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| SMG-8046 RF-electronics preparatory 1

Small Exam 1, 19.09.10 at 9.15
Answer to three of the four guestions. Each guestion gives maximum of 5 points.

1. (a) i Assume that the voltage over Rs is 1V. What is value of I then?

currents of all the other resistors.
Let Ri = 2.50, Ry = 1592, R3 = 109, R4 = 129, and R; = 50.

 

ii. Assume that / = 2A. Find voltage over Rs. Determine also the voltages and

 

 

 

 

 

 

 

 

 

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aN I O R3 97, Fa W Rs a

 

 

 

 

 

 

 

 

 

 

 

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(b) Are following true or not. Include proper reasoning to your answer.

 

i. Any five vectors in R5 form a basis for RP.
ii. A basis in a vector space is unigue.
| iii. £ A is a 3x3 matrix and det(A) hen the rows of A are linearly dependent
voctors of R3.

2. (a) Let us assume that space V has basis (v1, va,..., Vn). Let us also assume that
for two linear transformations Tj : V > W and 77 : V —> W holds that Tiv; =
Tyv; = w; for alli=1,2,...,n. Show that then for any vector v € V holds that
Tiv = Tov.

(b) i. Find the voltage U. Let R; = 19, R, = 199, L = 0,095H, € = 0,001F,

 

 

 

 

 

 

 

 

 

 

 

 

w = 100 rad/s, E =12 V. a 00!" : = 00
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+ Ro C 2 a
1 E 5 L) ,) 2 U
- L < 194 J
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ii. Comment on your result, does it seem to be a bit unexpected?

v

V 3. (a) Determine rank and nullity of matrix A.

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A=|-6 42 23 a PA
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(b) Let A € R"*" and let us define a transformation T4 : R" > R"" as D

TA(x) = Ax

where x € R”. Is T4 linear or not? Give a proper verification.

TURN OVER
 

4. (a) Solve the initial value problem
y" +3y' — 10y=0, yl0)=3, y(0)=-1.

Show also that you have obtained linearly independent solution(s).

(b) Look at the picture below which is related to solution of a 2nd order differential
eguation. Based on the picture, estimate what happens when time increases
further. How could you describe the system in guostion? Could you deduce and
write down a possible eguation that would characterize the solution?

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a.
/

=
: myy
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1000 1 £ £ E ]
0 5 10 1 2 25 30 35 40 45
time

 

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