WMA14 Pure Mathematics 4

Lessons in WMA14

1. Vectors in three dimensions
   |xi + yj + zk| = sqrt(x² + y² + z²); AB = b - a; unit vector = a/|a|.
2. Proof by contradiction
   Assume the negation, reason to an impossibility, then conclude the original statement is true.
3. Partial fractions
   One fraction per linear factor; a repeated factor (x + a)² needs A/(x + a) + B/(x + a)². Substitute roots, then compare coefficients.
4. Improper fractions and partial fractions
   Improper means degree of top ≥ degree of bottom. Divide first, or include a polynomial part of degree (top - bottom).
5. Parametric equations
   Eliminate t by substitution or with a trig identity. The range of t gives the domain of the Cartesian equation.
6. The binomial series for any rational n
   (1 + x)ⁿ = 1 + nx + n(n - 1)x²/2! + ...; for non-integer or negative n it is infinite and valid for |x| < 1. Bracket kx.
7. Expanding (a + bx)ⁿ and rational functions
   Take out aⁿ so the bracket starts with 1; validity |x| < |a/b|. Split rational functions into partial fractions and use the narrower range.
8. Parametric differentiation
   dy/dx = (dy/dt)/(dx/dt). Find the point and gradient from the same t, then use y - y1 = m(x - x1).
9. Implicit differentiation
   Differentiate y-terms and multiply by dy/dx; use the product rule for xy; collect dy/dx terms and factorise.
10. Connected rates of change
    Chain the rates: dA/dt = dA/dr × dr/dt. 'Proportional to' introduces a constant k; decreasing quantities get a minus sign.
11. Integration by substitution
    Replace dx using du/dx, write everything in u, change the limits, integrate.
12. Integration by parts
    ∫ u v' dx = uv - ∫ v u' dx. Let u be the polynomial, or ln x whenever it appears.
13. Integrating with partial fractions
    Split into partial fractions, then ∫ A/(ax + b) dx = (A/a)ln|ax + b|. Repeated factors give -1/(x + a) type terms.
14. Separable differential equations
    Separate the variables, integrate both sides with one constant, then use the condition for a particular solution.
15. Differential equations in context
    Form the equation, separate and solve, use one condition for the constant and another for k, then interpret.
16. Volumes of revolution
    V = π ∫ y² dx about the x-axis. Square y before integrating; subtract volumes for regions between graphs.
17. Areas and volumes with parametric equations
    Area = ∫ y (dx/dt) dt and volume = π ∫ y² (dx/dt) dt, with t-limits that match the x-limits.
18. Vector geometry and distance in 3D
    Parallel means a scalar multiple. Collinear means parallel through a shared point. Distance = sqrt(Δx² + Δy² + Δz²).
19. Vector equation of a line
    r = a + λb: a point plus a multiple of a direction. Check a point by finding one λ that works for every component.
20. Intersecting, parallel and skew lines
    Parallel: direction vectors are multiples. Otherwise solve two component equations; if the third holds they intersect, if not they are skew.
21. The scalar product
    a · b = a_1 b_1 + a_2 b_2 + a_3 b_3 = |a||b|cos θ. Zero means perpendicular; negative means the angle is more than 90°.
22. Angles between lines and perpendiculars
    Angle between lines: use direction vectors and take the acute angle. Foot of perpendicular: solve AP · b = 0.

Next exams for WMA14

• January 2027: Tue 19 Jan 2027, morning, 1h 30m
• Summer 2027: Mon 7 Jun 2027, afternoon, 1h 30m

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