Free QCE Specialist Mathematics Flashcard Review

For any complex number z = r(cos θ + i sin θ) and any positive integer n: zⁿ = rⁿ(cos(nθ) + i sin(nθ)) Equivalently: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

The nth roots of 1 are given by: z_k = cos(2πk/n) + i sin(2πk/n), where k = 0, 1, 2, …, n−1 There are exactly n distinct nth roots of unity, evenly spaced on the unit circle.

If a polynomial P(z) has real coefficients and a + bi (b ≠ 0) is a root, then a − bi is also a root. Complex roots of real polynomials always occur in conjugate pairs.

Proof by induction requires two steps: 1. Base case: Prove P(1) is true (or the smallest applicable value) 2. Inductive step: Assume P(k) is true, prove P(k+1) is true Conclusion: P(n) is true for all positive integers n.

For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃): a · b = a₁b₁ + a₂b₂ + a₃b₃ Alternatively: a · b = |a| |b| cos θ, where θ is the angle between them.

1. Commutative: a · b = b · a 2. Distributive: a · (b + c) = a · b + a · c 3. Scalar multiple: k(a · b) = (ka) · b = a · (kb) 4. a · a = |a|² 5. Perpendicular vectors: a · b = 0

For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃): a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) The result is perpendicular to both a and b with magnitude |a| |b| sin θ.

1. Anti-commutative: a × b = −(b × a) 2. Not associative: (a × b) × c ≠ a × (b × c) in general 3. Distributive: a × (b + c) = a × b + a × c 4. Parallel vectors: a × b = 0 5. |a × b| = area of parallelogram formed by a and b

Scalar triple product: a · (b × c) = det[a b c] Interpretation: |a · (b × c)| gives the volume of the parallelepiped formed by vectors a, b, c. Coplanar condition: a · (b × c) = 0

Vector form: r = a + λb where a is a position vector of a known point, b is the direction vector, and λ is a parameter (λ ∈ ℝ). Alternatively written: r = r₀ + t d

Cartesian form: (x − x₀)/l = (y − y₀)/m = (z − z₀)/n where (x₀, y₀, z₀) is a point on the line and (l, m, n) is the direction vector. Note: If a component is 0, that part becomes an equation (e.g., x = x₀).

Vector form: r = a + λu + μv where a is a position vector of a known point, u and v are non-parallel direction vectors, and λ, μ are parameters (λ, μ ∈ ℝ).

Cartesian form: ax + by + cz = d where (a, b, c) is the normal vector to the plane. Alternatively: a(x − x₀) + b(y − y₀) + c(z − z₀) = 0 using point (x₀, y₀, z₀).

For plane ax + by + cz = d, the normal vector is n = (a, b, c). This vector is perpendicular to every vector in the plane.

Vectors a and b are parallel if and only if a = kb for some scalar k ≠ 0. Equivalently: a × b = 0. Geometric: parallel vectors have the same or opposite direction.

Vectors a and b are perpendicular if and only if a · b = 0. Geometric: the angle between them is 90°.

Standard form: (x − h)² + (y − k)² + (z − l)² = r² where (h, k, l) is the centre and r is the radius. General form: x² + y² + z² + 2gx + 2fy + 2lz + c = 0, where centre is (−g, −f, −l).

Distance from point P(x₀, y₀, z₀) to plane ax + by + cz = d: d = |ax₀ + by₀ + cz₀ − d| / √(a² + b² + c²) Numerator is absolute value of plane equation evaluated at the point; denominator is magnitude of normal vector.

If r(t) = (x(t), y(t), z(t)), then: dr/dt = (dx/dt, dy/dt, dz/dt) The derivative represents the tangent vector (velocity in physical contexts).

If r(t) = (x(t), y(t), z(t)), then: ∫ r(t) dt = (∫ x(t) dt, ∫ y(t) dt, ∫ z(t) dt) + C where C is a vector constant of integration.