Mathematics IA — Last-Minute Revision
30 must-know points for Maths IA — rapid review covering every chapter. Read through this the night before or morning of your TGBIE 1st Year exam.
Bijection = one-one AND onto. Inverse exists only for bijections. fog ≠ gof in general.
Mathematical Induction: always start with base case (n=1), then show P(k) ⟹ P(k+1).
Matrices: multiplication AB is defined only if columns of A = rows of B; result order = rows(A) × cols(B).
Determinant: for 3×3 expand along row 1 — a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃.
Inverse of A: exists iff |A| ≠ 0. A⁻¹ = (1/|A|) · adj(A). Verify: A · A⁻¹ = I.
Rank of matrix = max order of non-zero minor. Use row-reduction to find rank quickly.
Cramer's rule: x = Δ₁/Δ, y = Δ₂/Δ, z = Δ₃/Δ. If Δ = 0 → no unique solution.
Unit vector: â = a⃗/|a⃗|. Always a unit vector has magnitude 1.
Section formula (internal): dividing AB in m:n → P = (m·b⃗ + n·a⃗)/(m+n).
Dot product perpendicularity: a⃗·b⃗ = 0. Parallel: a⃗ = λb⃗ for some scalar λ.
Cross product: a⃗×b⃗ = 0 means a⃗ and b⃗ are parallel (or one is zero vector).
Scalar triple product [a⃗ b⃗ c⃗] = 0 means the three vectors are coplanar.
Area of triangle with sides a⃗ and b⃗ from vertex: ½|a⃗×b⃗|.
sin(A+B) = sinAcosB + cosAsinB — this formula generates all compound angle results.
cos2A = 1 − 2sin²A = 2cos²A − 1 = cos²A − sin²A — all three forms needed.
sin3A = 3sinA − 4sin³A; cos3A = 4cos³A − 3cosA — memorise both.
Sum-to-product: sinC + sinD = 2sin((C+D)/2)cos((C−D)/2). Know all four.
Product-to-sum: 2sinAcosB = sin(A+B) + sin(A−B). Know all four.
General solution of sinθ = k: θ = nπ + (−1)ⁿ·sin⁻¹(k), n ∈ ℤ.
General solution of cosθ = k: θ = 2nπ ± cos⁻¹(k), n ∈ ℤ.
General solution of tanθ = k: θ = nπ + tan⁻¹(k), n ∈ ℤ.
sin⁻¹(x) + cos⁻¹(x) = π/2 for x ∈ [−1,1]. This is a frequently asked SAQ result.
tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x+y)/(1−xy)) when xy < 1.
cosh²x − sinh²x = 1 (hyperbolic Pythagorean identity). Unlike trig: cosh²x is the bigger one.
sinh⁻¹(x) = ln(x + √(x²+1)), cosh⁻¹(x) = ln(x + √(x²−1)) for x ≥ 1.
Sine rule: a/sinA = 2R. Use when given: two angles + one side, or two sides + opposite angle.
Cosine rule: cosA = (b²+c²−a²)/(2bc). Use when given: three sides, or two sides + included angle.
Half-angle: tan(A/2) = √((s−b)(s−c)/(s(s−a))). s = (a+b+c)/2.
Heron's formula: Δ = √(s(s−a)(s−b)(s−c)). In-radius r = Δ/s.
r₁ + r₂ + r₃ − r = 4R. This result is often asked as a prove question.