📐
SEC 2027 · K310 SEC 4

Sec 4 E Math
Visual Notes

Pictures first. Formulas boxed. Every exam keyword highlighted.
HOW TO STUDY WITH THIS
Read the 💡 box
the whole topic in 1 line
Study the diagram
cover it, redraw it
Learn the yellow
formulas & keywords
Redo the worked example
without looking
THE CHAPTERS
⭕
CH 1
Sets & Venn Diagrams
Shade the region the symbol means.
🔢
CH 2
Matrices
Row × column.
➡️
CH 3
Vectors
AB = b − a.
📈
CH 4
Cumulative Frequency & Box Plots
Median at n/2.
📊
CH 5
Standard Deviation
Spread → consistency.
🌳
CH 6
Combined Probability
AND ×, OR +.
📋
★
Formula Sheet
Given vs must-memorise.
🎯
★
Exam Strategy
Where the marks are.
🖨 PRINTCtrl+P → Save as PDF · Scale 100% · Margins None · Background graphics ON
E Math · Sec 4 · SEC K310
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1

Sets & Venn Diagrams

N8 · set language and notation
BIG IDEAA Venn diagram is a picture of sets. Each symbol = a region to shade.
ABξ
A ∪ B · union (either)
ABξ
A ∩ B · intersection (both)
ABξ
A′ · complement (not A)
ABξ
A ∩ B′ · A only
SymbolMeaningSymbolMeaning
∈is an element ofξuniversal set
∉not an element of∅ or { }empty set
⊆subsetn(A)number of elements
⊂proper subsetA′complement
Counting formula
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
30 students: 18 like tea, 15 coffee, 6 neither
n(T ∪ C) = 24 → n(T ∩ C) = 18 + 15 − 24 = 9
Fill a Venn
① start with the overlap
② then each "only" region
③ then outside (neither)
④ check the total = n(ξ)
⚠ TRAP{a} ⊂ A (a set) but a ∈ A (an element). The empty set ∅ has no elements — don't write {∅}.
E Math · Sec 4
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2

Matrices

N9 · data, addition, scalar & matrix multiplication
BIG IDEAA matrix is a table of numbers. To multiply: row × column, add up.
Order = rows × columns
(
123
456
) is a 2 × 3 matrix
Add / subtract / scalar
same order only, add matching entries
2(
13
02
) = (
26
04
)

Multiplying

Example
(
12
34
)(
5
6
) = (
1×5 + 2×6
3×5 + 4×6
) = (
17
39
)
Can you multiply?
(m × n)(n × p) → answer is m × p
inner numbers must match
AB ≠ BA in general

Matrices for real data

Shop example
Prices (
$3$5
) (pen, file) · Quantities (
10
4
)
Total = 3 × 10 + 5 × 4 = $50
✍ INTERPRETSay what each entry represents: "the element 50 represents the total cost in dollars of the pens and files sold."
⚠ TRAPWrite the matrices so the labels match: columns of the first = rows of the second.
E Math · Sec 4
➡️
3

Vectors in 2D

G7 · notation, magnitude, position vectors, geometry
BIG IDEAA vector has size AND direction. To go from A to B: AB = OB − OA = b − a.
xyabAB = b − aABO
position vectors from O
Column vector & magnitude
AB = (4, 3)ᵀ (4 right, 3 up)
|AB| = √(4² + 3²) = 5
negative vector = reverse direction
BA = −AB
Parallel vectors
one is a scalar multiple of the other
p = (2, −1)ᵀ, q = (6, −3)ᵀ = 3p → p ∥ q
Collinear points
AB = k·BC and they share point B
→ A, B, C on a straight line
Geometry routine
write every vector as a path through known vectors: PQ = PO + OQ = −p + q
midpoint M of AB: OM = ½(a + b)
⚠ TRAPVectors in exam answers must be bold or underlined (a̲). Magnitude is a number, not a vector.
E Math · Sec 4
📈
4

Cumulative Frequency & Box Plots

S1 · quartiles, percentiles, IQR
BIG IDEACumulative frequency = running total. Read Q1 at n/4, median at n/2, Q3 at 3n/4.
markscfQ1Q2Q3
n = 80 → read at 20, 40, 60
Read the curve
cf valuen = 80
Q1n/420
Mediann/240
Q33n/460
90th %ile0.9n72
IQR = Q3 − Q1

