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Use the subject cards above or the sidebar to navigate. Each subject has clickable topic tabs with full notes, formulas, diagrams and examples.

Mathematics

JSCNSSCONSSCAS

Number Systems

Numbers are classified into nested sets. Every natural number is also a whole number, every whole number is also an integer, and so on up to the real numbers.

Set	Symbol	Examples	Key Property
Natural numbers	ℕ	1, 2, 3, 4…	Counting numbers; no zero
Whole numbers	ℕ₀	0, 1, 2, 3…	Natural + zero
Integers	ℤ	…−2, −1, 0, 1, 2…	Whole + negatives
Rational numbers	ℚ	½, 0.75, −3, 6	Can be written as p/q (q ≠ 0); decimals terminate or recur
Irrational numbers	ℚ′	√2, π, e	Cannot be written as p/q; decimals are non-terminating, non-recurring
Real numbers	ℝ	All of the above	Every point on the number line

Proof that √2 is irrational

Assume √2 = p/q in lowest terms. Then 2q² = p², so p² is even, so p is even. Write p = 2k → 2q² = 4k² → q² = 2k², so q is also even. This contradicts p/q being in lowest terms. Therefore √2 is irrational.

Order of Operations — BODMAS

Key Rule

Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction

Worked Example 1

Evaluate: 3 + 4 × (2² − 1)

Brackets: 2² − 1 = 4 − 1 = 3
Multiply: 4 × 3 = 12
Add: 3 + 12 = 15

Worked Example 2

Evaluate: 18 ÷ (3² − 3) + 2³

Brackets: 3² − 3 = 9 − 3 = 6
Orders: 2³ = 8
Division: 18 ÷ 6 = 3
Add: 3 + 8 = 11

Indices (Laws of Exponents)

Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ Power rule: (aᵐ)ⁿ = aᵐⁿ Zero exponent: a⁰ = 1 (a ≠ 0) Negative exponent: a⁻ⁿ = 1/aⁿ Fractional exponent:aⁿ/ᵐ = ᵐ√(aⁿ) e.g. 8²/³ = (³√8)² = 4

Worked Example — Simplify

Simplify: (2x³y²)³ ÷ (4x²y)

Expand numerator: 2³ · x⁹ · y⁶ = 8x⁹y⁶
Divide: 8x⁹y⁶ ÷ 4x²y = 2x⁷y⁵

Surds

A surd is an irrational root that cannot be simplified to a rational number. You can simplify surds by finding the largest perfect-square factor.

√72 = √(36 × 2) = 6√2 Rules: √a × √b = √(ab) √a ÷ √b = √(a/b) (√a)² = a Rationalising the denominator: 1/√a = √a/a (multiply top and bottom by √a) 1/(a+√b) → multiply by (a−√b)/(a−√b) (conjugate)

Rationalise: 6/(2+√3)

Multiply by (2−√3)/(2−√3):

= 6(2−√3) / (4−3) = 6(2−√3) / 1 = 12 − 6√3

Fractions, Percentages & Ratio

To convert a fraction to a percentage, multiply by 100. To divide a quantity T in ratio a:b, each part = T/(a+b), then multiply by a or b respectively.

Ratio Problem

Divide N$2 400 in the ratio 3:5.

Total parts = 8. Each part = 2400/8 = 300.

Shares: 3×300 = N$900 and 5×300 = N$1 500.

Percentage Change

% change = (new − old) / old × 100. A positive result is an increase; negative is a decrease.

Standard Form (Scientific Notation)

A × 10ⁿ where 1 ≤ A < 10 and n ∈ ℤ Examples: 0.000 047 = 4.7 × 10⁻⁵ 93 000 000 = 9.3 × 10⁷ Multiplying: (3×10⁴) × (2×10³) = 6×10⁷ Dividing: (9×10⁸) ÷ (3×10⁵) = 3×10³ Adding: Convert to same power first, then add coefficients.

Number Patterns & Sequences

Sequence Type	Pattern	nth Term	Example
Arithmetic	Add constant difference d	T(n) = a + (n−1)d	3, 7, 11, 15… (a=3, d=4)
Geometric	Multiply by constant ratio r	T(n) = arⁿ⁻¹	2, 6, 18, 54… (a=2, r=3)
Quadratic	Second difference is constant	T(n) = an²+bn+c	1, 4, 9, 16… (perfect squares)

Arithmetic series sum: Sₙ = n/2 × (2a + (n−1)d) or Sₙ = n/2 × (first + last) Geometric series sum: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1 Infinite geometric sum: S∞ = a/(1−r) only when |r| < 1

Find the 20th term & sum of 5, 8, 11, 14…

a = 5, d = 3, n = 20

T(20) = 5 + 19×3 = 62

S₂₀ = 20/2 × (5 + 62) = 10 × 67 = 670

Expanding & Factorising

Expand brackets using the distributive law. For two binomials use FOIL (First, Outer, Inner, Last). Factorising is the reverse — find common factors or apply identities.

(a+b)(a−b) = a²−b² (difference of two squares) (a+b)² = a²+2ab+b² (a−b)² = a²−2ab+b² (a+b)³ = a³+3a²b+3ab²+b³

Factorising by grouping — 6x² + 9x − 4x − 6

Group: (6x²+9x) + (−4x−6)
Factor: 3x(2x+3) − 2(2x+3)
Result: (2x+3)(3x−2)

Solving Linear Equations

Isolate the unknown by performing identical inverse operations on both sides. Always substitute back to verify.

Linear with fractions — (x+2)/3 − (x−1)/4 = 2

Multiply by LCM=12: 4(x+2)−3(x−1)=24
Expand: 4x+8−3x+3=24 → x+11=24
x = 13

Quadratic Equations

A quadratic has the form ax²+bx+c=0 (a≠0). Three methods:

Method	When to use
Factorising	When integer roots are likely
Completing the square	Deriving vertex form; any quadratic
Quadratic formula	Always works; non-integer roots

Quadratic Formula: x = [ −b ± √(b²−4ac) ] / (2a) Discriminant Δ = b²−4ac: Δ > 0 → two distinct real roots Δ = 0 → one repeated root Δ < 0 → no real roots

Completing the square — x²−6x+5=0

x²−6x = −5; add (3)²=9: (x−3)²=4
x−3=±2 → x=5 or x=1

Formula — 2x²+3x−2=0

Δ=9+16=25, x=(−3±5)/4 → x=½ or x=−2

Simultaneous Equations

Elimination — 3x+2y=16 and 5x−2y=8

Add (2y cancels): 8x=24 → x=3
Sub in: 9+2y=16 → y=3.5

Linear & quadratic — y=x+1 and y=x²−3

Substitute: x+1=x²−3 → x²−x−4=0
Δ=1+16=17>0 → two intersection points

Inequalities

Rule: reverse sign when multiplying/dividing by negative. Linear: 2x−3>7 → 2x>10 → x>5 Quadratic: x²−x−6≤0 → (x−3)(x+2)≤0 → −2≤x≤3 Sketch the parabola to identify which region satisfies the inequality.

Algebraic Fractions

Simplify: (x²−4)/(x²−4x+4) = (x+2)(x−2)/(x−2)² = (x+2)/(x−2) Add: 2/(x+1)+3/(x−1) = [2(x−1)+3(x+1)]/[(x+1)(x−1)] = (5x+1)/(x²−1)

Logarithms

log_a(x)=y means aʸ=x. Logarithms are the inverse of exponential functions.

log(AB) = logA+logB (product) log(A/B)= logA−logB (quotient) log(Aⁿ) = n·logA (power) log_a(a)= 1, log_a(1)=0 Change of base: log_a(x) = log(x)/log(a) Natural log: ln(x)=log_e(x)

Solve 2^(x+1)=50

(x+1)log2=log50
x+1=log50/log2=5.644
x≈4.64

Angle Properties

Angles on a straight line sum to 180°
Angles at a point sum to 360°
Vertically opposite angles are equal
Alternate angles (Z-angles) are equal; co-interior (C-angles) sum to 180°
Corresponding angles (F-angles) are equal (parallel lines)

Triangle Properties

Triangle Type	Properties
Equilateral	All sides equal; all angles = 60°
Isosceles	Two equal sides; base angles equal
Scalene	No equal sides or angles
Right-angled	One angle = 90°; use Pythagoras & trig

Area of triangle = ½ × base × height Area = ½ab sinC (using two sides a, b and included angle C) Perimeter = sum of all sides

Pythagorean Theorem

a² + b² = c² where c is the hypotenuse (side opposite 90°)

/| / | c / | b / | /_____| a c is the hypotenuse (longest side).

Find the hypotenuse: a=6, b=8

c² = 6² + 8² = 36 + 64 = 100 → c = 10

(6-8-10 is a Pythagorean triple)

Trigonometry (SOHCAHTOA)

sin θ = Opposite / Hypotenuse (SOH) cos θ = Adjacent / Hypotenuse (CAH) tan θ = Opposite / Adjacent (TOA) Inverses: θ = sin⁻¹(opp/hyp), cos⁻¹(adj/hyp), tan⁻¹(opp/adj)

Angle	sin	cos	tan
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	undefined

Sine rule: a/sinA = b/sinB = c/sinC Cosine rule: c² = a² + b² − 2ab cosC cosC = (a²+b²−c²)/(2ab) Use sine rule when you know angle-side-angle or side-side-angle. Use cosine rule when you know three sides or two sides + included angle.

