HIGH SCHOOL MATHEMATICS MASTERBANK
HIGH SCHOOL MATHEMATICS MASTER BANK
Final-Week Revision Edition — FINAL EXAM HACKS
HOW TO USE THIS IN YOUR FINAL WEEK
Same system as the other Master Banks: organized by topic cluster, not by high school year. Math
papers (Paper 1 and Paper 2) draw from across all high school years in a single paper, often
combining topics within one question (e.g. a trigonometry question that also needs a bearing/vector
step).
Math punishes SLOPPY METHOD more than any other subject —so, ALWAYS show full
working, never skip steps, and never erase working even if you think you made an error (cross out
neatly instead).
For each cluster: core facts/formulae, worked examples (full working shown, exam style), and
exam traps.
PART 0: QUICK-REFERENCE FORMULA SHEET (memorize this
page cold)
Mensuration
Circle: circumference = 2πr = πd; area = πr²
Sector area = (θ/360) × πr²; arc length = (θ/360) × 2πr
Triangle area = ½ × base × height; also = ½ab sin C (two sides + included angle)
Trapezium area = ½(a+b)h
Parallelogram area = base × height
Cylinder: curved surface area = 2πrh; total surface area = 2πrh + 2πr²; volume = πr²h
Cone: curved surface area = πrl (l=slant height); total surface area = πrl + πr²; volume =
⅓πr²h
Sphere: surface area = 4πr²; volume = (4/3)πr³
Pyramid volume = ⅓ × base area × height
Prism volume = cross-sectional area × length
Algebra
Quadratic formula: x = [−b ± √(b²−4ac)] ÷ 2a, for ax²+bx+c=0
Difference of two squares: a²−b² = (a+b)(a−b)
Perfect square: a²+2ab+b² = (a+b)²; a²−2ab+b² = (a−b)²
Trigonometry
, HIGH SCHOOL MATHEMATICS MASTERBANK
SOH-CAH-TOA: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ =
opposite/adjacent
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b²+c²−2bc cos A (to find a side); cos A = (b²+c²−a²)/2bc (to find an angle)
Area of triangle = ½ab sin C
Statistics
Mean = Σx ÷ n (for raw data); Mean = Σfx ÷ Σf (for grouped/frequency data)
Median (grouped data) = L + [(n/2 − cf)/f] × c (L=lower class boundary, cf=cumulative freq
before median class, f=freq of median class, c=class width)
Mode (grouped) = L + [d1/(d1+d2)] × c (d1, d2 = differences in frequency from modal class
to adjacent classes)
Standard deviation (raw data) = √[Σ(x−x̄)²/n]
Range = highest value − lowest value
Sequences & Series
Arithmetic Progression (AP): nth term = a + (n−1)d; Sum of n terms, Sn = (n/2)[2a +
(n−1)d]
Geometric Progression (GP): nth term = arⁿ⁻¹; Sum of n terms, Sn = a(rⁿ−1)/(r−1) [r≠1]; Sum
to infinity (|r|<1) = a/(1−r)
Commercial Arithmetic
Simple Interest, I = PRT/100 (Principal × Rate × Time ÷ 100)
Compound Interest: Amount = P(1 + R/100)ⁿ
Profit/Loss % = (Profit or Loss ÷ Cost Price) × 100
Percentage change = (change ÷ original) × 100
Calculus
Differentiation of xⁿ: derivative = nxⁿ⁻¹
Gradient of a curve at a point = value of dy/dx at that point
Integration of xⁿ: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c [n≠−1]
Area under a curve = definite integral between the given limits
Coordinate Geometry
Gradient = (y₂−y₁)/(x₂−x₁)
Equation of a line: y = mx + c (m=gradient, c=y-intercept)
Distance between two points = √[(x₂−x₁)² + (y₂−y₁)²]
, HIGH SCHOOL MATHEMATICS MASTERBANK
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Parallel lines: same gradient. Perpendicular lines: m₁ × m₂ = −1
A. NUMBERS (Fractions, Decimals, Approximation, Number
Bases, Indices, Surds)
Core facts
Order of operations: BODMAS (Brackets, Of, Division, Multiplication, Addition,
Subtraction).
