Let P = ( 1 <msqrt> 2 </msqrt> </mfrac> , 1 <msqrt>

Willow Pratt 2022-07-13 Answered
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let P = ( 1 2 , 1 2 ) and Q = ( 1 2 , 1 2 ) be two vertices of a regular polygon having 12 sides such that PQ is a diameter of the circle circumscribing the polygon. Which of the following points is not a vertex of this polygon?
(A) ( 3 1 2 2 , 3 + 1 2 2 )
(B) ( 3 + 1 2 2 , 3 1 2 2 )
(C) ( 3 + 1 2 2 , 1 3 2 2 )
(D) ( 1 2 , 3 2 )
If P and Q are the end points of the diameter, it is quite clear that the equation of the circle must be
x 2 + y 2 = 1
Therefore, all the vertices must lie on this circle. Now, checking from the options, we find that every point given in the options satisfies the above equation. Now I am stuck.
How else should I tackle the sum?
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Answers (2)

Karla Hull
Answered 2022-07-14 Author has 20 answers
D) But how did I get that answer? Well, firstly have you drawn a picture of the diagram, with angles labelled? If so, then you’ll notice that the arg (angle from the x axis) of any point on the polygon must be 15 mod 30. The last option doesn’t satisfy this property (the angle it forms is 120 degrees).
However, another approach is to notice — again from a well drawn diagram — that the figure is symmetric in the x axis, the y axis, and the x=y line. Since the first three options are just reflections of each other in these lines, the answer must be D. Meta analysis is useful in multiple choice!

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Chant6j
Answered 2022-07-15 Author has 1 answers
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document A) ( 3 1 2 2 , 3 + 1 2 2 ) ( 1 2 , 1 2 ) = 3 2
B) 3 2
C) 1 2

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New questions

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
x ( k + 1 ) = A x ( k ) + b u ( k ) , y ( k ) = c x ( k )
where b = ( 0 , 1 ) T , c = ( 1 , 0 ) , A = [ 2 1 0 g ] for some g R
Find a feedback regulation (if there is any) of the form u ( k ) = K x ^ ( k ) where x ^ ( k ) is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders e ( k ) = x ( k ) x ^ ( k ) go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are λ 1 = 2 , λ 2 = g (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle | z | = 1.
The state observer equation is
[ x ( k + 1 ) e ( k + 1 ) ] T = [ A b K B k O A L C ] [ x ( k ) e ( k ) ] T
With characteristic equation
χ ( z ) = | z I A + b K | | z I A + L C | = χ K ( z ) χ L ( z )
Also consider
K = [ k 1 k 2 k 3 k 4 ]
and let a = k 1 + k 3 , β = k 2 + k 4
Then χ K ( z ) = ( z 2 ) ( z + g + β ) + a.
So we can select some eigenvalues inside the unit circle and determine a , β in terms of g. Choosing e.g. λ 1 , 2 = ± 1 / 2 we get a = 3 g + 33 / 8 , β = 9 / 4 g , g R
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose | λ 1 , 2 | < 1?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)