interval. Always check how many you should expect — a sine equation usually has two per 360°.
Reciprocal functions
cosec = 1/sin, sec = 1/cos, cot = 1/tan. Note that sec pairs with cos, not with sin — the naming is deliberately unhelpful and worth memorising directly.
Transformed graphs
y = a sin(bx) + c has amplitude a, period 360°/b, and is shifted up by c. Reading these off a given graph is a standard question.
The bit that catches people outYour calculator gives you one answer. The question almost always asks for all solutions in a stated range, such as 0° ≤ x ≤ 360°. Sketch the graph, draw the horizontal line, and count the crossings — that tells you how many answers you are looking for before you find them.
The grown-up words
What it means
What it is called
Note
Repeats every 360°
sine and cosine
Amplitude 1
Repeats every 180°
tangent
Undefined at 90°, 270°
sin²θ + cos²θ = 1
Pythagorean identity
The most used identity
1/sin, 1/cos, 1/tan
cosec, sec, cot
sec goes with cos
How far the wave goes from the middle
amplitude
The a in a sin(bx)
Length of one full cycle
period
360°/b
Check you have got it
Solve sin x = 0.5 for 0° ≤ x ≤ 360°.
x = 30° and x = 150°. The sine graph is symmetric about 90°, giving a second solution.
What is the period of y = cos(3x)?
360° ÷ 3 = 120°.
Edvia Free Resources · Additional Mathematics 4037 · Topic 10 — free to copy and share
Topic 11
Permutations and combinations
Counting arrangements when the numbers are too big to list — and the whole topic turns on one question: does order matter?
Picture itA three-digit padlock has 1,000 combinations. A three-card poker hand from 52 cards has 22,100. Both feel like 'choosing three things', but one cares about order and one does not — and that single difference changes the maths entirely.
Ask the order question first
If rearranging gives a genuinely different outcome, it is a permutation. If it is the same outcome either way, it is a combination. Deciding this before you reach for a formula is most of the work.
The formulas
Permutations: ⁿPᵣ = n!/(n−r)!. Combinations: ⁿCᵣ = n!/[r!(n−r)!]. The extra r! in the denominator divides out the orderings you do not want to count separately.
Factorial is just a descending product
5! = 5 × 4 × 3 × 2 × 1 = 120. And 0! = 1 — which looks strange but is needed for the formulas to work when r = n.
Restrictions are handled by dealing with them first
'How many arrangements with the two girls together?' — treat them as one block, arrange everything, then multiply by the arrangements within the block. 'Must start with a vowel?' — fix that position first, then arrange the rest.
The bit that catches people outA padlock 'combination' is a permutation — 1-2-3 does not open a lock set to 3-2-1. The everyday word is the mathematical opposite of what it describes, and that mismatch trips people up constantly.
The grown-up words
What it means
What it is called
Note
Order matters
permutation
ⁿPᵣ = n!/(n−r)!
Order does not matter
combination
ⁿCᵣ = n!/[r!(n−r)!]
n × (n−1) × … × 1
factorial
0! = 1
Items that must stay together
block
Treat as one, then arrange inside
Selecting without regard to order
choosing
Use ⁿCᵣ
Check you have got it
How many ways can 3 people be chosen from 8 for a committee?
Order does not matter, so ⁸C₃ = 56.
How many 3-letter arrangements can be made from 8 different letters?
Order matters, so ⁸P₃ = 336.
Edvia Free Resources · Additional Mathematics 4037 · Topic 11 — free to copy and share
Topic 12
Series
A series adds up a pattern — and if the pattern is regular enough, you can total a thousand terms without adding them.
Picture itThe story goes that a schoolboy Gauss was told to add every number from 1 to 100. He noticed 1+100, 2+99 and 3+98 all make 101, spotted there were fifty such pairs, and answered 5,050 almost immediately. That trick is the arithmetic series formula.
Arithmetic: add the same each time
Common difference d. The nth term is a + (n−1)d. The sum is Sₙ = n/2 [2a + (n−1)d], which is really just 'number of terms × average of first and last'.
Geometric: multiply by the same each time
Common ratio r. The nth term is arⁿ⁻¹. The sum is Sₙ = a(1 − rⁿ)/(1 − r). Growth and decay problems are almost always geometric.
Infinite geometric series can have a finite total
Only if |r| < 1, so the terms shrink towards zero. Then S∞ = a/(1 − r). If |r| ≥ 1 the terms do not shrink and the sum has no limit — and questions expect you to state that condition.
Binomial expansion
(1 + x)ⁿ expands using binomial coefficients, which are the ⁿCᵣ values from the previous topic. For a specific term, use the general term rather than expanding everything — much faster and far less error-prone.
The bit that catches people outCheck whether a sequence is arithmetic or geometric before choosing a formula. Arithmetic adds a constant; geometric multiplies by one. Test it: subtract consecutive terms, then divide consecutive terms. Whichever gives a constant tells you which it is.
The grown-up words
What it means
What it is called
Note
Same amount added each time
arithmetic series
Common difference d
Same factor multiplied each time
geometric series
Common ratio r
a + (n−1)d
nth term of an AP
a(1 − rⁿ)/(1 − r)
sum of a GP
a/(1 − r), only if |r| < 1
sum to infinity
Terms must shrink
Expanding (1 + x)ⁿ
binomial expansion
Coefficients are ⁿCᵣ
Check you have got it
A geometric series has first term 8 and ratio 0.5. Find its sum to infinity.
