Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

## Problems Marking scheme 1. Let A be a nonzero square matrix. Is it possible that a positive integer k exists such that ?? = 0 ? For example, find ?3 for the matrix [ 0 1 2 0 0 1 0 0 0 ] A square matrix A is nilpotent of index k when ? ≠ 0 , ?2 ≠ 0 , … . . , ??−1 ≠ 0, ??? ?? = 0. In this task you will explore nilpotent matrices. 1. The matrix in the example given above is nilpotent. What is its index? ( 2 marks ) 2. Use a software program to determine which of the following matrices are nilpotent and find their indices ( 12 marks ) A. [ 0 1 0 0 ] B. [ 0 1 1 0 ] C. [ 0 0 1 0 ] D. [ 1 0 1 0 ] E. [ 0 0 1 0 0 0 0 0 0 ] F. [ 0 0 0 1 0 0 1 1 0 ] 3. Find 3×3 nilpotent matrices of indices 2 and 3 ( 2 marks ) 4. Find 4×4 nilpotent matrices of indices 2, 3, and 4 ( 2 marks ) 5. Find nilpotent matrix of index 5 ( 2 marks ) 6. Are nilpotent matrices invertible? prove your answer ( 3 marks ) 7. When A is nilpotent, what can you say about ?? ? prove your answer ( 3 marks ) 8. Show that if ? is nilpotent , then ? − ? is invertible ( 4 marks ) 30% 2. A radio transmitter circuit contains a resisitance of 2.0 Ω, a variable inductor of 100 − ? ℎ?????? and a voltage source of 4.0 ? . find the current ? in the circuit as a function of the time t for 0 ≤ ? ≤ 100? if the intial curent is zero. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 3. An object falling under the influence of gravity has a variable accelertaion given by 32 − ? , where ? represents the velocity. If the object starts from rest, find an expression for the velocity in terms of the time. Also, find the limiting value of the velocity. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 4. When the angular displacement ? of a pendulum is small ( less than 60), the pendulum moves with simple harmonic motion closely approximated by ?′′ + ? ? ? = 0 . Here , ?′ = ?? ?? and ? is the accelertaion due to gravity , and ? is the length of the pendulum. Find ? as a function of time ( in s ) if ? = 9.8 ?/?2, ? = 1.0 ? ? = 0.1 and ?? ?? = 0 when ? = 0 . sketch the cuve using any graphical tool. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 5. Find the equation relating the charge and the time in a electric circuit with the following elements: ? = 0.200 ? , ? = 8.00 Ω , ? = 1.00 ?? , ? = 0. In this circuit , ? = 0 and ? = 0.500 ? when ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 6. A spring is stretched 1 m by ? 20 − ? Weight. The spring is stretched 0.5 m below the equilibrium position with the weight attached and the then released. If it is a medium that resists the motion with a force equal to 12?, where v is the velocity, sketch and find the displacement y of the weight as a function of the time. Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 7. A 20?? inductor, a 40.0 Ω resistor, a 50.0 ?? capacitor, and voltage source of 100 ?−100?are connected in series in an electric circuit. Find the charge on the capacitor as a function of time t , if ? = 0 and ? = 0 ?ℎ?? ? = 0 Correct solution 5% Graph the general solution 2.5% Graph the function and particular solution 2.5% 10% quality and neatness and using Math equations in MS word. –

