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SEC 2027 · K310 SEC 2

Sec 2 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
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THE CHAPTERS

🗺️

CH 1

Proportion & Map Scales

Direct: y = kx. Inverse: y = k/x.

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CH 2

Expansion & Factorisation

3 special identities + cross method.

➗

CH 3

Algebraic Fractions & Formulae

Factorise first, then cancel.

⛰️

CH 4

Quadratic Graphs & Equations

a > 0 smile, a < 0 frown.

✖️

CH 5

Simultaneous Equations

Eliminate one letter.

↔️

CH 6

Linear Inequalities

× or ÷ by negative → flip sign.

🔺

CH 7

Congruence & Similarity

Similar = same shape, sides in ratio.

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CH 8

Pythagoras & Trig

SOH CAH TOA.

🍦

CH 9

Pyramids, Cones, Spheres

Pointy solids = ⅓ × base × h.

📊

CH 10

Statistics

Mean, median, mode + charts.

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CH 11

Probability

favourable ÷ total.

🖨 PRINTCtrl+P → Save as PDF · Scale 100% · Margins None · Background graphics ON

E Math · Sec 2 · SEC K310

🗺️

Proportion & Map Scales

N2 · direct & inverse proportion, scales

BIG IDEADirect: one goes up, the other goes up by the same factor → y = kx. Inverse: one up, other down → y = k/x.

direct: straight line through origin

inverse: curve, never touches axes

Type	Equation	Example
y ∝ x	y = kx	5 pens cost $4 → 1 pen $0.80 → 12 pens $9.60
y ∝ x²	y = kx²	x doubles → y × 4
y ∝ 1/x	y = k/x	6 men take 10 days → 15 men take 6×10÷15 = 4 days

Write the relation with k

Substitute the given pair to find k

Write the full equation

Use it to answer

Map scales

Scale 1 : 50 000

1 cm on map = 50 000 cm = 0.5 km
length: 1 cm : 0.5 km
area: 1 cm² : 0.25 km²

Example

Lake is 6 cm² on the map → 6 × 0.25 = 1.5 km²

⚠ TRAPFor area, square the length scale — never use the length scale directly. Convert cm → km by ÷ 100 000.

E Math · Sec 2

🧮

Expansion & Factorisation

N5 · identities, quadratic factorisation, grouping

BIG IDEALearn the 3 identities both ways. Factorising = expanding in reverse.

Identity	Example
(a + b)² = a² + 2ab + b²	(x + 3)² = x² + 6x + 9
(a − b)² = a² − 2ab + b²	(2x − 1)² = 4x² − 4x + 1
(a + b)(a − b) = a² − b²	x² − 49 = (x + 7)(x − 7)

Cross method for ax² + bx + c

→ (2x + 1)(x + 3)

How

① split ax² into 2 factors (left)
② split c into 2 factors (right)
③ cross-multiply and add
④ must equal bx → read across the rows

Factorising checklist

Take out the common factor first

2 terms? → try a² − b²

3 terms? → identity or cross method

4 terms? → grouping

Grouping

ax + ay + bx + by
= a(x + y) + b(x + y) = (a + b)(x + y)

Common factor first

3x² − 12 = 3(x² − 4)
= 3(x + 2)(x − 2)

⚠ TRAP(a + b)² ≠ a² + b² — never forget the middle term 2ab.

E Math · Sec 2

➗

Algebraic Fractions & Formulae

N5 · simplify, ×, ÷, +, −, change the subject

BIG IDEAAlgebraic fractions work like number fractions. Factorise first, then cancel whole factors.

✂️ Simplify

(x² − 9)/(x² + 3x)
= (x + 3)(x − 3) / x(x + 3) = (x − 3)/x

✖️ Multiply / ➗ divide

÷ → flip the second fraction and ×
(2a/3b) ÷ (4a²/9b) = (2a/3b) × (9b/4a²) = 3/(2a)

➕ Add / subtract

use a common denominator
2/x + 3/(x + 1)
= [2(x + 1) + 3x] / x(x + 1) = (5x + 2)/x(x + 1)

⚠ TRAPYou can only cancel factors, never terms: (x + 3)/3 ≠ x. Put brackets round numerators when subtracting.