Circle Theorems

Angle at centre = 2 × angle at circumference (same arc)
Angles in a semicircle = 90°
Angles in the same segment are equal
Opposite angles in a cyclic quadrilateral sum to 180°
Tangent is perpendicular to the radius at the point of contact
Tangents from an external point are equal in length
Alternate segment theorem: angle between tangent and chord = angle in alternate segment

Coordinate Geometry

Distance between (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)²+(y₂−y₁)²] Midpoint: M = ((x₁+x₂)/2 , (y₁+y₂)/2) Gradient: m = (y₂−y₁)/(x₂−x₁) Perpendicular gradients: m₁×m₂ = −1

Mensuration — Areas & Volumes

Shape	Area	Volume / Surface Area
Rectangle	l×w	—
Triangle	½bh	—
Trapezium	½(a+b)h	—
Circle	πr²	C=2πr
Cylinder	—	V=πr²h; SA=2πr²+2πrh
Cone	—	V=⅓πr²h; SA=πr²+πrl (l=slant height)
Sphere	—	V=4/3πr³; SA=4πr²
Pyramid	—	V=⅓×base area×h

Exam Tip

For compound shapes: split into simpler shapes, calculate each area/volume separately, then add (or subtract for hollow shapes).

Transformations

Transformation	Key Features	What changes
Translation	Described by a column vector	Position only
Reflection	Described by mirror line	Position & orientation
Rotation	Centre, angle, direction	Position & orientation
Enlargement	Centre, scale factor k	Size (and position if k≠1)

What is a Function?

A function f maps every input x to exactly one output f(x). Written f: x ↦ f(x).

Types of Functions

Function	Equation	Graph Shape
Linear	y = mx + c	Straight line; gradient m, y-intercept c
Quadratic	y = ax² + bx + c	Parabola; opens up if a > 0
Cubic	y = ax³ + …	S-shaped curve
Exponential	y = aˣ	Rapid growth/decay curve
Reciprocal	y = k/x	Hyperbola with asymptotes

Gradient & y-intercept (Linear)

Gradient m = (y₂ − y₁) / (x₂ − x₁) Equation of line through (x₁,y₁) with gradient m: y − y₁ = m(x − x₁)

Vertex of a Parabola

y = ax² + bx + c Vertex x-coordinate: x = −b / (2a) The axis of symmetry passes through the vertex.

Transformations of Graphs

y = f(x) + a — translate up by a units
y = f(x + a) — translate left by a units
y = af(x) — stretch vertically by factor a
y = −f(x) — reflect in x-axis

Level

Calculus is examined at NSSCAS (Grade 11–12) level.

Differentiation — First Principles

f ′(x) = lim[h→0] [ f(x+h) − f(x) ] / h

Power Rule

d/dx [xⁿ] = nxⁿ⁻¹ Example: d/dx [x⁵] = 5x⁴

Standard Derivatives

f(x)	f ′(x)
xⁿ	nxⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	−sin x

Applications of Differentiation

Finding gradient at a point: substitute x into f ′(x)
Stationary points: solve f ′(x) = 0
Nature of stationary points: if f ″(x) > 0 → minimum; f ″(x) < 0 → maximum
Velocity v = ds/dt; Acceleration a = dv/dt

Integration (Reverse Differentiation)

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ −1) Definite integral: ∫[a→b] f(x) dx = F(b) − F(a) gives area under curve

Measures of Central Tendency

Mean x̄ = (Σx) / n (sum of all values ÷ number of values) Median = middle value when data ordered (average of two middle values if n is even) Mode = most frequently occurring value (can have more than one mode)

Data: 3, 7, 7, 8, 10, 12, 15

Mean = (3+7+7+8+10+12+15)/7 = 62/7 ≈ 8.86

Median = 4th value = 8

Mode = 7 (appears twice)

Measures of Spread

Range = max − min Lower quartile Q₁ = median of lower half Upper quartile Q₃ = median of upper half Interquartile Range (IQR) = Q₃ − Q₁ Variance σ² = Σ(x−x̄)² / n Standard Deviation σ = √[Σ(x−x̄)² / n] A larger standard deviation means data is more spread out from the mean.

Frequency Tables & Grouped Data

Mean from frequency table: x̄ = Σ(fx) / Σf (f = frequency, x = midpoint of class interval) Estimated mean from grouped data uses class midpoints. Modal class = class interval with highest frequency. Median class = class interval containing the (n/2)th value.

Grouped data — find estimated mean

Class	Midpoint (x)	Freq (f)	fx
10–19	14.5	4	58
20–29	24.5	10	245
30–39	34.5	6	207

x̄ = (58+245+207)/(4+10+6) = 510/20 = 25.5

Representing Data

Display	Best Used For	Key Feature
Bar chart	Discrete/categorical data	Gaps between bars
Histogram	Continuous grouped data	Area ∝ frequency; frequency density = f/class width
Pie chart	Proportions of a whole	Angle = (f/total)×360°
Box-and-whisker	Spread and quartiles	Shows min, Q₁, median, Q₃, max
Scatter diagram	Correlation between two variables	Line of best fit (LOBF) through mean point
Cumulative frequency curve	Finding medians and quartiles from grouped data	S-shaped ogive

Correlation

Type	Description	Example
Strong positive	Points close to upward LOBF	Height vs shoe size
Weak positive	Points loosely around upward LOBF	Age vs blood pressure
Negative	Points around downward LOBF	Speed vs journey time
No correlation	Scattered randomly	Shoe size vs intelligence

Correlation ≠ Causation

A strong correlation means two variables are related, but does NOT prove one causes the other. A third (confounding) variable may be responsible.

Probability

P(event) = favourable outcomes / total equally likely outcomes [0 ≤ P ≤ 1] P(A') = 1 − P(A) (complement) P(A or B) = P(A)+P(B)−P(A and B) (addition rule) P(A and B) = P(A)×P(B) (independent events only) P(A|B) = P(A and B)/P(B) (conditional probability)

Tree diagram — Two draws from bag: 3R, 2B (without replacement)

P(RR) = 3/5 × 2/4 = 6/20 = 3/10

P(at least one B) = 1 − P(RR) = 1 − 3/10 = 7/10

Exam Tip

Use tree diagrams for sequential events. Multiply along branches (AND). Add between branches for different outcomes (OR).

Permutations & Combinations (NSSCAS)

Permutation (order matters): P(n,r) = n!/(n−r)! Combination (order doesn't matter): C(n,r) = n!/[r!(n−r)!] Example: Choose 3 from 5 people for a committee: C(5,3)=10 ways Arrange 3 from 5 in a line: P(5,3)=60 ways

Physics

NSSCONSSCAS

Newton's Laws of Motion

Law	Statement	Real-world example
1st (Inertia)	An object remains at rest or uniform motion unless a resultant force acts on it.	Passenger lurches forward when bus brakes suddenly
2nd	Resultant force = rate of change of momentum: F = ma	Heavier truck needs more force to accelerate at same rate
3rd	Every action has an equal and opposite reaction.	Rocket exhaust pushes down → rocket pushed up

F = ma (N = kg·m/s²) W = mg (weight; g = 9.8 m/s² on Earth) p = mv (momentum, kg·m/s) Fnet = Δp/Δt (Newton's 2nd in terms of momentum)

Free-body diagram — box on surface with applied force: ↑ N (Normal reaction = mg if horizontal) F → [BOX] ← f (friction = μN) ↓ W = mg Fnet (horizontal) = F − f = ma

Friction

f = μN (friction force = coefficient of friction × normal force) Static friction (μs): opposes start of motion; usually larger than kinetic. Kinetic friction (μk): opposes motion once sliding has begun.

Kinematic Equations (SUVAT)

Variables: s=displacement(m), u=initial vel(m/s), v=final vel(m/s), a=acceleration(m/s²), t=time(s) v = u + at s = ut + ½at² v² = u² + 2as s = ½(u+v)t These equations only apply when acceleration is CONSTANT.

Car braking — u=30 m/s, v=0, a=−5 m/s². Find s and t.

t = (v−u)/a = (0−30)/(−5) = 6 s

s = (v²−u²)/(2a) = (0−900)/(−10) = 90 m

Projectile Motion

A projectile has constant horizontal velocity and constant downward acceleration g. Treat horizontal and vertical components independently.

Horizontal: x = u·cosθ · t (no acceleration) Vertical: y = u·sinθ · t − ½gt² (a = −g) Time of flight: T = 2u·sinθ / g Range: R = u²·sin2θ / g (maximum at θ=45°) Max height: H = (u·sinθ)² / (2g)

Work, Energy & Power

W = Fs cosθ (work done; J) KE = ½mv² (kinetic energy; J) GPE = mgh (gravitational PE; J) Elastic PE = ½kx² (spring, k=spring constant, x=extension) Conservation of mechanical energy: KE₁ + GPE₁ = KE₂ + GPE₂ (no friction) P = W/t = Fv (power; W = J/s) Efficiency = (useful output / total input) × 100%

Roller-coaster — mass 500 kg drops h=20 m (no friction). Find speed at bottom.

