Advanced Further Maths

Start with the method you need. Work through the example, test a prediction, then try a fresh question without the solution beside you.

These topics occur in different core and optional papers. Choose your course to filter practice, or browse freely.

• Finding every root, without counting one twice

  Write z = r(cos θ + i sin θ) before taking powers or roots. Powers multiply the argument; taking an nth root divides the argument but also introduces n choices. An argument describes a direction, so adding a full turn leaves the same complex number.

• A complex equation can describe a whole line

  |z − a| is the distance from z to a on an Argand diagram. The equation |z − a| = |z − b| therefore describes the perpendicular bisector of the segment joining a and b, rather than just its midpoint.

• What survives a matrix transformation?

  An eigenvector is a nonzero vector whose image is a scalar multiple of itself: Av = λv. The eigenvalue λ gives the scale and sign along that direction. The zero vector satisfies the equation for every λ, which is why it cannot be an eigenvector.

• Forcing terms and repeated roots

  A first-order linear equation y′ + py = q uses the integrating factor e^(∫p dx). Multiplying by it turns the left side into the derivative of a product. Keep the integration constant until you apply the initial condition.

• Choose an interval that traces the region once

  The polar point (r, θ) has Cartesian coordinates (r cos θ, r sin θ). A negative r puts the point on the opposite ray; it is not a negative distance. A curve can retrace a region as θ increases, so a full-turn integral is not automatically the area of the union.

• A finite expansion still has a domain

  A Maclaurin polynomial matches derivatives at zero. Its xⁿ coefficient is f^(n)(0)/n!, not just the nth derivative. A truncated expansion is an approximation unless the original function is a polynomial of that degree.

• Use the exponential definitions to settle signs

  sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. Subtracting their squares gives cosh²x − sinh²x = 1. The minus sign differs from the circular identity.

• Perpendicularity identifies the closest point

  A line r = a + td has a closest point to P when the displacement a + td − P is perpendicular to d. Set their scalar product to zero. This gives one linear equation for t when d is nonzero.

Test an idea in a diagram

• Roots on the Argand diagram

  Predict the angular separation between neighbouring roots, as a coefficient of π.

• A matrix and its invariant directions

  For A = [a, b; b, a], predict the eigenvalue along (1, −1).

• Tracing a polar rose

  Predict the area of one petal as Kπ. Enter the exact coefficient K.

• Newton iteration and its tangent

  Predict the first iterate for x² − c, starting at x₀. Enter an exact fraction.

Use several results in one linked problem · Return to questions that need another attempt