Changing the subject of a formula

Clear fractions (× denominator)

Expand brackets

Get all the new subject terms on one side

Factorise it out, then divide

Example 1

Make r the subject: V = πr²h
r² = V/(πh) → r = √(V/(πh))

Example 2 (subject twice)

Make x the subject: y = (x + 2)/(x − 1)
y(x − 1) = x + 2 → yx − x = y + 2
x(y − 1) = y + 2 → x = (y + 2)/(y − 1)

🧠 HOOKUndo in the reverse order of what was done to the letter: last thing done = first thing undone.

E Math · Sec 2

⛰️

Quadratic Graphs & Equations

N6 · N7 · parabolas, factorisation

BIG IDEAy = ax² + bx + c is a parabola: a > 0 → ∪ (minimum), a < 0 → ∩ (maximum). It is symmetric about its turning point.

y = x² − 2x − 3 = (x + 1)(x − 3)

Read the graph

roots: where it cuts the x-axis (y = 0)
y-intercept: c
line of symmetry: halfway between roots
turning point: on that line

Solve by factorisation

Make one side = 0

Factorise

Set each factor = 0

Solve both

Example

x² − 2x = 3 → x² − 2x − 3 = 0
(x + 1)(x − 3) = 0 → x = −1 or x = 3

⚠ TRAPNever divide both sides by x — you lose the root x = 0. x² = 5x → x(x − 5) = 0 → x = 0 or 5.

🧠 HOOKSketch checklist: shape (∪/∩), y-intercept, x-intercepts, line of symmetry. Label all of them.

E Math · Sec 2

✖️

Simultaneous Linear Equations

N7 · elimination, substitution, graphical

BIG IDEA2 unknowns need 2 equations. Eliminate one letter, solve, then substitute back.

solution = point of intersection

Elimination

x + 2y = 7 …(1)
x − y = 1 …(2)
(1) − (2): 3y = 6 → y = 2
sub into (2): x = 3

Substitution

from (2): x = y + 1
sub into (1): y + 1 + 2y = 7 → 3y = 6 → y = 2, x = 3

Which method?

one letter already alone → substitution
same coefficient → elimination (same sign subtract, different signs add)

Graphs of ax + by = c

Quick plot

put x = 0 → find y; put y = 0 → find x; join the 2 points.
2x + 3y = 12 → (0, 4) and (6, 0)

⚠ TRAPLabel equations (1) and (2) and show which you add/subtract. Check the answer in the other equation.

🧠 HOOKWord problems: let the 2 unknowns be x and y, write 2 sentences as 2 equations.

E Math · Sec 2

↔️

Linear Inequalities

N7 · solving, number lines

BIG IDEASolve like an equation, but when you × or ÷ by a negative, flip the sign.

x > 0 → open circle (not included)

x ≤ 1 → filled circle (included)

Example

5 − 2x ≥ 11
−2x ≥ 6 → x ≤ −3 (÷ −2, flip!)

Integer answers

"largest integer x with 3x < 10" → x < 3.33 → x = 3

⚠ TRAPMultiplying/dividing by a negative reverses the inequality. Swapping sides also reverses it: 4 > x means x < 4.

🧠 HOOK< > → open circle ○ · ≤ ≥ → filled circle ●.

E Math · Sec 2

🔺

Congruence & Similarity

G2 · congruent & similar figures, enlargement

BIG IDEACongruent = same shape AND size. Similar = same shape, sides in the same ratio, angles equal.

congruent: equal sides & angles

similar: scale factor k = 8/4 = 2

Enlargement

Scale factor k

k > 1 → enlargement
0 < k < 1 → reduction
new length = k × old length

Finding a missing side

match corresponding sides (opposite equal angles)
x / 12 = 5 / 8 → x = 7.5

⚠ TRAPMatch the sides by their equal angles, not by their position on the page. Redraw the triangles the same way round.

🧠 HOOKSec 3 adds the formal tests (SSS, SAS, ASA, RHS for congruence; AA, SSS, SAS for similarity) and area/volume ratios.