GPE lost = KE gained: mgh = ½mv²

v = √(2gh) = √(2×9.8×20) = √392 ≈ 19.8 m/s

Momentum & Impulse

Impulse J = Ft = Δp = mv − mu (N·s) Conservation of momentum (isolated system): m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ Elastic collision: KE conserved AND momentum conserved Inelastic collision: momentum conserved; KE NOT conserved Perfectly inelastic: objects stick together after collision

Two trolleys collide and stick. m₁=3 kg, u₁=4 m/s; m₂=1 kg, u₂=0.

Total momentum before = 3×4 + 1×0 = 12 kg·m/s

v = 12/(3+1) = 3 m/s

KE before=24 J; KE after=18 J → 6 J lost to heat/sound (inelastic)

Circular Motion

v = 2πr/T = rω (speed; ω = angular velocity in rad/s) a = v²/r = rω² (centripetal acceleration, directed toward centre) F = mv²/r = mrω² (centripetal force) Centripetal force is not a new force — it is the resultant of existing forces (tension, gravity, friction) directed toward the centre.

Wave Properties

Transverse waves: oscillations perpendicular to direction of travel (e.g. light, water, EM waves)
Longitudinal waves: oscillations parallel to direction of travel — compressions & rarefactions (e.g. sound)

Quantity	Symbol	Unit	Definition
Amplitude	A	m	Maximum displacement from equilibrium
Wavelength	λ	m	Distance of one complete cycle
Frequency	f	Hz	Number of cycles per second
Period	T	s	Time for one complete cycle; T=1/f
Wave speed	v	m/s	v = fλ

Transverse wave: ___ ___ / \ / \ ← amplitude A (above equilibrium) ---/-----\-----/-----\--- equilibrium line \ / \ / \_/ \_/ |← λ (wavelength) →|

Find the speed of a wave with f=500 Hz, λ=0.68 m

v = fλ = 500 × 0.68 = 340 m/s (speed of sound in air)

The Electromagnetic Spectrum

Type	Wavelength (approx.)	Uses
Radio waves	>10 cm	Broadcasting, communications
Microwaves	1 mm – 10 cm	Cooking, satellite, radar
Infrared	700 nm – 1 mm	Remote controls, thermal imaging
Visible light	400 – 700 nm	Vision, photography
Ultraviolet	10 – 400 nm	Sterilisation, sun tanning
X-rays	0.01 – 10 nm	Medical imaging
Gamma rays	<0.01 nm	Cancer treatment, sterilising food

All EM waves

Travel at c = 3×10⁸ m/s in a vacuum. They are transverse waves and do NOT require a medium.

Reflection, Refraction & Diffraction

Phenomenon	Definition	Key rule
Reflection	Wave bounces off surface	Angle of incidence = angle of reflection (both from normal)
Refraction	Wave changes speed at a boundary	Slows down → bends toward normal; speeds up → bends away
Diffraction	Wave spreads through a gap or around obstacle	Most pronounced when gap width ≈ λ
Interference	Two waves superpose	Constructive: crests align; Destructive: crest + trough

Snell's Law of Refraction

n₁ sinθ₁ = n₂ sinθ₂ n = c/v (refractive index = speed in vacuum / speed in medium) Critical angle C: sinC = n₂/n₁ (light going from dense → less dense) If angle of incidence > C → Total Internal Reflection (TIR) Applications of TIR: optical fibres, diamonds, periscopes

Glass (n=1.5) to air (n=1.0). Find critical angle C.

sinC = 1.0/1.5 = 0.667 → C = sin⁻¹(0.667) = 41.8°

Sound Waves

Speed of sound in air ≈ 340 m/s; in water ≈ 1500 m/s; in steel ≈ 5000 m/s
Pitch depends on frequency; loudness depends on amplitude
Echo: reflected sound wave; used in sonar and ultrasound
Resonance: object vibrates at maximum amplitude when driven at its natural frequency

Echo distance: d = v×t/2 Divide by 2 because the sound travels to the object AND back.

Doppler Effect

When a source moves toward an observer, wavefronts are compressed → higher observed frequency (blue shift). When moving away → lower frequency (red shift).

f_observed = f_source × (v ± v_observer) / (v ∓ v_source) Use + when moving toward, − when moving away. Application: police speed cameras, astronomy (measuring star motion), weather radar.

Fundamental Quantities

Q = It (charge = current × time; C = A·s) I = Q/t (current; A) V = W/Q (potential difference; V = J/C) R = V/I (resistance; Ω — Ohm's Law) P = VI = I²R = V²/R (power; W)

Ohm's Law

V ∝ I (at constant temperature). A conductor obeys Ohm's Law if its resistance stays constant. Components like diodes and filament bulbs are non-ohmic (resistance changes with temperature/current).

Resistance & Resistivity

R = ρL/A ρ = resistivity (Ω·m), L = length, A = cross-sectional area Longer wire → more resistance; thicker wire → less resistance Temperature ↑ → resistance of metals ↑ (more collisions)

Copper wire: ρ=1.7×10⁻⁸ Ω·m, L=2 m, diameter=1 mm

A = π(0.0005)² = 7.85×10⁻⁷ m²

R = (1.7×10⁻⁸ × 2)/(7.85×10⁻⁷) = 0.043 Ω

Series vs Parallel Circuits

Property	Series	Parallel
Current	Same through all components	Splits at junctions; I_total=I₁+I₂+…
Voltage	Splits: V_total=V₁+V₂+…	Same across each branch
Resistance	R_T = R₁+R₂+…	1/R_T = 1/R₁+1/R₂+…
If one component fails	Whole circuit breaks	Others still work

Series: Parallel: +--[R1]--[R2]--+ +--[R1]--+ | | | | [V] [ ] [V] +--[R2]--+ | | | | | +--------------+ +---+--------+

Parallel — R₁=6Ω, R₂=3Ω. Find R_total and currents if V=12V.

1/R_T = 1/6+1/3 = 1/6+2/6 = 3/6 → R_T = 2 Ω

I_total=12/2=6A; I₁=12/6=2A; I₂=12/3=4A ✓ (2+4=6A)

Kirchhoff's Laws

KCL & KVL

KCL (Junction rule): Sum of currents entering a node = sum leaving. (Conservation of charge)

KVL (Loop rule): Sum of EMFs around any closed loop = sum of potential drops. (Conservation of energy)

Internal Resistance — EMF ε, internal resistance r, external R

Terminal voltage: V = ε − Ir

If ε=12V, r=0.5Ω, R=5.5Ω: I=12/6=2A; V=12−2(0.5)=11V

Capacitors

Q = CV (charge = capacitance × voltage; C = F·V) Energy = ½CV² = ½QV = Q²/(2C) Series: 1/C_T = 1/C₁+1/C₂+… Parallel: C_T = C₁+C₂+… (opposite to resistors!) Capacitors store charge. Charging/discharging time depends on RC time constant τ = RC.

Electromagnetic Induction

A changing magnetic flux through a conductor induces an EMF (Faraday's Law). The induced current opposes the change causing it (Lenz's Law).

EMF = −N × ΔΦ/Δt (N = number of turns, Φ = flux in Wb) Φ = BA cosθ (B = field strength, A = area) Transformer: V_s/V_p = N_s/N_p = I_p/I_s (ideal) Step-up: N_s > N_p → voltage increases Step-down: N_s < N_p → voltage decreases

Fleming's Rules

Left-Hand Rule (motor effect): thuMb=Motion, Index=field, Middle=current.
Right-Hand Rule (generator/dynamo): thuMb=Motion, Index=field, Middle=induced current.

