Engineering Sciences (XE) - sectional paper
What XE tests
The Engineering Sciences (XE) sectional paper begins with a compulsory Engineering Mathematics section (Section A) that supplies the common analytical toolkit for every specialised section. Questions test whether you can set up and execute a calculation cleanly: rank and consistency of linear systems, eigenvalues and quadratic forms, partial differentiation and multiple integrals, first- and second-order ordinary differential equations, analytic functions and residues, probability distributions, and numerical iteration.
Core rules
- A linear system Ax=b is consistent iff rank(A)=rank([A|b]); if the common rank equals the number of unknowns the solution is unique.
- For a symmetric matrix, positive definiteness follows from positive leading principal minors; eigenvalues of a real symmetric matrix are real.
- In a change of variables for multiple integrals, multiply by the absolute Jacobian determinant.
- For dy/dx + P(x)y = Q(x), use the integrating factor exp(∫P dx).
- A complex function is analytic where the Cauchy–Riemann equations hold and partial derivatives are continuous; residues give contour integrals.
- For a continuous random variable, normalise the density before computing E[X] or Var(X); Var(X)=E[X²]−μ².
Worked example
A continuous random variable has density f(x)=k x² on 0≤x≤1 and zero elsewhere. Normalisation requires ∫₀¹ k x² dx = k/3 = 1, so k=3. Then E[X]=∫₀¹ x·3x² dx = 3/4. Also E[X²]=∫₀¹ x²·3x² dx = 3/5, so Var(X)=3/5−(3/4)²=3/80. The order matters: normalise first; otherwise every moment is scaled by the wrong constant.
Traps
- Treating a candidate density as a probability without checking that it integrates to 1.
- Using sample variance with n instead of n−1 when the question specifies a sample.
- Declaring a system inconsistent from det(A)=0 alone; compare ranks.
- Omitting the Jacobian, or using its signed value rather than absolute value.
- Applying Newton–Raphson without checking that the derivative is nonzero and the iterate is converging.