Moja Lab
Maths Hub Every Gr 12 CAPS paper topic · visual, interactive, exam-ready

Function GrapherUNIT 4 · PAPER 1

Drag the sliders and watch the graph move. This is exactly how the parameters a, p, q affect each function.

1.0
0.0
0.0
Functiony = x²
y-intercept(0; 0)
Shape / key pointmin (0; 0)

Calculus SandboxUNIT 7 · PAPER 1

Enter a cubic. Watch the derivative get built from first principles. See the tangent line at any point.

f(x)—
f'(x)—
Stationary points—
First principles: f'(x) = limh→0 [f(x+h) − f(x)] ÷ h.
Power rule: if f(x) = axⁿ then f'(x) = naxⁿ⁻¹.
At a stationary point: f'(x) = 0. Solve for x, then substitute back into f(x) to get the y-value.

Finance CalculatorUNIT 6 · PAPER 1

Compound interest, future value, present value, annuities and loans — all in one place.

Trig ExplorerUNIT 5 · UNITS 10–11

See how amplitude, period, and phase shift change the sine wave. Then use the sine, cosine and area rules to solve triangles.

Graph: y = a·sin(bx + p) + q

1.0
1.0
0°
0.0
Amplitude1.0
Period360°
Range[-1; 1]

Triangle solver

Sine Rule

a / sin A = b / sin B = c / sin C

Use when you have: two angles + one side, OR two sides + non-included angle

Cosine Rule

a² = b² + c² − 2bc·cos A

Use when you have: three sides, OR two sides + included angle

Area Rule

Area = ½ · ab · sin C

Use when you have: two sides + included angle

Sequence GeneratorUNIT 3 · PAPER 1

Enter the first few terms and see whether it's arithmetic, geometric or quadratic — plus the general term Tₙ.

Type—
Common diff/ratio—
General term Tₙ—
T₁₀—

Quadratic MachineUNIT 2 · PAPER 1

Enter a, b, c and see the discriminant, roots, and the nature of the roots — the exact wording your exam expects.

Discriminant Δ—
Roots—
Nature of roots—

Factorisation (if roots are rational)

Probability SimulatorUNIT 8 · PAPER 1

Flip coins, roll dice, draw cards — see theoretical vs experimental probability converge in real time.

Law of large numbers: The more trials you run, the closer the experimental probability gets to the theoretical probability. Try 500 then 5 000 — see the bars stabilise.

Analytical GeometryUNIT 9 · PAPER 2

Distance, midpoint, gradient, inclination, equation of a line, circle equation — all calculated live.

Distance AB—
Midpoint M—
Gradient m—
Inclination θ—
Equation of AB—

Circle equation

Equation—
Formula sheet reminders:
Distance = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint = ((x₁+x₂)/2 ; (y₁+y₂)/2)
Gradient m = (y₂−y₁)/(x₂−x₁)
Inclination: m = tan θ
Circle: (x−a)² + (y−b)² = r²

Euclidean GeometryUNIT 12 · PAPER 2

The theorem proofs you must know. Learn the structure of each proof — the exam requires them from memory.

Theorem 7 — Proportionality

If DE ∥ BC in ∆ABC, then AD/DB = AE/EC

Proof uses: area of triangles with common altitude.

Theorem 9 — Equiangular ∆s

If ∆ABC and ∆DEF are equiangular, then AB/DE = BC/EF = AC/DF

Proof uses: construction + SAS congruence.

Theorem 10 — Sides in proportion

If AB/DE = BC/EF = AC/DF, then ∆ABC ∥∥ ∆DEF

Proof uses: constructing an equiangular triangle.

Theorem 11 — Pythagoras

In a right-angled triangle, BC² = AB² + AC²

Proof uses: similar triangles created by the altitude from the right angle.

Exam tip: When proving triangles similar, always state the three pairs of equal angles with a reason (given, common, alternate, corresponding, etc.). Then conclude: "∴ ∆ABC ∥∥ ∆DEF (∠∠∠)". The order of vertices matters!

Statistics WorkshopUNIT 13 · PAPER 2

Paste any dataset (comma-separated). Get mean, median, mode, quartiles, IQR, variance, standard deviation, and a box plot.

Exponents & SurdsUNIT 1 · PAPER 1

The laws of exponents and surds — every rule you need to simplify any expression.

Laws of exponents

LawExample
aᵐ × aⁿ = aᵐ⁺ⁿ3² × 3⁴ = 3⁶ = 729
aᵐ ÷ aⁿ = aᵐ⁻ⁿ5⁷ ÷ 5³ = 5⁴ = 625
(aᵐ)ⁿ = aᵐⁿ(2³)⁴ = 2¹² = 4096
a⁰ = 117⁰ = 1
a⁻ⁿ = 1 / aⁿ2⁻³ = 1/8
a^(m/n) = ⁿ√(aᵐ)8^(2/3) = (³√8)² = 4
(ab)ⁿ = aⁿ × bⁿ(2×5)³ = 8 × 125 = 1000
(a/b)ⁿ = aⁿ / bⁿ(2/3)³ = 8/27

Surd rules

RuleExample
√a × √b = √(ab)√3 × √7 = √21
√a ÷ √b = √(a/b)√20 ÷ √5 = √4 = 2
√a + √a = 2√a (like surds)3√5 + 2√5 = 5√5
√(a²) = |a|√(25) = 5
Rationalise: 1/√a × √a/√a1/√2 = √2/2
Conjugate: 1/(√a+√b) × (√a−√b)/(√a−√b)1/(√3+1) = (√3−1)/2
Common mistakes:
❌ √(a + b) ≠ √a + √b — e.g. √(9+16) = √25 = 5, NOT 3+4 = 7
❌ (a + b)² ≠ a² + b² — e.g. (2+3)² = 25, NOT 4+9 = 13
❌ a^(m+n) ≠ aᵐ + aⁿ
❌ 3⁻² ≠ −9 — it's 1/9

Number PatternsUNIT 3 · PAPER 1

First difference. Second difference. This is how you identify any pattern and derive the general term Tₙ.

  1. Arithmetic: T₂ − T₁ = T₃ − T₂ = d (constant). Then Tₙ = a + (n−1)d.
  2. Geometric: T₂ ÷ T₁ = T₃ ÷ T₂ = r (constant). Then Tₙ = a·rⁿ⁻¹.
  3. Quadratic: First differences not equal, but second differences are. Then Tₙ = an² + bn + c, where 2a = second difference.
  4. To find a, b, c for a quadratic: 2a = second diff; 3a + b = T₂ − T₁; a + b + c = T₁.
Worked example: 5; 8; 13; 20; …
1st differences: 3, 5, 7 (not constant)
2nd differences: 2, 2 (constant ✓ so quadratic)
2a = 2 → a = 1
3a + b = 3 → b = 0
a + b + c = 5 → c = 4
Tₙ = n² + 4