Gravitational Fields

g = GM/r² (gravitational field strength) F = Gm₁m₂/r² (Newton's Law of Gravitation) G = 6.67 × 10⁻¹¹ N m² kg⁻²

Electric Fields

F = kQ₁Q₂/r² (Coulomb's Law) E = F/q = kQ/r² (electric field strength) k = 8.99 × 10⁹ N m² C⁻²

Magnetic Fields & Forces

F = BIL sin θ (force on current-carrying conductor) F = Bqv sin θ (force on moving charge) Direction: Fleming's Left-Hand Rule for motors

Temperature & Thermal Energy

Q = mcΔT (heat energy) Q = mL (latent heat) ΔU = Q − W (First Law of Thermodynamics)

Gas Laws

Law	Constant	Equation
Boyle's	Temperature	pV = constant → p₁V₁ = p₂V₂
Charles's	Pressure	V/T = constant → V₁/T₁ = V₂/T₂
Pressure Law	Volume	p/T = constant → p₁/T₁ = p₂/T₂
Ideal Gas	—	pV = nRT (R = 8.31 J mol⁻¹ K⁻¹)

Important

Always use absolute temperature (Kelvin): T(K) = T(°C) + 273

Nuclear Structure & Notation

²³⁸₉₂U → mass number (A) on top, atomic number (Z) below Protons = Z; Neutrons = A − Z; Electrons = Z (neutral atom) Nuclear radius ≈ 10⁻¹⁵ m (nucleus is ~10,000× smaller than atom)

Radioactive Decay

Radiation	Symbol	Nature	Penetration	Ionising power
Alpha	α (⁴₂He)	2 protons + 2 neutrons	Stopped by paper / few cm air	Highest
Beta-minus	β⁻ (e⁻)	Fast electron (neutron → proton)	Few mm aluminium	Medium
Beta-plus	β⁺ (e⁺)	Positron (proton → neutron)	Few mm aluminium	Medium
Gamma	γ	High-energy EM photon	Reduced by thick lead/concrete	Lowest

Alpha decay: ᴬ_Z X → ᴬ⁻⁴_(Z-2) Y + ⁴₂He Beta⁻ decay: ᴬ_Z X → ᴬ_(Z+1) Y + ⁰_(-1)e + antineutrino Gamma: follows alpha or beta; nucleus loses energy only (no change in A or Z)

Alpha decay of Radium-226

²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He

Check: A: 226=222+4 ✓; Z: 88=86+2 ✓

Half-Life

N = N₀ × (½)^(t/t½) A = A₀ × (½)^(t/t½) (activity also halves) N₀ = initial number of nuclei; t½ = half-life

t½ = 5 years. After 20 years, what fraction remains?

t/t½ = 20/5 = 4 half-lives

Fraction remaining = (½)⁴ = 1/16 = 6.25%

Uses of Radioactivity

Medical: Technetium-99m (diagnostic imaging); Iodine-131 (thyroid cancer treatment)
Industrial: Thickness gauges (beta radiation); smoke detectors (alpha — Americium-241)
Archaeological: Carbon-14 dating (t½=5730 years; measures age of organic material)

Nuclear Reactions: Fission & Fusion

Fission: heavy nucleus splits → releases huge energy ²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + energy (chain reaction possible → nuclear power/weapons) Fusion: light nuclei combine → even more energy per unit mass ²₁H + ³₁H → ⁴₂He + ¹₀n + energy (requires ~10⁷ K) (powers stars; future clean energy — ITER project)

Mass-energy equivalence: E = mc² c = 3×10⁸ m/s Mass defect (Δm) = mass of reactants − mass of products Binding energy = Δm × c² The binding energy per nucleon is greatest for iron-56 — it is the most stable nucleus.

Photoelectric Effect

E = hf (photon energy; h=6.63×10⁻³⁴ J·s) hf = φ + ½mv²_max (Einstein's equation; φ=work function) Threshold frequency f₀ = φ/h No electrons emitted if f < f₀, regardless of intensity. Above f₀, KE of electrons increases with frequency, not intensity. Intensity only affects the NUMBER of emitted electrons.

Wave-Particle Duality

Light behaves as a wave in diffraction/interference; as a particle (photon) in the photoelectric effect.
Matter (electrons) also has wave properties: λ = h/(mv) — de Broglie wavelength.
Evidence: electron diffraction patterns confirm wave nature of electrons.

Energy Levels & Atomic Spectra

Electrons in atoms occupy discrete energy levels. When an electron drops to a lower level, it emits a photon of specific energy (frequency). When it absorbs a photon, it jumps to a higher level.

ΔE = hf = hc/λ (energy of photon emitted or absorbed) Line emission spectra: bright lines on dark background (electrons falling to lower levels) Line absorption spectra: dark lines on continuous spectrum (electrons absorbing specific frequencies) Application: identifying elements in stars from their absorption spectra.

Biology

JSCNSSCO

Cell Theory

All living things are made of cells
The cell is the basic unit of structure and function in all organisms
All cells arise from pre-existing cells (cell division)

Cell Ultrastructure

Organelle	Function	Animal	Plant
Cell membrane	Controls entry/exit of substances; partially permeable	✓	✓
Cell wall (cellulose)	Provides rigid support; fully permeable	✗	✓
Nucleus	Contains DNA; controls cell activities	✓	✓
Mitochondria	Site of aerobic respiration; produces ATP	✓	✓
Ribosomes	Site of protein synthesis	✓	✓
Endoplasmic reticulum (ER)	Rough ER: protein transport; Smooth ER: lipid synthesis	✓	✓
Golgi apparatus	Packages and secretes proteins	✓	✓
Chloroplasts	Site of photosynthesis; contain chlorophyll	✗	✓
Large central vacuole	Stores water; maintains turgor pressure	✗	✓
Lysosomes	Contain digestive enzymes; break down waste	✓	✗

Prokaryotes vs Eukaryotes

Feature	Prokaryote (bacteria)	Eukaryote (animals, plants, fungi)
Nucleus	No membrane-bound nucleus; DNA in nucleoid region	True nucleus with membrane
Size	1–10 μm (smaller)	10–100 μm (larger)
Organelles	No membrane-bound organelles	Has mitochondria, ER, Golgi, etc.
Ribosomes	70S (smaller)	80S (larger)
Cell wall	Present (peptidoglycan)	Plants: cellulose; Animals: absent

Osmosis, Diffusion & Active Transport

Transport Mechanisms Compared

Diffusion: Net movement of molecules from high → low concentration (passive; no energy needed). Affected by: concentration gradient, temperature, surface area, distance.

Osmosis: Movement of water molecules through a partially permeable membrane from dilute (high water potential) → concentrated (low water potential) solution.

Active transport: Movement of molecules against a concentration gradient using ATP energy and carrier proteins (e.g. reabsorption of glucose in kidney tubules; sodium-potassium pump in nerve cells).

Osmosis in plant cells

Turgid cell: in dilute solution → water enters by osmosis → vacuole swells → cell wall resists → high turgor pressure (plant stays upright)
Plasmolysed cell: in concentrated solution → water leaves by osmosis → vacuole shrinks → membrane pulls away from cell wall (plasmolysis)

Enzymes

Enzymes are biological catalysts — protein molecules that speed up metabolic reactions without being consumed. Each enzyme has an active site complementary in shape to its specific substrate (lock-and-key model).

Enzyme + Substrate → Enzyme-Substrate Complex → Enzyme + Products

Factor	Effect on enzyme activity
Temperature	Activity increases up to optimum (~37°C in humans); above optimum → denaturation (active site changes shape irreversibly)
pH	Each enzyme has optimum pH (pepsin pH 2; amylase pH 7); extreme pH → denaturation
Substrate concentration	Activity increases until all active sites are occupied (saturation point)
Enzyme concentration	Activity increases if substrate is in excess

Cell Division: Mitosis & Meiosis

Feature	Mitosis	Meiosis
Purpose	Growth, repair, asexual reproduction	Sexual reproduction — produces gametes
Number of divisions	1	2 (meiosis I and II)
Daughter cells produced	2 diploid (2n) cells	4 haploid (n) cells
Genetic content	Identical to parent cell (clones)	Genetically unique (crossing over + independent assortment)
Location in body	All dividing body cells	Gonads (testes, ovaries)

Stages of Mitosis (PMAT)

Prophase: chromosomes condense; spindle forms
Metaphase: chromosomes line up at equator
Anaphase: sister chromatids pulled to opposite poles
Telophase: nuclear envelopes reform; cytoplasm divides (cytokinesis)

Nutrients & Their Roles

Nutrient	Role	Deficiency disease	Source
Carbohydrates	Primary energy supply; structural (cellulose)	Kwashiorkor (combined with protein deficiency)	Bread, rice, potatoes, maize
Proteins	Growth, repair, enzymes, antibodies, hormones	Kwashiorkor, Marasmus	Meat, fish, eggs, legumes
Fats (lipids)	Long-term energy store; insulation; cell membranes; fat-soluble vitamins	Rare; excess leads to obesity	Oils, butter, nuts, avocado
Vitamin C	Collagen synthesis; immune function	Scurvy (bleeding gums)	Citrus fruits, peppers
Vitamin D	Calcium absorption; bone formation	Rickets (soft bones in children)	Sunlight, fish oils, dairy
Iron	Component of haemoglobin (carries O₂)	Anaemia (fatigue, pale skin)	Red meat, spinach, legumes
Calcium	Bone and teeth formation; muscle contraction	Osteoporosis	Dairy, leafy greens
Water	Universal solvent; transport; temperature regulation; chemical reactions	Dehydration	Fluids, food

Food Tests

Starch: Iodine solution → blue-black if positive
Glucose: Benedict's solution + heat → brick-red precipitate if positive
Protein: Biuret reagent → purple/violet if positive
Fats: Ethanol emulsion test → milky-white emulsion if positive

Photosynthesis

Green plants manufacture glucose from carbon dioxide and water using light energy absorbed by chlorophyll.

6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ (light energy, chlorophyll) Light-dependent stage (thylakoid membranes): Light splits water (photolysis): H₂O → H⁺ + O₂ Produces: ATP, NADPH, O₂ (released as by-product) Light-independent stage / Calvin Cycle (stroma): CO₂ fixed by RuBisCO enzyme into 3-carbon compounds Uses ATP + NADPH to produce G3P → glucose

Factor Limiting Photosynthesis	Effect when increased
Light intensity	Rate increases until another factor limits
CO₂ concentration	Rate increases until another factor limits
Temperature	Rate increases up to enzyme optimum; decreases above (denaturation)
Water availability	Shortage closes stomata → less CO₂ entry → rate decreases

Compensation point

The light intensity at which the rate of photosynthesis exactly equals the rate of respiration — no net gas exchange occurs. Above this point, the plant gains biomass.