Box-and-whisker plot

minQ1medianQ3max
box = middle 50% of data
Compare 2 data sets
① compare medians (average)
② compare IQRs (spread)
smaller IQR = more consistent
⚠ TRAPDraw lines across to the curve, then down with a ruler and show them. Plot cf at the upper boundary of each class.
E Math · Sec 4
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5

Standard Deviation

S1 · grouped & ungrouped data, comparing
BIG IDEAMean tells you the average. Standard deviation tells you how spread out the data is.
Formulas (given)
x̄ = Σfx / Σf
SD = √[ Σfx²/Σf − (Σfx/Σf)² ]
Calculator
use STAT mode: enter x (and freq) → read x̄ and σx
use σx, not sx
x24579Total
f132118
fx212107940
fx²448504981232
Working
x̄ = 40 ÷ 8 = 5
SD = √(232/8 − 5²) = √(29 − 25) = 2
Range vs IQR vs SD
range: max − min (affected by extremes)
IQR: middle 50% (ignores extremes)
SD: uses all values
✍ COMPARE"Class A has a higher mean, so on average they scored better. Class B has a lower standard deviation, so their scores are more consistent."
⚠ TRAPGrouped data: use the mid-value of each class as x. Always comment on both average and spread when comparing.
E Math · Sec 4
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6

Combined Probability

S2 · possibility diagrams, tree diagrams
BIG IDEAAND → multiply along branches. OR → add the results.
123456123456234567345678456789567891067891011789101112
2 dice: P(sum = 7) = 6/36 = 1/6
RBRBRB 3/52/52/42/43/41/4= 6/20= 6/20= 6/20= 2/20
3 red, 2 blue, take 2 without replacement
From the tree
P(both red) = 3/5 × 2/4 = 3/10
P(one of each) = 6/20 + 6/20 = 3/5
check: all ends add to 1
Rules
P(A or B) = P(A) + P(B) (mutually exclusive)
P(A and B) = P(A) × P(B) (independent)
⚠ TRAPWithout replacement: the second branch fractions change (top and bottom both go down by 1). With replacement they stay the same.
🧠 HOOK"At least one" = 1 − P(none). Much faster than adding many branches.
E Math · Sec 4
📋
★

Formula Sheet

given vs must-memorise (K310)
BIG IDEAThe exam gives you some formulas. Everything else you must memorise.
📄 GIVEN in the paper
compound interest A = P(1 + r/100)ⁿ
cone curved SA πrl · volume ⅓πr²h
sphere SA 4πr² · volume (4/3)πr³
triangle area ½ab sin C
arc s = rθ · sector ½r²θ
sine rule · cosine rule
mean Σfx/Σf · standard deviation
🧠 MEMORISE
quadratic formula [−b ± √(b² − 4ac)]/2a
(a ± b)², a² − b²
laws of indices
gradient, length, y = mx + c
Pythagoras, SOH CAH TOA
prism / cylinder / pyramid volume
π rad = 180°, segment area
similar figures k, k², k³
circle theorems · polygon angle sums
simple interest PRT/100
E Math · Sec 4
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★

Exam Strategy

Paper 1 & Paper 2
BIG IDEAMarks come from working, accuracy and reasons — not just the final answer.
RuleWhy
Give answers to 3 s.f., angles to 1 d.p.lose accuracy marks otherwise
Keep full calculator values in workingrounding early = wrong final answer
Write the formula, then substitutemethod marks even if the answer is wrong
State reasons in geometry"alt. ∠s", "∠ in semicircle" etc.
Check DEG / RAD modemost common lost marks in trig
Draw graphs with a smooth curve, label axespresentation marks
Read units (cm vs m, min vs h)convert before calculating
🧠 HOOKShow "shown" questions fully: start from the given information and end with the exact expression asked for.
✍ CHECKLast 10 minutes: check signs, units, s.f., and that you answered what the question asked (e.g. "total", "remaining", "each").
E Math · Sec 4