Mathematics IB — Last-Minute Revision
30 must-know points for Maths IB — rapid review covering every chapter. Read through this the night before or morning of your TGBIE 1st Year exam.
Locus: the locus of a point equidistant from two fixed points is the perpendicular bisector of the segment joining them.
Transformation: shifting origin to (h,k) → new X = x−h, Y = y−k. Substitute back to get original equation.
Slope of a line: m = (y₂−y₁)/(x₂−x₁). Slope of ax+by+c=0 is −a/b.
Parallel lines have equal slopes. Perpendicular lines: m₁·m₂ = −1.
Intercept form: x/a + y/b = 1. If a = b, the line makes equal intercepts on both axes.
Perpendicular distance from (x₁,y₁) to ax+by+c=0: d = |ax₁+by₁+c|/√(a²+b²). Memorise this.
Three lines are concurrent if the determinant of coefficients = 0 (3×3 determinant).
For pair of lines ax²+2hxy+by²=0: angle tanθ = 2√(h²−ab)/(a+b).
Condition for perpendicular pair of lines: a + b = 0.
Pair of bisectors of ax²+2hxy+by²=0: (x²−y²)/(a−b) = xy/h.
Second degree equation represents a pair of lines if Δ = abc+2fgh−af²−bg²−ch² = 0.
Direction cosines l, m, n satisfy: l² + m² + n² = 1 (always).
Angle between two lines with DC (l₁,m₁,n₁) and (l₂,m₂,n₂): cosθ = |l₁l₂+m₁m₂+n₁n₂|.
Section formula in 3D: same as 2D but with z-coordinate: z = (mz₂+nz₁)/(m+n).
Equation of plane: ax+by+cz+d=0. Normal vector to the plane is (a, b, c).
lim(x→0) sinx/x = 1 only when x is in radians. This is the most-used standard limit.
lim(x→0) (eˣ−1)/x = 1; lim(x→0) (aˣ−1)/x = logₑa. These appear in SAQs.
A function is continuous at a if: f(a) is defined, lim exists, and lim = f(a).
Differentiability implies continuity — but continuity does NOT imply differentiability.
Product rule: d/dx(uv) = u'v + uv'. Quotient rule: d/dx(u/v) = (u'v − uv')/v².
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).
Parametric: if x=f(t), y=g(t), then dy/dx = (dy/dt)/(dx/dt) provided dx/dt ≠ 0.
Logarithmic differentiation: take log both sides when function is raised to a function.
Slope of tangent at point P = value of dy/dx at P. Normal is perpendicular: slope = −1/(dy/dx).
Angle between two curves: find slopes at intersection point, then use tanθ formula.
Rolle's theorem conditions: f continuous on [a,b], differentiable on (a,b), f(a)=f(b). Then ∃c: f'(c)=0.
LMVT: f continuous on [a,b], differentiable on (a,b). Then ∃c: f'(c) = (f(b)−f(a))/(b−a).
For maxima/minima: find f'(x)=0 (critical points). Use f''(x) to classify: f''>0 → min, f''<0 → max.
Increasing function: f'(x) > 0 for all x in the interval. Decreasing: f'(x) < 0.
Error approximation: dy = (dy/dx)·Δx. Percentage error = (dy/y)×100.