Aerobic Respiration

Aerobic respiration releases energy (ATP) from glucose in the presence of oxygen. It is the most efficient respiratory pathway.

C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O + ~38 ATP Stage 1: Glycolysis (cytoplasm) Glucose (6C) → 2× Pyruvate (3C) + 2 ATP + 2 NADH Stage 2: Krebs Cycle / Citric Acid Cycle (mitochondrial matrix) Pyruvate → Acetyl-CoA → enters cycle Produces: CO₂, NADH, FADH₂, 2 ATP per glucose Stage 3: Oxidative Phosphorylation / Electron Transport Chain (inner mitochondrial membrane) NADH, FADH₂ donate electrons → H⁺ gradient → ATP synthase → ~34 ATP Oxygen is the final electron acceptor → forms H₂O

Anaerobic Respiration

In animals / humans (e.g. sprinting): Glucose → Lactic acid + 2 ATP (lactic acid causes muscle fatigue; repaid by O₂ debt during recovery) In yeast (fermentation): Glucose → Ethanol + CO₂ + 2 ATP (used in baking — CO₂ makes bread rise; brewing — ethanol is alcohol) Anaerobic respiration only produces 2 ATP per glucose vs ~38 for aerobic.

The Digestive System

Organ	Function	Enzymes / Secretions
Mouth	Mechanical digestion (teeth); chemical digestion starts	Salivary amylase (starch → maltose)
Stomach	Churns food; protein digestion	Pepsin (protein→peptides); HCl (pH 2; kills bacteria)
Small intestine	Main site of digestion and absorption	Pancreatic amylase, lipase, proteases; bile (emulsifies fats)
Large intestine	Water reabsorption; formation of faeces	Bacteria produce some vitamins
Liver	Produces bile; detoxification; glycogen storage; urea production	Bile salts (not enzymes)

Villi Adaptations

Villi in the small intestine increase surface area for absorption. Each villus has microvilli (brush border), a thin wall (one cell thick), a dense capillary network (glucose, amino acids), and lacteals (fatty acids, glycerol).

The Human Circulatory System

A double circulatory system: pulmonary (heart ↔ lungs) and systemic (heart ↔ body).

Vessel	Carries	Wall
Artery	Blood away from heart	Thick, muscular, elastic
Vein	Blood toward heart	Thin, has valves
Capillary	Exchange of gases/nutrients	One cell thick

Blood Components

Red blood cells: carry O₂ via haemoglobin; no nucleus
White blood cells: immune defence (phagocytes, lymphocytes)
Platelets: blood clotting
Plasma: liquid; transports dissolved substances

Transpiration in Plants

Water moves up a plant by the transpiration stream: roots → xylem → leaves → evaporates through stomata.

Exam Tip

Factors increasing transpiration: high temperature, low humidity, high wind speed, high light intensity.

DNA Structure

DNA is a double helix of two polynucleotide strands. Each nucleotide consists of a deoxyribose sugar, phosphate group, and a nitrogenous base (A, T, G, C).

Complementary Base Pairing

Adenine (A) pairs with Thymine (T) — 2 H-bonds
Guanine (G) pairs with Cytosine (C) — 3 H-bonds

Mendelian Genetics

Term	Definition
Allele	Alternative form of a gene
Homozygous	Two identical alleles (AA or aa)
Heterozygous	Two different alleles (Aa)
Dominant	Expressed when one or two copies present
Recessive	Only expressed when homozygous
Phenotype	Observable characteristic
Genotype	Genetic make-up

Punnett Square

Parents: Tt × Tt (T = tall, t = short) T t +--------+--------+ T | TT | Tt | +--------+--------+ t | Tt | tt | +--------+--------+ Ratio: 3 tall : 1 short (phenotype)

Ecosystems & Biomes

An ecosystem includes all living organisms (biotic) and their non-living environment (abiotic factors: temperature, water, light, soil) interacting in an area.

Food Chains & Food Webs

Grass → Locust → Lizard → Snake → Eagle (Producer)→(Primary)→(Secondary)→(Tertiary)→(Quaternary) Energy decreases ~90% at each trophic level

Energy & Nutrient Cycles

Carbon cycle: photosynthesis (fixes CO₂), respiration & combustion (release CO₂)
Nitrogen cycle: fixation → nitrification → assimilation → denitrification
Water cycle: evaporation, condensation, precipitation, run-off

Human Impact

Deforestation reduces biodiversity and increases CO₂
Eutrophication: excess fertiliser → algal bloom → O₂ depletion
Greenhouse effect: CO₂, CH₄, N₂O trap heat
Desertification: overgrazing, drought, land degradation (relevant to Namibia)

Diseases & Pathogens

Pathogen	Examples	Treatment
Bacteria	TB, cholera, typhoid	Antibiotics
Virus	HIV/AIDS, influenza	Antivirals, vaccines
Fungi	Ringworm, thrush	Antifungals

The Immune System

Non-specific defence: skin, mucus, stomach acid
Phagocytosis: white blood cells engulf pathogens
Lymphocytes: B-cells produce antibodies; T-cells destroy infected cells
Vaccination: introduces antigens to stimulate memory cells

Namibia Context

HIV/AIDS remains a significant public health challenge. Understanding transmission and prevention (condoms, ART, education) is essential.

Chemistry

JSCNSSCO

Sub-atomic Particles

Particle	Mass (amu)	Charge	Location
Proton	1	+1	Nucleus
Neutron	1	0	Nucleus
Electron	≈ 1/1840	−1	Electron shells / orbitals

Atomic number (Z) = number of protons Mass number (A) = protons + neutrons Number of neutrons = A − Z Number of electrons = Z (in neutral atom)

Isotopes & Relative Atomic Mass

Isotopes are atoms of the same element with the same proton number but different neutron numbers (different mass numbers). They have identical chemical properties but slightly different physical properties.

Relative Atomic Mass (Aᵣ) = weighted average of isotope masses Example — Chlorine: ³⁵Cl (75%) and ³⁷Cl (25%) Aᵣ = (75×35 + 25×37)/100 = (2625 + 925)/100 = 35.5

Electron Configuration

Electrons fill shells in order of increasing energy: shell 1 holds max 2; shell 2 holds max 8; shell 3 holds max 8 (at NSSCO level). The arrangement determines chemical reactivity.

Element	Z	Config	Outer electrons	Ion formed
Hydrogen	1	1	1	H⁺
Carbon	6	2,4	4	Forms covalent bonds
Sodium	11	2,8,1	1	Na⁺
Magnesium	12	2,8,2	2	Mg²⁺
Chlorine	17	2,8,7	7	Cl⁻
Argon	18	2,8,8	8	No ion (stable)

The Periodic Table

Elements are arranged in order of increasing atomic number. Vertical columns are groups (same number of outer electrons → similar chemistry). Horizontal rows are periods (same number of shells).

Trends Across a Period (left → right)

Atomic radius decreases (more protons pull electrons closer)
Ionisation energy increases (harder to remove electron)
Electronegativity increases
Metallic character decreases; non-metallic character increases

Trends Down a Group (top → bottom)

Atomic radius increases (more electron shells)
Ionisation energy decreases (outer electrons further from nucleus)
Reactivity of metals increases (easier to lose electrons)
Reactivity of non-metals decreases (harder to gain electrons)

Important Groups

Group	Name	Key Properties
Group 1	Alkali metals (Li,Na,K…)	Soft; low density; react vigorously with water → metal hydroxide + H₂; reactivity increases down group
Group 7	Halogens (F,Cl,Br,I…)	Diatomic molecules; form −1 ions; coloured gases/liquids; reactivity decreases down group; more reactive halogen displaces less reactive
Group 0	Noble gases (He,Ne,Ar…)	Full outer shells; inert; monatomic; very low boiling points
Transition metals	Fe, Cu, Zn, Cr…	High melting points; conduct electricity; variable oxidation states; coloured compounds; good catalysts

Halogen displacement — More reactive halogen displaces less reactive

Cl₂(aq) + 2KBr(aq) → 2KCl(aq) + Br₂(aq)

Chlorine (higher up Group 7) displaces bromine. The solution turns orange-brown.

Types of Chemical Bonding

Bond	Between	How	Example
Ionic	Metal + non-metal	Transfer of electrons	NaCl
Covalent	Non-metal + non-metal	Sharing electron pairs	H₂O, CO₂
Metallic	Metal + metal	Delocalised electrons	Cu, Fe

Ionic (Na + Cl): Na• + •Cl: → [Na]⁺ [Cl]⁻ Covalent (H₂): H• + •H → H:H (shared pair = H—H)

Intermolecular Forces

London dispersion: weak, in all molecules
Dipole-dipole: polar molecules
Hydrogen bonding: strongest; N–H, O–H, F–H groups; explains water's high boiling point

Balancing Equations

Law of Conservation of Mass

Atoms are neither created nor destroyed — the number of each type of atom must be equal on both sides.

