Engineering Mathematics
Linear algebra, calculus, differential equations, complex analysis, probability and numerical methods.
About this subject
Engineering Mathematics is common to all engineering papers and supports every other section: linear algebra, calculus and vector calculus, differential equations, complex analysis, probability and statistics, and numerical methods.
The questions are usually short and exact, which makes them excellent marks for anyone who has practised. Nearly all of them have a quick check - the trace of a matrix, the total of the probabilities - that catches arithmetic slips in seconds.
Solved questions
Each question is a practice problem written for this portal at GATE standard. Try it first, then open the full derivation, and read the note on the idea behind it.
- MCQ Eigenvalues of a 2x2 matrix from trace and determinant Linear algebra · 1 mark
- NAT Probability two balls drawn without replacement match Probability · 2 marks
- NAT Contour integral of a function with two enclosed poles Complex analysis · 2 marks
- NAT Divergence of a vector field evaluated at a point Vector calculus · 1 mark
Key formulas
The results these questions lean on most. Know what each symbol means, and when each one applies.
| What | Formula |
|---|---|
| Eigenvalues of a 2 × 2 matrix | $\lambda^2 - (\text{tr}\,A)\lambda + \det A = 0$ |
| Checks on eigenvalues | $\sum \lambda_i = \text{tr}\,A$, $\;\prod \lambda_i = \det A$ |
| Divergence | $\nabla\cdot\vec F = \partial F_x/\partial x + \partial F_y/\partial y + \partial F_z/\partial z$ |
| Residue theorem | $\oint_C f(z)\,dz = 2\pi j \sum \text{Res}$ (poles inside $C$) |
| Residue at a simple pole | $\text{Res}_{z_0} = \lim_{z\to z_0} (z - z_0) f(z)$ |
| Without replacement | $P(A \cap B) = P(A)\,P(B \mid A)$ |
| Combinations | $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$ |
How to prepare
- Use the trace and determinant to check every eigenvalue answer.
- In probability, decide at the start whether draws are with or without replacement.
- For contour integrals, find every pole, decide which are inside the contour, and only then compute residues.
- Pair each derivative with the right variable in vector calculus: divergence uses $\partial F_x/\partial x$, not $\partial F_x/\partial y$.
Common traps
Each of these costs marks every year. The questions above are built to catch them.
- Sign errors in a determinant.
- Squaring single-draw probabilities when the draws are without replacement.
- Assuming an integral is non-zero because poles are enclosed.
- Mixing up the pairings of divergence and curl.