S∞ = a/(1−r) = 8/0.5 = 16.
Why does a geometric series with r = 2 have no sum to infinity?
The terms grow rather than shrink, so the total increases without limit. A sum to infinity needs |r| < 1.
Edvia Free Resources · Additional Mathematics 4037 · Topic 12 — free to copy and share
Topic 13
Vectors in two dimensions
A vector carries both size and direction — which lets you describe movement and prove geometry without coordinates.
Picture itA plane flying at 800 km/h into a 100 km/h headwind travels at 700 km/h over the ground. Add the vectors and the answer appears. Add just the speeds and you get nonsense, because direction was carrying half the information.
Notation and the basics
Written as a column with x on top and y below, or in i and j form. Add vectors by adding components. Multiplying by a scalar stretches or shrinks it, and a negative scalar reverses its direction.
Magnitude is Pythagoras
The size of a vector is √(x² + y²). A unit vector has magnitude 1 and is found by dividing a vector by its own magnitude — useful when you want direction without size.
Position vectors locate points
The position vector of A is the vector from the origin to A. The vector from A to B is then b − a — destination minus start. Getting that order the right way round is essential.
Proving things with vectors
If one vector is a scalar multiple of another, the two are parallel. If they are parallel and share a common point, the points are collinear — they lie on one straight line. Almost every vector proof question uses one of these two facts.
Relative velocity
The velocity of A relative to B is v_A − v_B. Questions about closest approach or interception are usually asking you to work in the relative frame, where one object appears stationary.
The bit that catches people outThe vector from A to B is b − a, not a − b. Think of it as 'where you end minus where you start'. Reversing it points the vector the wrong way and every subsequent deduction inherits the error.
The grown-up words
What it means
What it is called
Note
Size and direction
vector
Column or i, j form
√(x² + y²)
magnitude
Pythagoras on the components
Vector of magnitude 1
unit vector
Divide by the magnitude
From the origin to a point
position vector
One is a multiple of the other
parallel vectors
Key to most proofs
Points on one straight line
collinear
Parallel vectors sharing a point
Check you have got it
A has position vector (2, 1) and B has (7, 13). Find the vector AB and its magnitude.
AB = b − a = (5, 12). Magnitude = √(25 + 144) = 13.
How do you show three points are collinear using vectors?
Show that one vector between two of them is a scalar multiple of another, and that they share a common point.
Edvia Free Resources · Additional Mathematics 4037 · Topic 13 — free to copy and share
Topic 14
Calculus
Differentiation finds the rate of change; integration reverses it — and between them they solve almost anything about curves and motion.
Picture itA speedometer performs differentiation. It takes your position, which is changing, and reports how fast it is changing right now. An odometer performs integration — it takes your speed and accumulates it into total distance.
Differentiation: the gradient at a point
The derivative dy/dx gives the gradient of the curve at any x. Differentiate xⁿ by bringing the power to the front and reducing it by one: n xⁿ⁻¹. That single rule covers most of the syllabus.
Stationary points are where the gradient is zero
Set dy/dx = 0 and solve. Then use the second derivative to classify: if d²y/dx² is positive it is a minimum, if negative a maximum. Positive-minimum feels backwards to most people, so learn it deliberately.
The chain, product and quotient rules
For a function inside a function, use the chain rule: differentiate the outside, then multiply by the derivative of the inside. Use the product rule for things multiplied and the quotient rule for things divided. Identify the structure before choosing.
Integration: going backwards
Add one to the power and divide by the new power. Always add + c for an indefinite integral — and find c if you are given a point on the curve. For a definite integral, substitute the limits and subtract.
What integration gives you
The area under a curve between two limits. Also, in kinematics: differentiate displacement to get velocity and again for acceleration; integrate acceleration to get velocity and again for displacement. The whole chain runs both ways.
The bit that catches people outForgetting the + c on an indefinite integral is the most-penalised single error in calculus. It matters because infinitely many curves have the same gradient function — they differ only by a vertical shift, and c is that shift.
The grown-up words
What it means
What it is called
Note
Rate of change / gradient at a point
derivative
dy/dx
Bring the power down, reduce by one
differentiating xⁿ
n xⁿ⁻¹
Where the gradient is zero
stationary point
Maximum, minimum or inflexion
Tells you max or min
second derivative
Positive = minimum
Function inside a function
chain rule
Outside, then times inside
Reverse of differentiation
integration
Do not forget + c
Area under a curve between limits
definite integral
Substitute and subtract
Check you have got it
Find the stationary point of y = x² − 6x + 5 and say whether it is a maximum or minimum.
dy/dx = 2x − 6 = 0 gives x = 3, y = −4. Second derivative is 2, which is positive, so it is a minimum.
A particle's velocity is v = 3t². How do you find the distance travelled between t = 0 and t = 2?
Integrate: ∫3t² dt = t³. Evaluate between the limits: 8 − 0 = 8 metres.
Edvia Free Resources · Additional Mathematics 4037 · Topic 14 — free to copy and share
Like how this is taught?
Every handout starts with the idea in plain English and only then the formal version. That is how every class at Edvia College works — for two full years of Cambridge A Levels.