Significant figures: count from the first non-zero digit; leading zeros are never significant,
trailing zeros after a decimal point ARE significant.
Standard form (scientific notation): A × 10ⁿ, where 1 ≤ A < 10.
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.
Surds: simplify by finding the largest perfect square factor; rationalize the denominator by
multiplying by the conjugate (for denominators of the form a+√b, multiply by a−√b).
Number bases: converting to/from base 10 uses place values (powers of the base);
addition/subtraction in a non-decimal base carries/borrows according to that base's value,
not 10.
Worked examples
1. Express 0.00456 in standard form. → 4.56 × 10⁻³
2. Simplify: 2³ × 2⁴ ÷ 2⁵. → 2^(3+4−5) = 2² = 4
3. Rationalize: 6/(3+√5). → Multiply by (3−√5)/(3−√5): [6(3−√5)] ÷ [(3)²−(√5)²] =
(18−6√5)/(9−5) = (18−6√5)/4 = (9−3√5)/2
4. Convert 235 (base 10) to base 2. → Divide repeatedly by 2, noting remainders: 235÷2=117
r1; 117÷2=58 r1; 58÷2=29 r0; 29÷2=14 r1; 14÷2=7 r0; 7÷2=3 r1; 3÷2=1 r1; 1÷2=0 r1. Read
remainders bottom to top: 11101011₂
Exam traps
Standard form: the coefficient A MUST be between 1 and 10 (not 10 or more, not less than
1) — a very common formatting slip.
Leading zeros (e.g. in 0.0056) are NEVER significant — only count from the first non-zero
digit onward.
When rationalizing, multiply by the CONJUGATE (same terms, opposite sign in the
middle), not by the original expression again.
B. ALGEBRA (Equations, Simplification, Factorization)
Core facts
HIGH SCHOOL MATHEMATICS MASTER BANK
Final-Week Revision Edition — FINAL EXAM HACKS
HOW TO USE THIS IN YOUR FINAL WEEK
Same system as the other Master Banks: organized by topic cluster, not by high school year. Math
papers (Paper 1 and Paper 2) draw from across all high school years in a single paper, often
combining topics within one question (e.g. a trigonometry question that also needs a bearing/vector
step).
Math punishes SLOPPY METHOD more than any other subject —so, ALWAYS show full
working, never skip steps, and never erase working even if you think you made an error (cross out
neatly instead).
For each cluster: core facts/formulae, worked examples (full working shown, exam style), and
exam traps.
PART 0: QUICK-REFERENCE FORMULA SHEET (memorize this
page cold)
Mensuration
Circle: circumference = 2πr = πd; area = πr²
Sector area = (θ/360) × πr²; arc length = (θ/360) × 2πr
Triangle area = ½ × base × height; also = ½ab sin C (two sides + included angle)
Trapezium area = ½(a+b)h
Parallelogram area = base × height
Cylinder: curved surface area = 2πrh; total surface area = 2πrh + 2πr²; volume = πr²h
Cone: curved surface area = πrl (l=slant height); total surface area = πrl + πr²; volume =
⅓πr²h
Sphere: surface area = 4πr²; volume = (4/3)πr³
Pyramid volume = ⅓ × base area × height
Prism volume = cross-sectional area × length
Algebra
Quadratic formula: x = [−b ± √(b²−4ac)] ÷ 2a, for ax²+bx+c=0
Difference of two squares: a²−b² = (a+b)(a−b)
Perfect square: a²+2ab+b² = (a+b)²; a²−2ab+b² = (a−b)²
Trigonometry
, HIGH SCHOOL MATHEMATICS MASTERBANK
SOH-CAH-TOA: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ =
opposite/adjacent
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b²+c²−2bc cos A (to find a side); cos A = (b²+c²−a²)/2bc (to find an angle)