E Math · Sec 2

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Pythagoras & Trigonometry

G4 · right-angled triangles

BIG IDEARight-angled triangle only: a² + b² = c² (c = hypotenuse, opposite the right angle) and SOH CAH TOA.

labels depend on θ

SOH CAH TOA

sin θ = opp / hyp
cos θ = adj / hyp
tan θ = opp / adj
angle → use sin⁻¹, cos⁻¹, tan⁻¹

Pythagoras example

legs 6 and 8 → c = √(36 + 64) = 10
hyp 13, leg 5 → √(169 − 25) = 12

Is it right-angled?

check if (longest)² = sum of other two²
7, 24, 25: 49 + 576 = 625 ✓

Trig example

θ = 35°, hyp = 10 → opp = 10 sin 35° = 5.74

Find an angle

opp 4, adj 7 → θ = tan⁻¹(4/7) = 29.7°

⚠ TRAPCalculator must be in DEG mode. Give angles to 1 d.p.. Label opp/adj/hyp from the angle you are using.

E Math · Sec 2

🍦

Pyramids, Cones & Spheres

G5 · volume & surface area

BIG IDEAPointy solids are ⅓ of the matching prism: V = ⅓ × base area × height.

Pyramid

Cone

Sphere

Solid	Volume	Surface area
Pyramid	⅓ × base × h	base + all triangles
Cone	⅓πr²h	πrl + πr² (curved = πrl)
Sphere	(4/3)πr³	4πr²
Hemisphere	(2/3)πr³	2πr² + πr² = 3πr² (solid)

Slant height

l² = r² + h²
r = 3, h = 4 → l = 5

Example

Cone r = 3 cm, h = 4 cm
V = ⅓ × π × 9 × 4 = 37.7 cm³
Total SA = π(3)(5) + π(9) = 75.4 cm²

⚠ TRAPCurved surface of a cone uses l (slant height), volume uses h (vertical height). A solid hemisphere includes its flat circle.

🧠 HOOKCone, sphere & ½ab sin C formulas are given in the exam formula list — but you must know which to use.

E Math · Sec 2

📊

Statistics

S1 · dot diagrams, histograms, stem-and-leaf, averages

BIG IDEAMean = total ÷ count · Median = middle value (in order) · Mode = most common.

Dot diagram — mode = 2

Histogram — bars touch, equal class widths

🌿 Stem-and-leaf

1 | 2 5 8
2 | 0 3 3 7
3 | 1 4

key: 1 | 2 means 12
median = 23 (5th of 9)

Which average?

mean: uses all data, affected by extreme values
median: best with outliers
mode: for categories / most popular

Mean of grouped data — use mid-values

Mass (kg)	Mid x	f	fx
40 – 50	45	4	180
50 – 60	55	10	550
60 – 70	65	6	390
Total		20	1120

Answer

mean = Σfx ÷ Σf = 1120 ÷ 20 = 56 kg

⚠ TRAPMedian position = (n + 1) ÷ 2 after sorting. Grouped mean is an estimate because we use mid-values.

E Math · Sec 2

🎲

Probability

S2 · single events

BIG IDEAP(event) = favourable outcomes ÷ total outcomes — always between 0 (impossible) and 1 (certain).

🎲 One die

P(even) = 3/6 = ½
P(more than 4) = 2/6 = ⅓
P(7) = 0

🃏 Bag of 3 red, 5 blue

P(red) = 3/8
P(not red) = 1 − 3/8 = 5/8

🧠 HOOKP(not A) = 1 − P(A)

⚠ TRAPAnswers as a fraction in simplest form (or decimal) — never as a ratio like 3 : 8.

✍ METHODList the outcomes (sample space) when unsure — e.g. tossing 2 coins: HH, HT, TH, TT → P(one head) = 2/4 = ½.

🧠 HOOKSec 4 adds combined events: possibility diagrams, tree diagrams, "and" (×) / "or" (+) rules.

E Math · Sec 2

Sec 2 E MathVisual Notes

Proportion & Map Scales

Map scales

Expansion & Factorisation

Cross method for ax² + bx + c

Factorising checklist

Algebraic Fractions & Formulae

Changing the subject of a formula

Quadratic Graphs & Equations

Solve by factorisation

Simultaneous Linear Equations

Graphs of ax + by = c

Linear Inequalities

Congruence & Similarity

Enlargement

Pythagoras & Trigonometry

Pyramids, Cones & Spheres

Statistics

Mean of grouped data — use mid-values

Probability

Sec 2 E Math
Visual Notes