Hoofdstuk 1
Meetkunde en Lineaire
Algebra
Vraag 1.1 Het trapoppervlak is een afwikkelbaar oppervlak met
oneindig veel singuliere punten.
vals
Vraag 1.2 Het schroefoppervlak is een afwikkelbaar oppervlak
met oneindig veel singuliere punten.
waar
Vraag 1.3 De parameterkrommen op de wig van Wallis zijn
rechten en (eventueel ontaarde) ellipsen.
waar
Vraag 1.4 Het oppervlak met cartesiaanse vergelijking 2z =
x2 − y 2 is een regeloppervlak met twee stellen beschrijvenden,
waarbij elk van beide stellen zijn eigen richtvlak heeft.
waar
1
Vraag 1.5 Er bestaan hermitische matrices in C2×2 waarvan
het spoor 0 is en de determinant positief.
vals
Vraag 1.6 Er bestaan nilpotente matrices die diagonaliseerbaar
zijn.
vals
Vraag 1.7 Zij u ∈ Rn×1 , u 6= 0, n > 1. Dan is 0 ∈ σ(u uT )
met algebra¨ısche en meetkundige multipliciteit gelijk aan n−1.
waar
Vraag 1.8 Zij u ∈ Cn×1 , u 6= 0, n > 1. Dan bevat σ(u u† )
twee verschillende elementen.
waar
Vraag 1.9 Zij x, y twee van nul verschillende, orthogonale elementen in Rn×1 . Dan bezit xy T enkel de eigenwaarde nul.
waar
Vraag 1.10 Zij A ∈ Cn×n . Dan hebben de matrices A + I
en A − I dezelfde eigenvectoren en zijn de spectra van beide
matrices over een afstand 2 t.o.v. elkaar verschoven.
waar
2
Hoofdstuk 2
Natuurkunde I
Vraag 2.1 Each of two small non-conducting spheres is charged
positively, the combined charge being 40 µC. When the two
spheres are 50 cm apart, each sphere is repelled from the other
by a force of magnitude 2.0 N. Determine the magnitude of the
smaller of the two charges.
1.4 µC
Vraag 2.2 In the figure, if Q = 30 µC, q = 5.0 µC, and d = 30
cm, what is the magnitude of the electrostatic force on q?
7.5 N
Vraag 2.3 If Q = 20 µC and L = 60 cm, what is the magnitude
of the electrostatic force on any one of the charges shown?
9.1 N
3
Vraag 2.4 A uniformly charged rod (length = 2.0 m, charge
per unit length = 5.0 nC/m) is bent to form one quadrant of a
circle. What is the magnitude of the electric field at the center
of the circle?
50 N/C
Vraag 2.5 When gravitational, magnetic and any forces other
than static electric forces are not present, electric field lines in
the space surrounding a charge distribution show
tangents to the directions in which either static or moving charges
would accelerate when passing through points on those lines.
Vraag 2.6 A positively charged particle is moving in the +ydirection when it enters a region with a uniform electric field
pointing in the +x-direction. Which of the diagrams below
shows its path while it is in the region where the electric field
exists. The region with the field is the region between the plates
bounding each figure. The field lines always point to the right.
The x-direction is to the right; the y-direction is up.
4
Vraag 2.7 Three pith balls supported by insulating threads hang
from a support. We know that ball X is positively charged.
When ball X is brought near balls Y and Z without touching
them, it attracts Y and repels Z. We can conclude that
Z has a positive charge.
Vraag 2.8 Two identical pith balls supported by insulating threads
hang side by side and close together, as shown below. One is
positively charged; the other is neutral. We can conclude that
some of the field lines leaving the positively charged pith ball end
on the neutral pith ball.
Vraag 2.9 Two imaginary spherical surfaces of radius R and
2R respectively surround a positive point charge Q located at
the center of the concentric spheres. When compared to the
number of field lines N1 going through the sphere of radius R,
the number of electric field lines N2 going through the sphere
of radius 2R is
N2 = N1
Vraag 2.10 A uniform electric field E~I is present in the region
between infinite parallel plane plates A and B and a uniform
electric field E~II is present in the region between infinite parallel plane plates B and C. When the planes are vertical, E~I is
directed to the right and E~II to the left. The signs of the charges
on plates A, B and C may be
5
any one of the above
6