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Midterm 1 – Math 363, instructor: Pawel Gladki Friday, August 3rd, 2007. Time: 60 minutes 1. Prove the following theorem:

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Midterm 1 – Math 363, instructor: Pawel Gladki Friday, August 3rd, 2007.

Time: 60 minutes 1. Prove the following theorem:

n

X

k=1

(ak + b) = n

2[an + (a + 2b)], where a, b are some fixed real numbers.

2. Solve the equation

(4 − 3i)z2− (2 + 11i)z − (5 + i) = 0.

3. Solve the following matrix equation:

X

1 1 −1

2 1 0

1 −1 1

=

1 −1 3

4 3 2

1 −2 5

4. Let R be the relation in the set Q defined as follows: for two rational numbers x and y, xRy if and only if xy = t2 for some rational number t.

Show that R is an equivalence relation, and has infinitely many equivalence classes.

5. In the set R of real numbers we define the “addition” as follows: x ⊕ y = p3

x3+ y3. Find a neutral element e of this addition and show that (R, ⊕, e) is an Abelian group.

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