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AERN 45350 Avionics Name: _______________________________ 1 | P a g e Homework Set One (40 Points) Due: 25 September 2015 General Instructions: Answer the following questions, submitting your answers on Blackboard. SHOW YOUR WORK on any math problems. Consider the following voltage signal: V t 12sin377t 1. What is the peak voltage of the signal [Volts]? 2. What is the angular frequency [rad/sec]? 3. What is the frequency of the signal [Hz]? 4. What is the period of the signal [sec/cycle]? In a heterodyne receiver, the intermediate frequency of the receiver is 21.4 MHz. 5. What is the local oscillator frequency (f1) if the tuned frequency (f2) is 120.9 MHz? 6. If the local oscillator frequency (f1) is 145.7 MHz, what is the tuned frequency (f2)? The gain of a power amplifier is 5. 7. If 30W are coming in, what is the power going out? 8. What is the gain in decibels (dB)? The attenuation of a voltage attenuator is 10. 9. If 120V are coming in, what is the voltage going out? 10. What is the loss in decibels (dB)? 11. What is the component of the ILS that provides the extended centerline of the runway? 12. What is the component of the ILS that provides vertical guidance to the runway? 13. If the aircraft is on the correct trajectory, the airplane will arrive at the outer marker on the ILS corresponding to intercepting what? 14. If the aircraft is on the correct trajectory, the airplane will arrive at the middle marker on the ILS corresponding to reaching what? 15. All marker beacons transmit at what frequency? 16. Why doesn’t this cause problems (all marker beacons transmitting on the same frequency)? 17. What are the four components to an ILS? 18. What is the most common ILS category? 19. Which ILS category requires aircraft with the “auto-land” feature? 20. An attenuator leads to a power ratio of 0.5. What is that in decibels (dB)?

## AERN 45350 Avionics Name: _______________________________ 1 | P a g e Homework Set One (40 Points) Due: 25 September 2015 General Instructions: Answer the following questions, submitting your answers on Blackboard. SHOW YOUR WORK on any math problems. Consider the following voltage signal: V t 12sin377t 1. What is the peak voltage of the signal [Volts]? 2. What is the angular frequency [rad/sec]? 3. What is the frequency of the signal [Hz]? 4. What is the period of the signal [sec/cycle]? In a heterodyne receiver, the intermediate frequency of the receiver is 21.4 MHz. 5. What is the local oscillator frequency (f1) if the tuned frequency (f2) is 120.9 MHz? 6. If the local oscillator frequency (f1) is 145.7 MHz, what is the tuned frequency (f2)? The gain of a power amplifier is 5. 7. If 30W are coming in, what is the power going out? 8. What is the gain in decibels (dB)? The attenuation of a voltage attenuator is 10. 9. If 120V are coming in, what is the voltage going out? 10. What is the loss in decibels (dB)? 11. What is the component of the ILS that provides the extended centerline of the runway? 12. What is the component of the ILS that provides vertical guidance to the runway? 13. If the aircraft is on the correct trajectory, the airplane will arrive at the outer marker on the ILS corresponding to intercepting what? 14. If the aircraft is on the correct trajectory, the airplane will arrive at the middle marker on the ILS corresponding to reaching what? 15. All marker beacons transmit at what frequency? 16. Why doesn’t this cause problems (all marker beacons transmitting on the same frequency)? 17. What are the four components to an ILS? 18. What is the most common ILS category? 19. Which ILS category requires aircraft with the “auto-land” feature? 20. An attenuator leads to a power ratio of 0.5. What is that in decibels (dB)?

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Chapter 14 Practice Problems (Practice – no credit) Due: 11:59pm on Friday, May 16, 2014 You will receive no credit for items you complete after the assignment is due. Grading Policy Harmonic Oscillator Equations Learning Goal: To derive the formulas for the major characteristics of motion as functions of time for a horizontal spring oscillator and to practice using the obtained formulas by answering some basic questions. A block of mass is attached to a spring whose spring constant is . The other end of the spring is fixed so that when the spring is unstretched, the mass is located at . . Assume that the +x direction is to the right. The mass is now pulled to the right a distance beyond the equilibrium position and released, at time , with zero initial velocity. Assume that the vertical forces acting on the block balance each other and that the tension of the spring is, in effect, the only force affecting the motion of the block. Therefore, the system will undergo simple harmonic motion. For such a system, the equation of motion is , and its solution, which provides the equation for , is . Part A At what time does the block come back to its original equilibrium position ( ) for the first time? Express your answer in terms of some or all of the variables: , , and . You did not open hints for this part. ANSWER: m k x = 0 A t = 0 a(t) = − x(t) km x(t) x(t) = Acos( t) km −−  t1 x = 0 A k m