WFM03 Further Pure Mathematics 3

Lessons in WFM03

1. Hyperbolic functions: definitions and graphs
   sinh x = (e^x - e^-x)/2, cosh x = (e^x + e^-x)/2, tanh x = sinh x/cosh x. cosh x ≥ 1 and is even; sinh and tanh are odd; -1 < tanh x < 1.
2. Hyperbolic identities
   cosh² x - sinh² x ≡ 1; sech² x ≡ 1 - tanh² x; sinh 2x ≡ 2 sinh x cosh x; cosh 2x ≡ 2cosh² x - 1 ≡ 1 + 2sinh² x. Prove identities from the exponential defi
3. Solving hyperbolic equations
   Substitute u = e^x to get a quadratic, or use cosh² x - sinh² x = 1 to get a quadratic in one function. Reject e^x ≤ 0 and cosh x < 1. Give answers as exact l
4. Inverse hyperbolic functions
   arsinh x = ln(x + sqrt(x² + 1)) for all x; arcosh x = ln(x + sqrt(x² - 1)) for x ≥ 1; artanh x = (1/2) ln((1 + x)/(1 - x)) for |x| < 1.
5. The ellipse
   Ellipse: x²/a² + y²/b² = 1, with parametric form x = a cos t, y = b sin t. Intercepts (±a, 0) and (0, ±b).
6. The hyperbola
   Hyperbola: x²/a² - y²/b² = 1, vertices (±a, 0), asymptotes y = ±(b/a)x. Parametric: (a sec t, b tan t) or (a cosh t, b sinh t) for the right branch.
7. Foci, directrices and eccentricity
   PS = e PM. Ellipse: b² = a²(1 - e²), e < 1. Hyperbola: b² = a²(e² - 1), e > 1. Foci (±ae, 0), directrices x = ±a/e. Ellipse PS + PS' = 2a.
8. Tangents and normals to conics
   Ellipse gradient -b^2x/(a^2y); hyperbola b^2x/(a^2y). Tangent at (x1, y1): x x1/a² ± y y1/b² = 1. Normal gradient = -1/(tangent gradient).
9. When is a line a tangent?
   Substitute and set the discriminant to zero. Ellipse: c² = a^2m2 + b². Hyperbola: c² = a^2m2 - b².
10. Loci with conics
    Write the point in terms of t, isolate the trig or hyperbolic functions, and eliminate t with the matching identity.
11. Differentiating hyperbolic functions
    d/dx sinh x = cosh x; d/dx cosh x = sinh x (no minus); d/dx tanh x = sech² x. Use the chain, product and quotient rules as usual.
12. Differentiating inverse functions
    Write x = f(y), find dx/dy, express it in x with an identity, then flip. arcsin: 1/sqrt(1 - x²); arctan: 1/(1 + x²); arsinh: 1/sqrt(x² + 1); arcosh: 1/sqrt(x
13. Integrating hyperbolic functions
    ∫ sinh = cosh, ∫ cosh = sinh, ∫ sech² = tanh, ∫ tanh = ln cosh. Use cosh 2x identities for squares and cosh² - sinh² = 1 for odd powers.
14. Integrals that give inverse functions
    1/(a² + x²) gives (1/a) arctan(x/a); 1/sqrt(a² - x²) gives arcsin(x/a); 1/sqrt(x² + a²) gives arsinh(x/a); 1/sqrt(x² - a²) gives arcosh(x/a). Complete t
15. Integrating inverse trig and hyperbolic functions
    Write f(x) as 1 × f(x) and integrate by parts with u = f(x). Then the remaining integral is an f'/f or f'/sqrt(f) type.
16. Substitutions and quadratic surds
    sqrt(a² - x²): x = a sin u. sqrt(x² + a²): x = a sinh u. sqrt(x² - a²): x = a cosh u. Change dx and the limits; complete the square for quadratic surds.
17. Deriving reduction formulae
    Integrate by parts (or use an identity) to link I_n to I_(n-1) or I_(n-2), then work down to a starting value you can integrate directly.
18. Using reduction formulae
    If one round of parts does not reduce the integral to I_(n-1) or I_(n-2), do a second round. Start values first, exact values throughout.
19. Arc length
    s = ∫ sqrt(1 + (dy/dx)²) dx or s = ∫ sqrt((dx/dt)² + (dy/dt)²) dt. Use identities to make the square root a perfect square.
20. Area of a surface of revolution
    S = 2π ∫ y ds about the x-axis, 2π ∫ x ds about the y-axis, with ds = sqrt(1 + (dy/dx)²) dx or sqrt((dx/dt)² + (dy/dt)²) dt.
21. The vector product
    a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1), perpendicular to a and b. |a × b| is the parallelogram area; a triangle is half. b × a = -(a × b).
22. The triple scalar product
    a · (b × c) = determinant of the rows a, b, c. Parallelepiped volume |a · (b × c)|; tetrahedron (1/6)|a · (b × c)|; zero means coplanar.
23. Equations of lines in 3D
    r = a + λb; (r - a) × b = 0; (x - a1)/b1 = (y - a2)/b2 = (z - a3)/b3. Directions that are multiples mean parallel lines.
24. Equations of planes
    r · n = p with p = a · n; cartesian n1x + n2y + n3z = p; r = a + sb + tc with n = b × c. Distance from O to the plane is |p|/|n|.
25. Points, lines and planes
    Point to plane: |ax1 + by1 + cz1 - d|/sqrt(a² + b² + c²). Line meets plane: substitute the general point. b · n = 0 means parallel to the plane or in it.
26. Two planes: line of intersection
    Direction = n1 × n2. A point: set z = 0 and solve (or x = 0 or y = 0 if that fails). Check the line in both plane equations.
27. Shortest distances between lines
    Skew lines: |(c - a) · (b × d)|/|b × d|. Point to line or parallel lines: |(p - a) × b|/|b|. Zero distance means the lines meet.
28. Linear transformations in three dimensions
    Columns are the images of i, j, k. A then B is the matrix BA. Linear maps fix the origin. det = volume scale factor.
29. Transpose and 3 × 3 determinants
    det A = a(ei - fh) - b(di - fg) + c(dh - eg). Singular means det = 0. det(AB) = det A det B; det(kA) = k³ det A; (AB)^T = B^T A^T.
30. Inverse of a 3 × 3 matrix
    A^-1 = (1/det A) × transpose of the cofactor matrix. Remember the sign pattern. (AB)^-1 = B^-1A-1. Use the inverse to undo a transformation.
31. Eigenvalues and eigenvectors
    Solve det(A - λI) = 0 for λ, then (A - λI)x = 0 for x. Eigenvalues sum to the trace and multiply to the determinant. Normalise by dividing by the length.
32. Diagonalising symmetric matrices
    For symmetric A: eigenvectors for different eigenvalues are perpendicular. Normalise them, make them the columns of P, then P^T A P = D with eigenvalues in matc

Next exams for WFM03

• January 2027: Thu 21 Jan 2027, afternoon, 1h 30m
• Summer 2027: Tue 8 Jun 2027, afternoon, 1h 30m

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