Unit 9 of 10
Operations research and numerical analysis
Seven or eight questions, roughly half linear programming and half numerical methods.
- 7–8questions expected next time
- 8asked in 2025
- 7asked in 2026
- 2ideas asked in both papers
Most LPP questions only ask the type of solution. Learn to spot infeasible and unbounded fast.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Nature of an LPP solution
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Objective parallel to a binding edge: many optima. Region open in the improving direction: unbounded. Empty region: infeasible.
Interpolation: Lagrange, divided differences, Hermite
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n + 1 points fix one polynomial of degree at most n. Its n-th divided difference is the leading coefficient.
Asked in 2026
7 questions. Tap one for the answer and working.
Q35LP constraints x+2y<=6, 2x+y<=6; objective ax+by optimal on edge joining two vertices; find ratio a:bLikely bonus marksA
Answer: 1:2
Vertices (0,0),(3,0),(2,2),(0,3). Alternate optima need objective parallel to an edge: x+2y=6 edge gives 1:2, 2x+y=6 edge gives 2:1. Both edges have integral (assignment-polytope-like) endpoints, so the clause does not discriminate. Picked 1:2.
Not sure
Defective: by the x↔y symmetry of the region, 1:2 (A) and 2:1 (B) both fit. Both solves picked A.
Q59Gauss-Seidel on diagonally dominant PDE-discretised system: convergence rate vs JacobiB
Answer: Faster
For diagonally dominant systems GS typically converges faster than Jacobi (rho_GS = rho_J^2 for consistently ordered matrices).
Fairly sure
"faster" is the usual textbook claim, not universal for all diagonally dominant systems.
Q65Primal has alternate optima, dual region non-degenerate: implication for dualLikely bonus marksA
Answer: dual unique solution
Primal alt. optima iff dual degenerate at optimum, so hypothesis (non-degenerate dual) is contradictory; dual is feasible and bounded (C, D out). Between unique/multiple, picked unique.
Not sure
The premises contradict each other: primal alternate optima force a degenerate dual. Both solves picked A.
Q69Modified Euler unstable on stiff equation; condition for stability improvement with RK4D
RK4 has a bounded stability region (|h*lambda| < ~2.78) so stiff problems need small step size to stay stable
Sure
Q78Phase I objective zero, artificial variables in basis at zero level, dual feasible: implicationD
Phase I optimum 0 with artificials still basic at zero level, feasible original problem -> degenerate basic feasible solution
Sure
Q88Which gives higher accuracy for smooth functionsD
Hermite uses derivative data as well as values (osculatory, higher accuracy for smooth functions)
Sure
Q94Unbalanced transportation, non-degenerate BFS, MODI shows multiple optima: geometric implicationB
Multiple optima in transportation LP: optimal solutions lie on an edge (face) of the feasible polytope
Sure
Asked in 2025
8 questions. Tap one for the answer and working.
Q80Nature of LPP solution (feasible/bounded/many)D
x1>=x2>=3+3x1 forces x1<=-1.5, contradicting x1>=0: infeasible, so none of the options
Matches the official key
Q81Which set in R² is not convexB
Annulus 1<=r^2<=4 is not convex (the others are a disc, a disc, a half-plane)
Matches the official key
Q82NR iteration formulaA
Newton-Raphson: x_{n+1} = x_n - f(x_n)/f'(x_n)
Matches the official key
Q83Max value of objective subject to constraintsC
Vertices: (0,0)->0, (3.5,0)->17.5, (7/3,5/3)->23.33, (0,4)->28 (feasible); max 28
Matches the official key
Q84LPP: none/unbounded/alternative/unique optimumB
Take x2=x1, x1->inf: feasible, z=x1->inf, so unbounded
Matches the official key
Q85Third divided difference of 1/x at a,b,c,dD
Third divided difference of 1/x = (-1)^3/(abcd) = -1/(abcd)
Matches the official key
Q86Interpolating polynomial from tableA
x^3+x^2-x+2 gives 2,3,12,147 at x=0,1,2,5
Matches the official key
Q87Rate of convergence of NR (quadratic)A
Newton-Raphson converges quadratically (order 2)
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Simplex or Big-M by hand
- Game theory
- Simpson and trapezoidal rules
- Picard's method
- Gauss elimination
- Spline interpolation
Full syllabus for this unit
LPP by graph and simplex, Big-M, two-phase, duality, convex sets, transportation, Hungarian method, game theory; Newton–Raphson, fixed point, Gauss elimination, Gauss–Seidel, finite differences, interpolation, numerical calculus, Picard, Euler, Runge–Kutta.