Combustion of methane

Unbalanced: CH₄ + O₂ → CO₂ + H₂O

Balanced: CH₄ + 2O₂ → CO₂ + 2H₂O

Iron + chlorine

2Fe + 3Cl₂ → 2FeCl₃ (check: 2Fe, 6Cl each side ✓)

Types of Reactions

Type	General form	Example
Synthesis / Combination	A + B → AB	2H₂ + O₂ → 2H₂O
Decomposition	AB → A + B	CaCO₃ →(heat) CaO + CO₂
Thermal decomposition	Heat breaks compound	Cu(NO₃)₂ → CuO + NO₂ + O₂
Displacement	More reactive displaces less reactive	Fe + CuSO₄ → FeSO₄ + Cu
Redox	Oxidation + reduction simultaneously	Zn + H₂SO₄ → ZnSO₄ + H₂
Precipitation	Two solutions form an insoluble solid	Pb(NO₃)₂ + 2KI → PbI₂↓ + 2KNO₃

Oxidation & Reduction (Redox)

OIL RIG — Oxidation Is Loss (of electrons), Reduction Is Gain (of electrons) Oxidation state rules: Pure element: 0 Monatomic ion: = charge (e.g. Na⁺ = +1) Oxygen in compound: −2 (except peroxides: −1) Hydrogen in compound: +1 (except metal hydrides: −1) Sum in compound: = overall charge Example — find Mn in KMnO₄: +1 + Mn + 4(−2) = 0 → Mn = +7

Oxidising & Reducing Agents

Oxidising agent: causes oxidation of something else; itself gets reduced (gains electrons).
Reducing agent: causes reduction of something else; itself gets oxidised (loses electrons).

Reactivity Series of Metals

Most reactive (top): K — Potassium (reacts explosively with water) Na — Sodium (reacts vigorously with water) Ca — Calcium (reacts slowly with water) Mg — Magnesium (reacts with steam) Al — Aluminium Zn — Zinc Fe — Iron Pb — Lead Cu — Copper Ag — Silver Au — Gold Least reactive (bottom)

Mnemonic

"Please Stop Calling Me A Zebra In London, Call A Soldier Gold" (K, Na, Ca, Mg, Al, Zn, Fe, Pb, Cu, Ag, Au)

Moles & Stoichiometry

n = m/M (moles = mass ÷ molar mass) n = cV (moles = concentration × volume in litres) n = V_gas/22.4 (moles = volume ÷ 22.4 L/mol at STP) Avogadro: 1 mol = 6.02×10²³ particles

Mass of CO₂ produced when 10 g CaCO₃ decomposes

CaCO₃ → CaO + CO₂ (1:1 molar ratio)

M(CaCO₃)=100 g/mol → n=10/100=0.1 mol CO₂

m(CO₂)=0.1×44=4.4 g

Titration — 25 cm³ of NaOH neutralised by 20 cm³ of 0.1 mol/L HCl

n(HCl)=0.02×0.1=0.002 mol; NaOH+HCl→NaCl+H₂O (1:1)

n(NaOH)=0.002 mol; c(NaOH)=0.002/0.025=0.08 mol/L

Rates of Reaction

Rate = change in concentration (or mass) ÷ time. Factors that increase rate:

Temperature ↑: particles have more kinetic energy → more frequent, more energetic collisions
Concentration ↑: more particles per unit volume → more frequent collisions
Surface area ↑: more particles exposed for reaction (e.g. powder vs lump)
Catalyst: provides alternative pathway with lower activation energy
Pressure ↑ (gases): effectively increases concentration

Collision Theory

For a reaction to occur, colliding particles must: (1) actually collide, and (2) have energy ≥ the activation energy (Eₐ).

Acids, Bases & pH

pH = −log[H⁺] pH < 7 → acidic | pH = 7 → neutral | pH > 7 → alkaline

Acid-Base Reactions

Reaction	Products
Acid + metal	Salt + hydrogen
Acid + base	Salt + water
Acid + carbonate	Salt + water + CO₂

Strong vs Weak Acids

Strong acids: fully dissociate (HCl, H₂SO₄, HNO₃)
Weak acids: partially dissociate (CH₃COOH, citric acid)

Electrolysis

Rules

Cathode (−): cations reduced. Anode (+): anions oxidised.

Homologous Series

Series	Formula	Group	Example
Alkanes	CₙH₂ₙ₊₂	C–C	CH₄ methane
Alkenes	CₙH₂ₙ	C=C	C₂H₄ ethene
Alcohols	CₙH₂ₙ₊₁OH	–OH	C₂H₅OH ethanol
Carboxylic acids	CₙH₂ₙ₊₁COOH	–COOH	CH₃COOH acetic

Addition vs Substitution

Addition: alkenes + Br₂ → dibromoalkane (decolourises bromine water — test for C=C)
Substitution: alkanes + Cl₂ (UV) → chloroalkane + HCl

Polymers

Addition polymerisation: alkene monomers join to form a long-chain polymer. E.g. ethene → poly(ethene) (polythene).

Flame Tests

Ion	Flame Colour
Li⁺	Crimson red
Na⁺	Yellow/orange
K⁺	Lilac/violet
Ca²⁺	Brick red
Cu²⁺	Blue-green

Testing for Gases

Gas	Test	Positive Result
H₂	Burning splint	Squeaky pop
O₂	Glowing splint	Splint relights
CO₂	Limewater	Turns milky
Cl₂	Damp litmus	Bleaches/turns white
NH₃	Damp red litmus	Turns blue

Computer Science

NSSCONSSCAS

The CPU

ALU: performs calculations and logical operations
CU: fetches, decodes and executes instructions
Registers: ultra-fast temporary storage (PC, ACC, MAR, MDR, CIR)
Cache: faster than RAM; stores frequently used data close to CPU

Fetch-Decode-Execute Cycle

1. FETCH: MAR ← PC MDR ← memory[MAR] CIR ← MDR PC ← PC + 1 2. DECODE: CU decodes instruction in CIR 3. EXECUTE: ALU or CU carries out operation

Memory: RAM vs ROM

	RAM	ROM
Full name	Random Access Memory	Read-Only Memory
Volatile?	Yes (lost on power off)	No (permanent)
Writable?	Yes	No (or limited)
Use	Running programs, OS	Firmware/BIOS

Secondary Storage

Device	Type	Pros	Cons
HDD	Magnetic	Large capacity, cheap	Slow, mechanical
SSD	Flash	Fast, silent, durable	More expensive/GB
Optical (CD/DVD)	Laser	Portable	Small capacity, slow

Binary Number System

Convert 1011₂ to decimal: 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8+0+2+1 = 11₁₀ Convert 25₁₀ to binary: 25÷2=12 r1 | 12÷2=6 r0 | 6÷2=3 r0 3÷2=1 r1 | 1÷2=0 r1 → read up = 11001₂

Hexadecimal

Base-16: digits 0–9 and A(10)–F(15). Each hex digit = 4 bits.

Binary 1010 1101₂ = AD₁₆ 1010 = A (10), 1101 = D (13)

Representing Data

Data Type	Representation
Integers	Two's complement for negative numbers
Characters	ASCII (7-bit, 128 chars) or Unicode
Floating point	Sign + mantissa + exponent
Images	Pixels × bit depth = file size
Sound	Sample rate × bit depth × duration

Flowchart Symbols

[Oval] → START / END [Rectangle] → Process (assignment, calculation) [Diamond] → Decision (yes/no branch) [Parallelogram] → Input / Output

Sorting Algorithms

Algorithm	Best	Worst	Method
Bubble sort	O(n)	O(n²)	Swap adjacent out-of-order elements
Selection sort	O(n²)	O(n²)	Find minimum, place at start, repeat
Merge sort	O(n log n)	O(n log n)	Divide, sort, merge (recursive)

Searching Algorithms

Linear search: check each element — O(n)
Binary search: sorted list; halve search space — O(log n)

Binary Search — Find 42 in [10, 21, 35, 42, 58, 72, 90]

Mid = index 3 = 42 → Found!
If target < mid → search left half; if target > mid → search right half

Variables & Data Types

Type	Description	Example
Integer	Whole number	42
Float/Real	Decimal number	3.14
String	Text	"Hello"
Boolean	True or False	TRUE
Char	Single character	'A'

Pseudocode

INPUT name OUTPUT "Hello, " & name FOR i ← 1 TO 10 DO OUTPUT i ENDFOR WHILE count < 10 DO count ← count + 1 ENDWHILE IF score >= 50 THEN OUTPUT "Pass" ELSE OUTPUT "Fail" ENDIF

Procedures & Functions

PROCEDURE greet(name) OUTPUT "Hello " & name ENDPROCEDURE FUNCTION square(n) RETURN n * n ENDFUNCTION

OOP Concepts

Class: blueprint for objects
Object: instance of a class
Encapsulation: data + methods bundled; access via getters/setters
Inheritance: subclass inherits from superclass
Polymorphism: same method name behaves differently in different classes

Network Types

Type	Scope	Example
PAN	Personal (~10 m)	Bluetooth
LAN	Local (building)	School network
WAN	Wide (country/world)	The Internet

TCP/IP Model

Layer 4: Application (HTTP, FTP, SMTP, DNS) Layer 3: Transport (TCP – reliable; UDP – fast) Layer 2: Internet (IP – addressing & routing) Layer 1: Network Access (Ethernet, Wi-Fi)

Cybersecurity Threats

Threat	Description	Prevention
Malware	Viruses, worms, ransomware	Antivirus, updates
Phishing	Fake emails/sites	Awareness, 2FA
SQL injection	Malicious SQL in forms	Input validation, prepared statements
DDoS	Overwhelm server with traffic	Firewalls, rate limiting

Relational Databases

Data is organised in tables (relations). Each table has a primary key uniquely identifying each record. Tables link via foreign keys.

