Mathematical Ability - Algebra and Geometry: Identities, Mensuration, Coordinates, Trigonometry
Algebra
Commit to memory: (a ± b)², a² − b², a³ ± b³, and x² + 1/x² = (x + 1/x)² − 2. For ax² + bx + c = 0, the roots sum to −b/a and multiply to c/a, while the discriminant b² − 4ac decides whether roots are real, equal or imaginary. Use the substitution x + 1/x = t to tame symmetric expressions. Surds are rationalised by multiplying by the conjugate.
Mensuration
Cuboid: volume lbh, surface 2(lb + bh + hl). Cube: volume a³, surface 6a². Cylinder: πr²h, curved surface 2πrh. Cone: (1/3)πr²h, curved surface πrl where l = √(r² + h²). Sphere: (4/3)πr³, surface 4πr². Hemisphere: (2/3)πr³, total surface 3πr². When a solid is melted and recast, volume is conserved; when a solid is immersed, the rise in water level gives the displaced volume.
Coordinate geometry and trigonometry
Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; section formula x = (m₁x₂ + m₂x₁)/(m₁ + m₂); slope = (y₂ − y₁)/(x₂ − x₁); area of a triangle = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Standard values: sin 30° = ½, cos 30° = √3/2, tan 45° = 1, tan 60° = √3, and sin²θ + cos²θ = 1.
Worked example
If a + b = 10 and ab = 21, then a² + b² = (a + b)² − 2ab = 100 − 42 = 58. The numbers are 7 and 3, but the identity reaches the answer without finding them - the faster route under time pressure.
Traps
- Substituting the diameter where the formula needs the radius.
- Dropping the ⅓ in cone and pyramid volumes.
- Omitting the absolute value in the coordinate area formula.
- Interchanging tan 30° (1/√3) with tan 60° (√3).