WFM02 Further Pure Mathematics 2

Lessons in WFM02

1. Inequalities with algebraic fractions
   Never multiply by an expression whose sign you do not know. Bring everything to one side, find critical values from the top and bottom, then test each interval.
2. Inequalities with the modulus sign
   Sketch both sides, solve on each piece, keep only roots on the right piece, then read the intervals from the sketch.
3. The method of differences
   Write the term as f(r) - f(r + 1), list rows to see the cancelling, and keep only the terms that survive at each end.
4. Method of differences: gaps, three factors and limits
   If the gap in the partial fractions is k, then k terms survive at each end. Write enough rows to see it.
5. Euler's relation and exponential form
   reⁱθ = r(cos θ + i sin θ). Multiply: multiply moduli and add arguments. Divide: divide moduli and subtract arguments.
6. De Moivre's theorem
   (r eⁱθ)ⁿ = rⁿ eⁱⁿθ: raise the modulus to the power n and multiply the argument by n. Always evaluate the final answer.
7. De Moivre: cos nθ and sin nθ in powers
   Expand (c + is)ⁿ, equate real or imaginary parts, then use s² + c² = 1 to write the result in one function.
8. Powers of sin and cos as multiple angles
   zⁿ + z^-n = 2cos nθ and zⁿ - z^-n = 2i sin nθ. Expand (z ± 1/z)ⁿ, pair the terms from the ends, then divide.
9. Roots of complex numbers
   zⁿ = Reⁱφ has n roots R¹/ne^(i(φ + 2kπ)/n), equally spaced round a circle of radius R¹/n.
10. Loci: circles, bisectors and half-lines
    |z - a| = r: circle centre a. |z - a| = |z - b|: perpendicular bisector of ab. arg(z - a) = θ: half-line from a.
11. Loci: Apollonius circles and arcs
    |z - a| = k|z - b| (k ≠ 1) is a circle: go Cartesian and complete the square. arg((z - a)/(z - b)) = θ is an arc through a and b, ends excluded.
12. Regions in the Argand diagram
    Inside or outside, nearer to a or to b, between two arguments: draw each boundary, test a point, and shade where every condition holds.
13. Transformations of the complex plane
    Make z the subject, substitute into the locus of z, then simplify in w (or in u and v).
14. Transformations w = (az + b)/(cz + d)
    Rearrange to z = (b - dw)/(cw - a), substitute, use |A/B| = |A|/|B|, then decide: equal moduli give a line, unequal give a circle.
15. First order equations: separating variables
    Separate, integrate both sides with one constant, then use the condition. The general solution is a family of curves.
16. First order linear equations: the integrating factor
    Standard form first, IF = e^∫P dx, then d/dx(y × IF) = Q × IF. Never forget + c.
17. First order equations by substitution
    Express y and dy/dx in the new variable, substitute, solve the simpler equation, then substitute back.
18. Second order equations: the complementary function
    Solve am² + bm + c = 0. Distinct real roots: two exponentials. Repeated root: (A + Bx)e^αx. Complex p ± qi: e^px(A cos qx + B sin qx).
19. Second order equations: particular integrals
    General solution = CF + PI. Try a PI like f(x), always include both cos and sin, and multiply by x if the trial is already in the CF.
20. Second order equations: conditions and complete solutions
    CF + PI, then differentiate the whole thing and use both conditions. Always end with y = ....
21. Second order equations by substitution
    With x = e^u: x dy/dx = dy/du and x² d^2y/dx² = d^2y/du² - dy/du. Solve in u, then put e^u = x.
22. Higher derivatives and Maclaurin series
    f(x) = Σ f^r(0)x^r/r!. Find the derivatives, evaluate at 0, quote the formula, then substitute.
23. Using the standard Maclaurin series
    Substitute into, multiply or combine the standard series instead of differentiating. Check the new range of validity.
24. Taylor series
    f(x) = Σ f^r(a)(x - a)^r/r!. Evaluate the derivatives at a and keep the answer in powers of (x - a).
25. Series solutions of differential equations
    Use the equation to get y'', differentiate it implicitly for higher derivatives, evaluate at the given point, then write y = Taylor series.
26. Polar coordinates and polar curves
    x = r cos θ, y = r sin θ, r² = x² + y². Plot only where r ≥ 0, and use symmetry and key angles to sketch.
27. Areas enclosed by polar curves
    A = ½∫r² dθ. Sketch for the limits, use double angle formulae, and keep separate integrals for separate pieces.
28. Tangents to polar curves
    y = r sin θ and x = r cos θ. Parallel to the initial line: dy/dθ = 0. Perpendicular: dx/dθ = 0. Check r ≥ 0 and special points.

Next exams for WFM02

• January 2027: Wed 20 Jan 2027, morning, 1h 30m
• Summer 2027: Thu 3 Jun 2027, afternoon, 1h 30m

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