Well, I don't know how it goes over there, but here in Algeria, Engineers go through two common years (after which they chose a specialty in the third year, and then, in the fourth and fifth year, a specialty of specialty).
All Engineers go through both years, except Computer Science who don't do the common second year and they directly go to Computer Science.
In these two years, everyone goes through this (maybe it'll give you some ideas on what you want to add):
I'll only list the "Maths" we take first and second year:
First year:
- Algebra: (a long course, bottom up. From Boole's algebra, to groups, sigma-algebra, yadda yadda), linear algebra(vector spaces, etc)..
- Probabilities and Statistics.
- Analysis: (Taylor series (Lagrange, Laplace, Young, Cauchy, Maclaurin), integrals, differentiations, different series, convergence/divergence kung fu), Riemann overall, proofs, etc.. Functions, multivariable, real and complex, etc.
Second year:
- Analysis I - Numerical Analysis (Equation systems, Gauss-Seidel, different algorithms(also calculating their speeds), Newton-Raphson, extrapolation, interpolation, etc).
- Analysis II - Integrals(up to 3rd - curves, areas/surfaces(Green) and volumes (Ostrogradsky)), Differential equations (Wronskian, etc).
This is the minimum (to be able to function in other modules, and some other modules are needed before you can function in these, so there's sort of bootstrapping of sort).
And then it depends what you take as specialty (if it's something involving Signal Processing, for instance, or Control Systems, you also need to do stuff).
Hope that helps and you can find some things.
PS: None of these are done with computers, so computing stuff with Newton algorithm and operations on big matrices are all done by hand. It takes a lot of time.
PPS: We don't have multiple answer questions. There's a question, and you answer it (and some answers take multiple pages).
Also, most tests are designed in a way that even if you have the answer sheet right next to you, it still takes you more time to copy the answers than the time of the exam itself. i.e: Even if you don't think and only "write", the time-frame is too tight.