Area of triangle = ½ab sin C
Statistics
Mean = Σx ÷ n (for raw data); Mean = Σfx ÷ Σf (for grouped/frequency data)
Median (grouped data) = L + [(n/2 − cf)/f] × c (L=lower class boundary, cf=cumulative freq
before median class, f=freq of median class, c=class width)
Mode (grouped) = L + [d1/(d1+d2)] × c (d1, d2 = differences in frequency from modal class
to adjacent classes)
Standard deviation (raw data) = √[Σ(x−x̄)²/n]
Range = highest value − lowest value
Sequences & Series
Arithmetic Progression (AP): nth term = a + (n−1)d; Sum of n terms, Sn = (n/2)[2a +
(n−1)d]
Geometric Progression (GP): nth term = arⁿ⁻¹; Sum of n terms, Sn = a(rⁿ−1)/(r−1) [r≠1]; Sum
to infinity (|r|<1) = a/(1−r)
Commercial Arithmetic
Simple Interest, I = PRT/100 (Principal × Rate × Time ÷ 100)
Compound Interest: Amount = P(1 + R/100)ⁿ
Profit/Loss % = (Profit or Loss ÷ Cost Price) × 100
Percentage change = (change ÷ original) × 100
Calculus
Differentiation of xⁿ: derivative = nxⁿ⁻¹
Gradient of a curve at a point = value of dy/dx at that point
Integration of xⁿ: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c [n≠−1]
Area under a curve = definite integral between the given limits
Coordinate Geometry
Gradient = (y₂−y₁)/(x₂−x₁)
Equation of a line: y = mx + c (m=gradient, c=y-intercept)
Distance between two points = √[(x₂−x₁)² + (y₂−y₁)²]
, HIGH SCHOOL MATHEMATICS MASTERBANK
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Parallel lines: same gradient. Perpendicular lines: m₁ × m₂ = −1
A. NUMBERS (Fractions, Decimals, Approximation, Number
Bases, Indices, Surds)
Core facts
Order of operations: BODMAS (Brackets, Of, Division, Multiplication, Addition,
Subtraction).
Significant figures: count from the first non-zero digit; leading zeros are never significant,
trailing zeros after a decimal point ARE significant.
Standard form (scientific notation): A × 10ⁿ, where 1 ≤ A < 10.
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.
Surds: simplify by finding the largest perfect square factor; rationalize the denominator by
multiplying by the conjugate (for denominators of the form a+√b, multiply by a−√b).
Number bases: converting to/from base 10 uses place values (powers of the base);
addition/subtraction in a non-decimal base carries/borrows according to that base's value,
not 10.
Worked examples
1. Express 0.00456 in standard form. → 4.56 × 10⁻³
2. Simplify: 2³ × 2⁴ ÷ 2⁵. → 2^(3+4−5) = 2² = 4
3. Rationalize: 6/(3+√5). → Multiply by (3−√5)/(3−√5): [6(3−√5)] ÷ [(3)²−(√5)²] =
(18−6√5)/(9−5) = (18−6√5)/4 = (9−3√5)/2
4. Convert 235 (base 10) to base 2. → Divide repeatedly by 2, noting remainders: 235÷2=117
r1; 117÷2=58 r1; 58÷2=29 r0; 29÷2=14 r1; 14÷2=7 r0; 7÷2=3 r1; 3÷2=1 r1; 1÷2=0 r1. Read
remainders bottom to top: 11101011₂
Exam traps
Standard form: the coefficient A MUST be between 1 and 10 (not 10 or more, not less than
1) — a very common formatting slip.
Leading zeros (e.g. in 0.0056) are NEVER significant — only count from the first non-zero
digit onward.
When rationalizing, multiply by the CONJUGATE (same terms, opposite sign in the
middle), not by the original expression again.
B. ALGEBRA (Equations, Simplification, Factorization)
Core facts
