Mathematics — High School Test Prep Revision Cheat Sheet
1) Core Skills & Key Formulas
- Linear equations: ax + b = 0 ⇒ x = -b/a (if a ≠ 0)
- Quadratic: ax² + bx + c = 0
- Factor (if possible): (px + q)(rx + s) = 0
- Quadratic formula: x = (-b ± √(b² − 4ac)) / (2a)
- Discriminant: Δ = b² − 4ac
- Δ > 0: 2 real roots
- Δ = 0: 1 real (double) root
- Δ < 0: no real roots
- Functions: f(x) = mx + c (linear), f(x) = ax² + bx + c (quadratic)
- Indices / exponents:
- a^m · a^n = a^(m+n)
- a^m / a^n = a^(m−n) (a ≠ 0)
- (a^m)^n = a^(mn)
- (ab)^n = a^n b^n
- a^0 = 1 (a ≠ 0), a^(-n) = 1/a^n
- Surds (square roots):
- √(ab) = √a · √b (a,b ≥ 0)
- √(a/b) = √a / √b (b > 0)
- √(x²) = |x|
- Algebraic expansion:
- (a+b)² = a² + 2ab + b²
- (a−b)² = a² − 2ab + b²
- (a+b)(a−b) = a² − b²
- Right-triangle:
- a² + b² = c² (Pythagoras)
- sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
- Area/Perimeter:
- Rectangle: A = lw, P = 2(l+w)
- Triangle: A = (1/2)bh
- Circle: C = 2πr, A = πr²
- Coordinate geometry:
- Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
- Probability:
- P(A) = favourable/total
- Complement: P(A) = 1 − P(A’)
- Independent: P(A∩B) = P(A)P(B)
- Statistics:
- Mean: Σx / n
- Median: middle value (sorted)
- Mode: most frequent
- Range: max − min
2) Worked Micro-Examples (3–5)
- Micro 1: Solve linear equation
Solve: 3x − 7 = 11
3x = 18 ⇒ x = 6
- Micro 2: Factorise then solve quadratic
Solve: x² − 5x + 6 = 0
Factor: (x−2)(x−3)=0 ⇒ x=2 or x=3
- Micro 3: Quadratic formula
Solve: 2x² + 3x − 2 = 0
a=2, b=3, c=−2
Δ = 3² − 4(2)(−2) = 9 + 16 = 25
x = (−3 ± √25)/(2·2) = (−3 ± 5)/4
Solutions: x = 1/2 or x = −2
- Micro 4: Simplify indices
Simplify: 2x³ · 5x⁴
Coefficients: 2·5=10, powers: x^(3+4)=x⁷
Answer: 10x⁷
- Micro 5: Trig ratio
Right triangle: hypotenuse 10, opposite to θ is 6. Find sin θ.
sin θ = opp/hyp = 6/10 = 3/5
3) Common Traps (Avoid These)
- Quadratic sign errors: use Δ = b² − 4ac carefully (especially when c is negative).
- Forgetting absolute value: √(x²)=|x| (not just x).
- Index mistakes:
- (a+b)² ≠ a² + b² (missing 2ab)
- a^m + a^n cannot be combined as a^(m+n).
- Expanding incorrectly: keep track of all terms and signs in (a−b)².
- Probability with “without replacement”: if sampling changes the totals, don’t use independent multiplication.
- Angle confusion in trig: “opposite” and “adjacent” are relative to the chosen angle.
4) Quick Concept Diagram (Quadratics)
flowchart TD
A[Quadratic: ax^2+bx+c=0] --> B[Compute discriminant Δ=b^2-4ac]
B --> C{Δ}
C -->|Δ>0| D[Two real roots]
C -->|Δ=0| E[One real repeated root]
C -->|Δ<0| F[No real roots (complex)]
D --> G[Roots: