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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?
- a)Four left half plane (LHP) roots
- b)One right half plane (RHP) root, one LHP root and two roots on the imaginary axis
- c)Two RHP roots and two LHP roots
- d)All four roots are on the imaginary axis
Correct answer is option 'C'. Can you explain this answer?
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Which one of the following options correctly describes the locations o...
⇒ 2 sign changes and 1 ROZ
⇒ 2 poles in right half of S-plane
And symmetrical poles in the LHS-plane.
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Which one of the following options correctly describes the locations o...
Explanation:
The given equation is s^4 + s^2 + 1 = 0. To find the locations of the roots of this equation on the complex plane, we can use the fact that the roots of a polynomial with real coefficients come in complex conjugate pairs. Therefore, we can first solve for the roots of the quadratic equation s^2 = (-1 ± √3i)/2 using the quadratic formula.
Solving for the roots of the quadratic equation:
s^2 = (-1 ± √3i)/2
s = ± sqrt[(-1 ± √3i)/2]
We can simplify this expression by rationalizing the denominator:
s = ± sqrt[-(1 ± √3i)]/sqrt(2)
s = ± (1/√2) sqrt(-(1 ± √3i))(√2/2 ± √2/2i)
s = ± (1/√2) (√2/2 ± √6/2i)
Therefore, the roots of the quadratic equation are:
s1 = (1/√2)(√2/2 + √6/2i) = (1/2)(1 + √3i)
s2 = (1/√2)(√2/2 - √6/2i) = (1/2)(1 - √3i)
Finding the locations of the roots on the complex plane:
Now, we can use these roots to find the locations of the roots of the original equation s^4 + s^2 + 1 = 0 on the complex plane. We can do this by factoring the original equation as follows:
s^4 + s^2 + 1 = (s^2 + s√3i + 1)(s^2 - s√3i + 1)
Using the roots of the quadratic equation we found earlier, we can write these factors as:
s^2 + s√3i + 1 = (s - s1)(s - s2)
s^2 - s√3i + 1 = (s + s1)(s + s2)
Therefore, the roots of the original equation are the roots of these two factors, which are:
s1 = (1/2)(1 + √3i) --> RHP root
s2 = (1/2)(1 - √3i) --> RHP root
-s1 = -(1/2)(1 + √3i) --> LHP root
-s2 = -(1/2)(1 - √3i) --> LHP root
Therefore, the locations of the roots of the equation s^4 + s^2 + 1 = 0 on the complex plane are two roots in the right half plane (RHP) and two roots in the left half plane (LHP).
The given equation is s^4 + s^2 + 1 = 0. To find the locations of the roots of this equation on the complex plane, we can use the fact that the roots of a polynomial with real coefficients come in complex conjugate pairs. Therefore, we can first solve for the roots of the quadratic equation s^2 = (-1 ± √3i)/2 using the quadratic formula.
Solving for the roots of the quadratic equation:
s^2 = (-1 ± √3i)/2
s = ± sqrt[(-1 ± √3i)/2]
We can simplify this expression by rationalizing the denominator:
s = ± sqrt[-(1 ± √3i)]/sqrt(2)
s = ± (1/√2) sqrt(-(1 ± √3i))(√2/2 ± √2/2i)
s = ± (1/√2) (√2/2 ± √6/2i)
Therefore, the roots of the quadratic equation are:
s1 = (1/√2)(√2/2 + √6/2i) = (1/2)(1 + √3i)
s2 = (1/√2)(√2/2 - √6/2i) = (1/2)(1 - √3i)
Finding the locations of the roots on the complex plane:
Now, we can use these roots to find the locations of the roots of the original equation s^4 + s^2 + 1 = 0 on the complex plane. We can do this by factoring the original equation as follows:
s^4 + s^2 + 1 = (s^2 + s√3i + 1)(s^2 - s√3i + 1)
Using the roots of the quadratic equation we found earlier, we can write these factors as:
s^2 + s√3i + 1 = (s - s1)(s - s2)
s^2 - s√3i + 1 = (s + s1)(s + s2)
Therefore, the roots of the original equation are the roots of these two factors, which are:
s1 = (1/2)(1 + √3i) --> RHP root
s2 = (1/2)(1 - √3i) --> RHP root
-s1 = -(1/2)(1 + √3i) --> LHP root
-s2 = -(1/2)(1 - √3i) --> LHP root
Therefore, the locations of the roots of the equation s^4 + s^2 + 1 = 0 on the complex plane are two roots in the right half plane (RHP) and two roots in the left half plane (LHP).
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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer?
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Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer?.
Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following options correctly describes the locations of the roots of the equation s4 + s2 + 1 = 0 on the complex plane?a)Four left half plane (LHP) rootsb)One right half plane (RHP) root, one LHP root and two roots on the imaginary axisc)Two RHP roots and two LHP rootsd)All four roots are on the imaginary axisCorrect answer is option 'C'. Can you explain this answer?.
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