SQL Basics

-- Select all students SELECT * FROM students; -- Filter records SELECT name, grade FROM students WHERE grade >= 60; -- Insert a record INSERT INTO students (name, grade) VALUES ('Anna', 85); -- Update a record UPDATE students SET grade = 90 WHERE name = 'Anna'; -- Delete a record DELETE FROM students WHERE name = 'Anna'; -- Join two tables SELECT students.name, subjects.subject_name FROM students INNER JOIN enrolments ON students.id = enrolments.student_id INNER JOIN subjects ON enrolments.subject_id = subjects.id;

Database Concepts

Term	Definition
Primary key	Unique identifier for each record
Foreign key	Field linking to primary key of another table
Normalisation	Organising data to reduce redundancy (1NF, 2NF, 3NF)
DBMS	Database Management System (e.g. MySQL, SQLite)

English

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Parts of Speech

Part of Speech	Function	Example
Noun	Names a person, place, thing, idea	Windhoek, freedom
Pronoun	Replaces a noun	he, she, they, it
Verb	Action or state of being	run, is, think
Adjective	Describes a noun	beautiful, large
Adverb	Modifies verb/adjective/adverb	quickly, very
Conjunction	Joins clauses/words	and, but, because
Preposition	Shows relationship	in, on, at, under
Interjection	Expresses emotion	Oh! Wow!

Tenses

Tense	Form	Example
Simple present	base / base+s	She walks to school.
Simple past	-ed / irregular	She walked / ran.
Present continuous	am/is/are + -ing	She is walking.
Present perfect	have/has + past participle	She has walked.
Future simple	will + base	She will walk.

Active vs Passive Voice

Rule

Active: "The teacher marked the papers."
Passive: "The papers were marked by the teacher."

Types of Writing

Type	Purpose	Key Features
Narrative	Tell a story	Plot, characters, setting, conflict
Descriptive	Paint a picture	Sensory details, figurative language
Expository	Inform or explain	Facts, clear structure, topic sentences
Argumentative	Convince reader	Thesis, evidence, counter-argument
Formal letter	Official communication	Address, salutation, sign-off

Essay Structure

Structure

Introduction: Hook + background + thesis statement
Body paragraphs: Topic sentence → Evidence → Explanation → Link back
Conclusion: Restate thesis + summarise + final thought

Figurative Language

Device	Definition	Example
Simile	Comparison using "like" or "as"	As brave as a lion
Metaphor	Direct comparison	Life is a journey
Personification	Human qualities to non-human	The wind howled angrily
Alliteration	Repeated initial consonant	Peter Piper picked…
Hyperbole	Extreme exaggeration	I've told you a million times
Irony	Opposite of what is meant	"What lovely weather" (in a storm)

Comprehension Question Types

Type	Description
Literal	Answer found directly in text
Inferential	Read between the lines
Vocabulary-in-context	Word meaning as used in passage
Summary	Condense in your own words

Answering Tip

Always answer in full sentences and refer back to the text. For quote-and-explain questions, write the quote then explain its effect.

Prose Fiction Analysis

Plot: exposition → rising action → climax → falling action → resolution
Character: protagonist, antagonist; round vs flat; static vs dynamic
Setting: time, place and atmosphere
Theme: central message (e.g. identity, justice, belonging)
Point of view: 1st person, 3rd limited, 3rd omniscient

Poetry Analysis

Rhyme scheme: label end sounds ABAB, AABB etc.
Rhythm/metre: pattern of stressed/unstressed syllables
Imagery: visual, auditory, tactile
Tone: poet's attitude (joyful, melancholic, critical)

PEE Structure

Point → Evidence (quote) → Explanation

Word Formation

Process	Example
Prefix	un- + happy = unhappy
Suffix	comfort + -able = comfortable
Compound	sun + flower = sunflower
Synonym	happy / joyful / elated
Antonym	happy ↔ sad
Homophone	there / their / they're

Spelling Rules

i before e except after c: believe, receive
Drop final e before vowel suffix: hope → hoping
Double consonant before -ing/-ed for short vowels: run → running

NSSCO English Exam Papers

Paper	Content	Tips
Paper 1	Reading comprehension + summary	Read passage twice; annotate key ideas
Paper 2	Language structures & vocabulary	Know grammar rules; re-read after filling blanks
Paper 3	Extended writing	Plan before writing; check spelling & tense

Common Mistakes

Copying straight from text in summaries (paraphrase instead)
Writing in point form instead of full sentences
Not proofreading for tense consistency

Entrepreneurship

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What is Entrepreneurship?

The process of identifying a need in society and creating a business to meet that need, taking on financial risk in the hope of profit.

Characteristics of a Successful Entrepreneur

Risk-taking ability
Innovation and creativity
Self-motivation and discipline
Leadership and communication
Resilience and perseverance
Financial literacy

SWOT Analysis

Letter	Meaning	Internal/External
S	Strengths	Internal (positive)
W	Weaknesses	Internal (negative)
O	Opportunities	External (positive)
T	Threats	External (negative)

The Business Plan

Executive Summary
Business Description (name, location, vision, mission)
Market Analysis (target market, competition)
Products / Services
Marketing Strategy
Financial Plan (start-up costs, income projection)
Management Plan

Break-Even Analysis

Break-even (units) = Fixed Costs / (Selling Price − Variable Cost per unit) Example: Fixed=N$5000, Price=N$50, Variable=N$25 Break-even = 5000/(50−25) = 200 units

Sources of Finance

Source	Type	Advantage	Disadvantage
Personal savings	Internal	No interest, full control	Limited amount
Bank loan	External	Large amounts	Interest, collateral required
Grants (NEFF/DBN)	External	No repayment	Competitive, conditions apply
Investors/Partners	External	Capital + expertise	Share ownership/profits

Cash Flow & Profit

Opening balance + Total receipts − Total payments = Closing balance Gross Profit = Revenue − Cost of Goods Sold Net Profit = Gross Profit − Operating Expenses

The 4 P's of Marketing

P	Description
Product	What you sell; features, quality, branding
Price	Cost-plus, competition-based, penetration pricing
Place	Distribution channels (shop, online, agents)
Promotion	Social media, flyers, radio, word-of-mouth

Market Research Methods

Primary: surveys, interviews, observation
Secondary: reports, internet, existing studies

Forms of Business Ownership

Type	Owners	Liability	Example
Sole Trader	1	Unlimited	Street vendor
Partnership	2–20	Unlimited	Law firm
Close Corporation (CC)	1–10	Limited	Small/medium enterprises
Private Company (Pty)	1–50	Limited	Most SMEs in Namibia
Public Company (Ltd)	Unlimited	Limited	MTC, Namibia Breweries

Namibia Regulatory Bodies

BIPA — registers businesses. NAMRA — tax registration.

Problem-Solving Process

Identify and define the problem
Gather information and analyse causes
Generate possible solutions (brainstorm)
Evaluate and choose the best solution
Implement the solution
Evaluate and review the outcome

Leadership Styles

Style	Description	Best for
Autocratic	Leader makes all decisions	Crisis, quick decisions
Democratic	Team involved in decisions	Creative work, skilled teams
Laissez-faire	Freedom given to team	Expert, self-motivated teams

Business Studies

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PEST Analysis

Factor	Examples
Political	Government policies, taxation, trade regulations
Economic	GDP, inflation, unemployment, exchange rates
Social	Demographics, consumer attitudes, lifestyle trends
Technological	Innovation, automation, e-commerce

Stakeholders

Anyone with an interest: owners, employees, customers, suppliers, government, community.

Stakeholder Conflicts

Shareholders want profits; employees want higher wages; customers want lower prices — management must balance these interests.

Management Functions (POLC)

Function	Description
Planning	Setting objectives and strategies
Organising	Allocating resources and assigning tasks
Leading	Motivating, directing and communicating
Controlling	Monitoring performance against targets

Motivation Theories

Theory	Key Idea
Maslow's Hierarchy	Physiological → Safety → Social → Esteem → Self-actualisation
Herzberg's Two-Factor	Hygiene factors vs Motivators
Taylor (Scientific)	Workers primarily motivated by money; time-and-motion efficiency

Recruitment Process

Identify vacancy → job description & person specification
Advertise (internal or external)
Shortlist applications
Interview (and tests)
Select and offer → induction

Training Types

Type	Description
On-the-job	Learning while working; shadowing, mentoring
Off-the-job	Courses, workshops away from workplace
Induction	Introduction for new employees to policies, safety, culture

Labour Laws in Namibia

The Labour Act (No. 11 of 2007): minimum wage, 45hrs/week max, leave entitlements, unfair dismissal protection, equal opportunity.

