Download 1) Solve the following system of equations: −3y2 + 2xy + x + 9 = 0 (1

1) Solve the following system of equations:
−3y 2 + 2xy + x + 9 = 0
x + 4y + 9 = 0
(1)
(2)
There is always more than one way to do things, but what I would do is
notice that there is an x in each equation given. Subtract (ii) from (i) to get
−3y 2 + 2xy − 4y = 0. What can you do with this? Factor — every term has a
y in it. Get
y(−3y + 2x − 4) = 0.
Use the fact that this implies either y = 0 or −3x + 2x − 4 = 0.
• If y = 0, plug that in to either original equation to get x = −9.
• If −3y + 2x − 4 = 0, make a system out of that and equation (ii) above.
Use substitution or elimination to solve this: get the solution x = −1,
y = −2.
Make sure that you present your solutions appropriately: you can’t mix and
match the x and y solutions you get...
2) Draw a graph of the following system of inequalities:
x2 + y 2
x+y
≥ 9
≤ 3
The first equation gives a circle centered at the origin of radius 3. The second
equation gives a line through the points (0, 3) and (3, 0). The shaded area is
the one that lies below and to the left of both curves.
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3) Given the complex number z =
√
3 − i, find the following:
(a) |z|
For any complex number x + iy, |z|2 = x2 + y 2 . In this case, then, |z| = 2.
(b) z 7 (leave your answer in polar form)
Use DeMoivre’s Theorem. First express z in polar form (z = 2(cos(−30 o ) +
i sin(−30o ))). Then
|z|7
= [2(cos(−30o ) + i sin(−30o ))]7
= 27 (cos(−210o ) + i sin(−210o))
√
1
− 3
+i )
= 128(
2
√2
= −64 3 + i64
(c) z 1/2
DeMoivre’s Theorem again:
|z|1/2
π
π
= [2(cos(− + 2πk) + i sin(− + 2πk))]1/2
6
6
√
π
π
=
2(cos(− + πk) + i sin(− + πk))
12
12
Use the values k = 0, 1 to find the exact answers. On the midterm, be
prepared for slightly nicer numbers and the necessity of evaluating completely
(to get an answer that looks like x + iy).
4) What is the value of i101 ?
You can use DeMoivre’s Theorem, or notice that since i4 = 1, and 101 has
remainder 1 when you divide by 4, i101 = i1 = i. A lot of people were off by
one on this — not a bad thing to review a little bit.
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