Written Examination Foundations of Algebra Wednesday, 24 August

Written Examination
Foundations of Algebra
Wednesday, 24 August 2016
Duration: 08:00–13:00
Centre for Mathematical Sciences
Mathematics, Faculty of Science
Turn the page for the Swedish text.
No aids. The results will be posted on Friday, 26 August, at 13:00. The viewing of marked
scripts takes place from 13:15 to 14:00 on the same day in Room 503. Oral examinations
are held on Tuesday afternoon, 30 August.
1.
Solve the inequality
2.
Show that
4x − 10
x
≤
.
x−1
x−3
n
X
k=2
1
3
1
1
= −
−
k2 − 1
4 2n 2(n + 1)
for all integers n ≥ 2.
3.
A subset A of R consists of four negative and six positive numbers.
a) How many subsets of A have exactly four elements?
b) How many of the subsets with exactly four elements have the property that the
product of their elements is positive?
4.
Find all complex roots of the equation
z 6 + (1 + i)z 3 + i = 0.
Express them in Cartesian form a + bi and plot them in the complex plane.
5.
Let f be a polynomial over C and let f ′ be its derivative. Show that f ′ | f if and only
if f = a(x − α)n for some complex numbers a and α and some positive integer n.
6.
Show that the points z1 , z2 and z3 in the complex plane lie on the same line if and
only if z1 z 2 + z2 z 3 + z3 z 1 ∈ R.
Tentamensskrivning
Algebrans grunder
Onsdag den 24 augusti 2016
Skrivtid: 08.00–13.00
Matematikcentrum
Matematik NF
Turn the page for the English text.
Inga hjälpmedel. Resultatet anslås på fredag den 26 augusti kl. 13.00. Skrivningsvisningen
äger rum från 13.15 till 14.00 samma dag i rum 503. Muntliga tentamina ges på tisdag
eftermiddag den 30 augusti.
1.
Lös olikheten
2.
Visa, att
4x − 10
x
≤
.
x−1
x−3
n
X
k=2
k2
3
1
1
1
= −
−
−1
4 2n 2(n + 1)
för alla heltal n ≥ 2.
3.
En delmängd A av R består av fyra negativa och sex positiva tal.
a) Hur många delmängder av A har exakt fyra element?
b) Hur många av delmängderna med exakt fyra element besitter egenskapen, att
produkten av deras element är positiv?
4.
Bestäm alla komplexa rötter till ekvationen
z 6 + (1 + i)z 3 + i = 0.
Uttryck dem i kartesisk form a + bi, och markera dem i det komplexa talplanet.
5.
Låt f vara ett polynom över C, och låt f ′ vara dess derivata. Visa, att f ′ | f om och
endast om f = a(x − α)n för några komplexa tal a och α och något positivt heltal n.
6.
Visa, att punkterna z1 , z2 och z3 i det komplexa talplanet ligger på samma linje om
och endast om z1 z 2 + z2 z 3 + z3 z 1 ∈ R.