Financial Statements

Income Statement: Revenue − Cost of Sales = Gross Profit Gross Profit − Expenses = Net Profit Balance Sheet: Assets = Liabilities + Equity

Ratio Analysis

Ratio	Formula	Meaning
Gross Profit Margin	(Gross Profit / Revenue) × 100	% revenue kept as gross profit
Net Profit Margin	(Net Profit / Revenue) × 100	% revenue kept as net profit
Current Ratio	Current Assets / Current Liabilities	Liquidity; >1 is healthy
Return on Capital	(Net Profit / Capital) × 100	Efficiency of investment

Product Life Cycle

Sales | /‾‾‾‾\ | / \ | / \___ | / |_____/ Intro Growth Maturity Decline → Time

Pricing Strategies

Strategy	Description
Cost-plus	Cost + fixed % mark-up
Penetration	Low initial price to gain market share
Price skimming	High initial price (innovative product)
Competitive	Match or beat competitor prices
Psychological	N$99.99 instead of N$100

Ansoff Matrix

	Existing Markets	New Markets
Existing Products	Market Penetration (lowest risk)	Market Development
New Products	Product Development	Diversification (highest risk)

Porter's Five Forces

Threat of new entrants
Bargaining power of suppliers
Bargaining power of buyers
Threat of substitute products
Rivalry among existing competitors

Exam Tip

For strategy questions, use SWOT, PEST or Porter's and relate to a real Namibian business context for higher marks.

Geography

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Plate Tectonics

Boundary	Movement	Features
Convergent	Plates collide	Mountains, trenches, volcanoes
Divergent	Plates move apart	Rift valleys, mid-ocean ridges
Transform	Plates slide past	Earthquakes, fault lines

River Systems

Erosion: hydraulic action, abrasion, attrition, solution
Upper course: V-shaped valleys, waterfalls
Middle course: meanders, ox-bow lakes
Lower course: flood plains, deltas

Coastal Landforms

Erosional	Depositional
Cliffs, wave-cut platforms	Beaches, spits, bars
Caves, arches, stacks	Tombolos, sand dunes

Namibia's Physical Geography

Region	Features
Namib Desert (west)	One of world's oldest deserts; Sossusvlei sand dunes
Central Plateau	1,000–2,000 m; most farming; Windhoek here
Kalahari (east)	Semi-arid savanna; red sand
Zambezi Region	Lush; Okavango, Zambezi, Kwando rivers

Water Scarcity

Namibia is among the most arid countries in sub-Saharan Africa. Windhoek recycles sewage water into drinking water — one of the first cities in the world to do so.

Factors Affecting Climate

Latitude: closer to equator = hotter
Altitude: higher = cooler (~6.5°C per 1000 m)
Benguela Current: cold current on west coast — brings fog but little rain; causes aridity
Distance from sea: continental interiors have more extreme temperatures

Namibia's Climate Zones

Zone	Rainfall	Temperature
North (Oshana, Kavango)	400–600 mm/year	Hot, distinct wet season
Central (Windhoek)	~350 mm/year	Hot summers, cool winters
South (Keetmanshoop)	<200 mm/year	Semi-arid
Namib coast (Swakopmund)	<25 mm/year	Cool, foggy

Population Concepts

Birth rate = (births/year / population) × 1000 Death rate = (deaths/year / population) × 1000 Natural increase = birth rate − death rate

Demographic Transition Model

Stage	Birth Rate	Death Rate	Population
1 (pre-industrial)	High	High	Stable/low
2 (early developing)	High	Falling	Rising rapidly
3 (late developing)	Falling	Low	Rising slowly
4 (developed)	Low	Low	Stable

Namibia

~3 million people; low density (~2.9/km²); rapid urbanisation; transitioning between stages 2–3.

Namibia's Natural Resources

Resource	Location	Significance
Diamonds	Namaqualand coast, Oranjemund	Major export; Namdeb
Uranium	Erongo (Rössing, Husab)	Among world's top producers
Copper/Zinc/Lead	Tsumeb, Otavi	Historical mining hub
Fish	Atlantic coast (Benguela)	Fishing industry, Walvis Bay
Tourism	National parks, Sossusvlei, Etosha	Major GDP contributor

Desertification

Causes: overgrazing, deforestation, climate change, poor farming. Highly relevant to Namibia.

Climate Change Effects on Namibia

Increased drought frequency and severity
Shifts in rainfall patterns affecting crops
Sea level rise threatening coastal areas
Flash flood risk increase in north

Conservation

Namibia leads in community-based natural resource management (CBNRM). Communal conservancies allow communities to benefit from wildlife tourism, reducing poaching.

Key Fact

Namibia's constitution (Article 95) mandates environmental protection — one of few countries worldwide to do so.

History

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Pre-colonial Namibia

Namibia was home to the San (Bushmen) — among the oldest peoples on Earth. Later, Khoikhoi, Damara, Herero, Ovambo, Nama and other groups inhabited the region.

German Colonial Rule (1884–1915)

Germany declared South West Africa a protectorate in 1884 (Berlin Conference)
1904–1908: Herero and Nama Genocide — estimated 65–80% of Herero and 50% of Nama people killed; one of the first genocides of the 20th century
Battle of Waterberg (1904) — decisive German victory forcing Herero into the Omaheke Desert

Significance

Germany officially acknowledged and apologised for the genocide in 2021, pledging €1.1 billion in reconstruction aid over 30 years.

South African Administration (1915–1990)

League of Nations mandate (1920); apartheid laws extended after 1948
UN terminated South Africa's mandate in 1966; South Africa refused to leave
SWAPO launched armed liberation struggle from 1966
Border War: SWAPO vs South African forces in northern Namibia and southern Angola (1966–1989)

Namibian Independence (1990)

Namibia became independent on 21 March 1990, following UN Resolution 435. Sam Nujoma became the first president.

The Scramble for Africa (1880s)

European powers divided Africa at the Berlin Conference (1884–85), with no African representation. By 1914, 90% of Africa was under European control.

Colonial Power	Territories (examples)
Britain	Egypt, Nigeria, Kenya, South Africa
France	Algeria, Senegal, Mali
Germany	Namibia, Tanzania, Cameroon
Portugal	Angola, Mozambique
Belgium	Congo (DRC)

Effects of Colonialism

Economic: resource extraction; infrastructure built to export
Social: disruption of African cultures, languages; missionary education
Political: artificial borders cutting across ethnic groups — still causes conflict today

Causes of WWI — MAIN

Letter	Cause	Detail
M	Militarism	European powers built up large armies/navies
A	Alliance System	Triple Alliance vs Triple Entente
I	Imperialism	Competition for colonies created tension
N	Nationalism	Ethnic groups sought self-rule (Balkans)

Immediate trigger: Assassination of Archduke Franz Ferdinand in Sarajevo, 28 June 1914.

Treaty of Versailles (1919)

Germany accepted full blame (War Guilt Clause)
Reparations: 132 billion gold marks
Lost 13% of territory; army limited to 100,000
Widely seen as sowing seeds of WWII

Key Turning Points

Event	Date	Significance
Battle of Britain	1940	RAF defeated Luftwaffe
Operation Barbarossa	1941	Germany invaded USSR; overextended
Pearl Harbor	Dec 1941	USA entered war
Battle of Stalingrad	1942–43	Decisive Soviet victory; Eastern Front turned
D-Day (Normandy)	June 1944	Allied invasion of Western Europe
Hiroshima/Nagasaki	Aug 1945	Atomic bombs → Japan surrendered

The Holocaust

Nazi Germany systematically murdered approximately 6 million Jews and millions of others in death camps including Auschwitz, Treblinka and Sobibor.

Origins of the Cold War

USA (West)	USSR (East)
Capitalism; free markets	Communism; planned economy
NATO alliance (1949)	Warsaw Pact (1955)
Marshall Plan	Comecon

Key Events

Berlin Blockade (1948–49): USSR blockaded West Berlin; USA airlifted supplies
Korean War (1950–53): USA-backed South Korea vs USSR/China-backed North Korea
Cuban Missile Crisis (1962): 13-day nuclear standoff
Vietnam War (1955–75): USA failed to prevent communist takeover
Fall of Berlin Wall (1989): symbolised end of Cold War
USSR dissolved (1991): Cold War ended

The Independence Wave (1950s–1990s)

Country	Independence	From
Ghana	1957	Britain (Kwame Nkrumah)
Nigeria	1960	Britain
Kenya	1963	Britain (Mau Mau uprising)
Zimbabwe	1980	Britain (liberation war)
Namibia	1990	South Africa (SWAPO; Sam Nujoma)
South Africa	1994	Apartheid ended; Mandela elected

Exam Tip

Structure answers as: What happened → Why it happened → Effects. Support every claim with specific dates, names and facts.