Extra Exam Exercises (+ solutions) FINAL Mathematics for Premaster

Exam exercises Mathematics for Premaster FINAL
This is a preview of the extra exam exercises that you can use to prepare for your final. I
have successfully passed the premaster in 2019 myself, and since I received a 9 on
mathematics, I tutored a bunch of students in 2020. For a crash course, I created these
30 exercises with extensive elaboration of the answers myself. This might be very
helpful, because you haven’t practiced with these before.


Please note that copyright belongs to the creator of this document.
It is forbidden to freely distribute this file.


If you have bought this document, you can skip the first 6 pages (until ‘END OF
PREVIEW’) as these first pages are adapted with black lines to serve the preview.


Part I


1. Consider a producer with cost function 𝐶(𝑦) = 2𝑦 3 − 6𝑦 2 + 12𝑦. Determine the
minimum price at which the firm produces (i.e., does not make a loss in the
optimum).


2. Determine the supply function of the producer from question 1.


3. Solve the following constrained extremum problem:
.

maximize 𝑧(𝑥, 𝑦) = 𝑥𝑦 + 𝑦 2
subject to 𝑥 + 3𝑦 = 32
where 𝑥 ≥ 0 and 𝑦 ≥ 0


4. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX


5. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX

,6. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


7. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX


8. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX


9. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX


10. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


11. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX


12. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX


13. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX


14. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


15. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX

,Part II


16. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX


17. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX


18. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


19. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX


20. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX


21. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX


22. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


23. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX


24. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXX

,25. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX


26. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX XXXXXX XXXXXXXXXXX XXX XXXX XXX XXXX XX
XXXXXXXXXXX XXX XXXX XXXXX XXXXXXXXXXXXXXXXXXXXXXX


27. XXXX XXX XX XXXX XXXXX XXXX XXXXX XXXX XXXXXXXXX XXXXXXXXXX XXX
XXXXXX XXXX XXXXXXXXX


28. Consider the system of linear equations with corresponding extended
coefficient matrix.
.

1 2 1 | 8
[ 0 5 −4 | 10 ].
2
0 −5 𝑡 | 𝑡−8
.

Determine all 𝑡 such that the system is inconsistent.


1 6 4
29. Determine, if possible, the inverse 𝐸 −1 of the matrix 𝐸 = [ −1 3 5 ].
2 3 −1


30. Solve the following maximalization problem:
.

maximize 𝑧(𝑥, 𝑦) = 8𝑥 + 𝑦
such that 4𝑥 + 12𝑦 ≤ 80
2𝑥 + 2𝑦 ≤ 20
𝑥≤7
𝑥 ≥ 0 and 𝑦 ≥ 0

,
,