We explore the arithmetic and geometric properties of complex numbers, representing them in both Cartesian and polar form, and applying De Moivre’s theorem to solve equations and model periodic behaviour.
A complex number can be expressed as z = a + bi = r(cos θ + i sin θ), where r = √(a² + b²) is the modulus and θ = tan⁻¹(b/a) is the argument.
Find the modulus and argument of the complex number z = 3 + 4i.
r = √(3² + 4²) = 5, θ = tan⁻¹(4/3) ≈ 53.1°
Express z = 1 − i in polar form, and find z².
r = √2, θ = −45° → z = √2(cos(−45°) + i sin(−45°)), z² = 2cis(−90°) = −2i
Differentiation allows us to find instantaneous rates of change and the gradient of curves. The derivative of y = f(x) at a point gives the gradient of the tangent line at that point.
Find the derivative of y = 3x² − 4x + 1 and determine the gradient when x = 2.
dy/dx = 6x − 4 → at x = 2, dy/dx = 8
The curve y = x³ − 6x² + 9x has a stationary point at x = a. Find the coordinates of this point and determine its nature.
dy/dx = 3x² − 12x + 9 = 0 → x = 1 or 3; at x = 1, min (y = 4); at x = 3, max (y = 0)
We represent quantities that have both magnitude and direction using vectors, and apply them to problems in motion, forces, and geometry.
For vectors a = (a₁, a₂) and b = (b₁, b₂), the scalar product is a · b = a₁b₁ + a₂b₂, and the magnitude is |a| = √(a₁² + a₂²).
Find the angle between vectors a = (3, 4) and b = (5, 0).
a·b = 15, |a| = 5, |b| = 5 → cosθ = 15/(5×5) = 0.6 → θ = 53.1°
A particle moves so that its displacement is given by r(t) = (3t²)i + (2t)j. Find its velocity and acceleration at t = 2.
v(t) = dr/dt = (6t)i + 2j → v(2) = (12i + 2j); a(t) = d²r/dt² = (6i)
Integration is the reverse of differentiation and is used to find accumulated quantities such as area under curves or total displacement from velocity functions.
Find the area under y = x² between x = 0 and x = 3.
∫₀³ x² dx = [x³/3]₀³ = 9
The velocity of a particle is v(t) = 4t − 2. Find its displacement after 5 seconds if s(0) = 0.
s = ∫(4t − 2)dt = 2t² − 2t → s(5) = 2(25) − 10 = 40 m
Differential equations describe systems where a rate of change depends on the current state. We solve separable equations and apply initial conditions to model real-world phenomena.
Solve the differential equation dy/dx = 3x² given y = 2 when x = 1.
∫dy = ∫3x²dx → y = x³ + C → 2 = 1 + C → C = 1 → y = x³ + 1
A population grows at a rate proportional to its size, with dP/dt = 0.2P. Find P after 5 years if P₀ = 1000.
P = 1000e^(0.2×5) = 1000e¹ = 2718 (approx.)