WMA12 Pure Mathematics 2

Lessons in WMA12

1. Geometric series
   u_n = arⁿ-1; S_n = a(1 - rⁿ)/(1 - r); S_inf = a/(1 - r) only when |r| < 1.
2. Proof and disproof
   Deduction: general algebra and logical steps. Exhaustion: check every case. Counter-example: one failure disproves.
3. Algebraic division
   Divide, multiply, subtract, bring down. Use 0 for missing powers. f(x) = divisor x quotient + remainder.
4. The factor and remainder theorems
   Remainder on dividing by (x - a) is f(a); f(a) = 0 means (x - a) is a factor; for (ax - b) use f(b/a).
5. The equation of a circle
   (x - a)² + (y - b)² = r² has centre (a, b) and radius r. Complete the square to convert the expanded form.
6. Circles: tangents, chords and lines
   Angle in a semicircle = 90°; perpendicular from centre bisects a chord; tangent is perpendicular to the radius; the discriminant counts intersections.
7. Sequences and recurrence relations
   u_n = f(n) gives terms directly; u_n+1 = f(u_n) needs a first term. Periodic sequences repeat with a fixed order.
8. Arithmetic sequences and series
   u_n = a + (n - 1)d; S_n = (n/2)(2a + (n - 1)d) = (n/2)(a + l); 1 + 2 + ... + n = n(n + 1)/2.
9. Sigma notation
   Write out the first few terms, count terms as top - bottom + 1, then use the arithmetic or geometric sum formula.
10. The binomial expansion
    The term in x^r of (a + bx)ⁿ is nCr aⁿ-r (bx)^r. Bracket the whole term before raising it to a power.
11. Binomial expansion: unknowns and approximations
    Set the general term equal to the given coefficient. For small x, the first few terms approximate (1 + x)ⁿ well.
12. Exponential graphs and logarithms
    log_a b = x means a^x = b. y = a^x passes through (0, 1) with asymptote y = 0.
13. Laws of logarithms
    Add logs: multiply. Subtract logs: divide. A multiple k becomes a power. Check every solution of a log equation.
14. Solving exponential equations
    a^x = b gives x = log b / log a. log_a b = log b / log a. Substitute y = a^x for hidden quadratics and reject y ≤ 0.
15. Trigonometric identities
    tan θ = sin θ / cos θ; sin² θ + cos² θ = 1. Use the quadrant to choose the sign.
16. Solving trigonometric equations
    Principal value, then symmetry (180° - x for sin, 360° - x for cos, x + 180° for tan). Adjust the interval before solving for 2x or x + a.
17. Quadratic trigonometric equations
    Factorise as a quadratic in sin x or cos x, use sin² + cos² = 1 or tan = sin/cos to get one function, and never divide by a trig function.
18. Stationary points and increasing functions
    dy/dx = 0 at stationary points. d^2y/dx² > 0 minimum, < 0 maximum, = 0 check either side. f' > 0 increasing, f' < 0 decreasing.
19. Optimisation problems
    Use the constraint to get one variable, set the derivative to 0, and confirm with the second derivative.
20. Definite integrals and areas
    ∫_a^{b} f(x) dx = F(b) - F(a). Areas below the x-axis give negative integrals: split at the roots and add the sizes.
21. Area between a curve and a line
    Find the intersections, then area = ∫ (top - bottom) dx between them.
22. The trapezium rule
    Area ≈ (h/2)(first + last + 2 x the rest), h = (b - a)/n. Convex curves give overestimates; more strips give better estimates.

Next exams for WMA12

• January 2027: Wed 13 Jan 2027, afternoon, 1h 30m
• Summer 2027: Tue 11 May 2027, afternoon, 1h 30m

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