WFM01 Further Pure Mathematics 1

Lessons in WFM01

1. Complex numbers: the basics
   i² = -1; z* = a - bi; z z* = a² + b²; divide by multiplying by the conjugate.
2. Dividing complex numbers and equating parts
   Divide by multiplying by the conjugate of the bottom. Equal complex numbers have equal real parts and equal imaginary parts. Square roots come from (a + bi)² a
3. Modulus and argument
   |z| = sqrt(a² + b²). The principal argument is in (-π, π]; always sketch z to choose the quadrant.
4. Modulus-argument form
   z = r(cos θ + i sin θ) with r = |z| and θ = arg z. Moduli multiply and divide: |z_1 z_2| = |z_1||z_2|.
5. The Argand diagram
   Sum: parallelogram rule. |z_1 - z_2| is the distance between two points. Multiplying by i rotates by 90° anticlockwise.
6. Quadratic equations with complex roots
   Negative discriminant gives roots p ± qi. A quadratic with roots α and α* is z² - 2Re(α)z + |α|² = 0.
7. Cubic equations with complex roots
   Real coefficients: complex roots come in conjugate pairs. Pair them into a real quadratic factor, then find the remaining factor.
8. Quartic equations
   Quartic with real coefficients: roots in conjugate pairs, so 0, 2 or 4 are real. Pair roots into real quadratic factors.
9. Sum and product of roots
   α + β = -b/a and αβ = c/a. A quadratic with given roots is x² - (sum)x + (product) = 0.
10. Expressions in the roots
    Rewrite every symmetric expression in terms of α + β and αβ, then substitute.
11. Equations with transformed roots
    New sum and new product, in terms of α + β and αβ; then x² - (sum)x + (product) = 0 with integer coefficients.
12. Locating roots and interval bisection
    Sign change of a continuous function locates a root. Bisection keeps the half-interval with the sign change; the width halves each time.
13. Linear interpolation
    Linear interpolation: (x - a)/|f(a)| = (b - x)/|f(b)| from similar triangles. Then check the sign of f at the estimate and repeat.
14. The Newton-Raphson process
    x_(n+1) = x_n - f(x_n)/f'(x_n). It fails when f'(x_n) = 0 and can misbehave near a stationary point.
15. The parabola y² = 4ax
    y² = 4ax: parametric (at², 2at), focus (a, 0), directrix x = -a. Every point is as far from the focus as from the directrix.
16. The rectangular hyperbola xy = c²
    xy = c²: parametric (ct, c/t); asymptotes are the axes; symmetric about y = ±x.
17. Tangents and normals to a parabola
    On y² = 4ax at (at², 2at): tangent gradient 1/t, tangent ty = x + at², normal y + tx = 2at + at³.
18. Tangents and normals to a rectangular hyperbola
    On xy = c² at (ct, c/t): gradient -1/t²; tangent x + t² y = 2ct; normal t³ x - ty = c(t⁴ - 1).
19. Matrix arithmetic
    Add entry by entry; multiply row by column. AB needs columns of A = rows of B, and usually AB ≠ BA.
20. Determinants and inverse matrices
    det(a b; c d) = ad - bc; zero means singular. Inverse = (1/det)(d -b; -c a). (AB)^-1 = B^-1A-1.
21. Linear transformations and matrices
    Image of (x, y) = M(x, y). The columns of M are the images of (1, 0) and (0, 1).
22. Standard transformation matrices
    Reflections, rotations (cos θ -sin θ; sin θ cos θ), stretches (k 0; 0 1) or (1 0; 0 k), enlargements (k 0; 0 k). Describe a transformation fully.
23. Combined transformations
    P then Q is QP: the first transformation goes on the right.
24. Inverse transformations and area scale factor
    Work back with M^-1. Image area = |det M| × object area; negative det means orientation reversed.
25. Sigma notation and standard sums
    Σr = n(n + 1)/2, Σr² = n(n + 1)(2n + 1)/6, Σr³ = n²(n + 1)²/4. Split sums; Σ from k to n = Σ to n minus Σ to k - 1.
26. Summing series with the standard results
    Expand the term, split the sum, use the standard results, factorise early. Check with n = 1.
27. Proof by induction: series
    Basis, assumption, inductive step to the n = k + 1 form, conclusion. For series: S(k) + f(k + 1) = S(k + 1).
28. Proof by induction: divisibility
    Divisibility: show f(1) is divisible, then f(k + 1) = [f(k + 1) - f(k)] + f(k) with both parts divisible.
29. Proof by induction: sequences
    Sequences: check u_1, assume u_k = g(k), put it into the recurrence and simplify to g(k + 1).
30. Proof by induction: matrices
    Matrices: check n = 1, assume M^k, then M^k+1 = M^k M; simplify each entry into the n = k + 1 form.
31. F1 exam technique
    Show method, write conclusions, sketch for arguments, multiply matrices in the right order, round only at the end.

Next exams for WFM01

• January 2027: Thu 14 Jan 2027, afternoon, 1h 30m
• Summer 2027: Thu 27 May 2027, afternoon, 1h 30m

From Pearson's final timetables. All exam dates. Always confirm your entries with your exams officer.