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1 DEPARTMENT OF MATHEMATICS MA7163 APPLIED MATHEMATICS FOR ELECTRICAL ENGINEERS (UNIVERSITY QUESTIONS) UNIT- I MATRIX THEORY PART A 1. Write a note on least square solution 2. Define singular value matrix 3. If A is a non singular matrix,then what is A +? 4. What is meant by singular value of a matrix? 5. Find the norms of,.also verify that x and y are orthogonal.find < x,y > 6. State Singular value decomposition theorem. 7. Determine the canonical basia for the matrix A =. 8. Explain least square solution. 9. Define Toeplitz matrix with an example. 10. Find the least square solution to the system 11. Explain singular value decomposition in matrix theory. 12. What is the advantage in matrix factorization methods? 13. Check whether the given matrix is positive definite or not 14. Write the necessary conditions for Cholesky decomposition of a matrix. 15. Find the Cholesky decomposition of

2 PART B 1. Find the QR decomposition of 2. Solve the following system of equations in the least square sense 3. Construct QR decomposition for the matrix 4. Construct the singular value decomposition for the matrix 5. Find the QR factorization of(i) ii) 6. Obtain the singular value decomposition of (i) ( ii ) iii) 7. Let.Compute the singular values of and singular value decomposition of A. Also find A Let.Compute using QR algorithm. 9. Determine the Cholesky decomposition of the matrix 10. Determine the Cholesky decomposition of the matrix

3 UNIT II CALCULUS OF VARIATION PART -A 1. Write Euler s equation for functional. 2. Define isoperimetric problems. 3. Define several dependent variables. 4. Write a formula for functional involving higher order derivatives. 5. Define ring method. 6. State Brachistochrone problem. 7. Write other forms of Euler s equation. 8. Write the ostrogradsky equation for the functional 9. State Diffusion Equation and Poisson Equation. 10. Define moving boundaries. 11. Define a functional. 12. State the Euler-Ostrogradsky equation Test for extremum the functional I[ y( x)] = ( y' y ) dx, y(0) = 0, y( π / 2) = Write Euler-Poisson equation Find the curve on which the functional ( y' + 12xy) dx, y(0) = 0, y(1) = 1 can be extremised? 1 0 π 2 PART B 1. Find the extremals of (i), (ii) (iii) 2. Solve the extremal 3. Show that the straight line is the sharpest distance between two points in a plane. 4. A curve c joining the points ( ) and ( ) is revolved about the x-axis. Find the shape of the curve, so that the surface area generated is a minimum.

4 5. Find the extremals of the functional =,, z(0)=0,. 6. Show that the curve which extremize given that 7. Determine the extremal of the functional that satisfies the boundary condition 8. Prove that the sphere is the solid figure of a revolution which for a given surface has maximum volume. 9. Find the curve on which an extremum of the function can be achieved if the second boundary point is permitted to move along the straight line. 10. Solve the boundary value problem by ring method.

5 UNIT III ONE DIMENSIONAL RANDOM VARIABLES PART-A 1. Define moment generating function of a random variable X. 2. If X is a Poisson variate such that P ( X =2 ) =9 P (X =4 ) +90 P ( X = 6 ).Find the variance of X. 3. If X is a continuous RV with p.d.f. f(x) =2x,0 <x< 1, then find the pdf of the RV Y = X 3 4. If the mean of a Poisson varaite is 2,then what is the standard deviation? 5. If X and Y are independent RVs with varainaces 2 and 3.Find thevariance of 3X + 4Y. 6. The first four moments of a distribution about 4 are 1,4,10 and 45 respectively. Show that the mean is 5 and variance is Find the value of X,then find the value of K 8. The random variable X has a Binomial distribution with parameters n =20,p =0.4.Determine P (X = 3). 9. If the probability is 0.05 that a certain kind of measuring device will show excessive drift, what is the probability that the sixth measuring device tested will be the first to show excessive drift? 10. If a RV has the probability density, find the probabilities that will take a value between 1 and A RV X has the p.d.f..find the CDF of X 12.If the RV X takes the values 1,2,3 and 4 such that 2P(X = 1)=3P( X =2 ) =P ( X =3 ) = 5 P (X =4 ) find the probability distribution. 13.State the memoryless property of an exponential distribution. 14.If a RV has the probability density, find the mean and variance of the RV X. 15. Obtain the moment generating function of Geometric distribution.

6 PART B 1.Buses arrive at a specific stop at 15 minutes intervals starting at 7a.m.If a passenger arrives at a random time that is uniformly distributed between 7 and 7.30 am, find the probability that he waits 1) less than 5 minutes for a bus and 2) atleast 12 minutes for a bus. 2. A manufacturer of certain product knows that 5 % of his product is defective.if he sells his product in boxes of 100 and guarantees hat not more than 10 will be defective,what is the approximate probability that a box will fail to meet the guaranteed quality? 3. If X is a discrete random variable with probability function p(x) =, x =1,2... (K constant ) then find the moment generating function,mean and variance.. 4. In a company,5% defective components are produced.what is the probability that at least 5 components are to be examined in order to get 3 defectives? Let the random variable X has the p.d.f. Find the mean and variance. 6. In a normal distribution, 31 % of the items are under 45 and 8% are over 64.Find the mean and variance of the distribution. 7. If X is Uniformly distributed over ( 0,10),find the probability that ( i) X < 2 (ii) X > 8 (iii) 3 <X <9? 8.A discrete RV X has the probability function given below X : P(x) : 0 a 2a 2a 3a a 2 2 a 2 7a 2 + a Find (i) Value of a (ii) p (X <6), P ( X 6 ), P ( 0 < X < 4 ) (iii) Distribution function. 9. The slum clearance authorities in a city installed 2000 electric lamps in a newly constructed township. If the lamps have an average life of 1000 burning hours with a standard deviation of 200 hours, 1) what number of lamps might be expected to fail in the first 700 burning hours? 2) After what period of burning hours would you expect 10 percent of the lamps would have been failed? ( Assume that the life of the lamps follows a normal law) 10. The daily consumption of milk in a city,in excess of 20,000 gallons,in approximately distributed as a Gamma variate with the parameters k = 2 and The city has a daily stock of 30,000 gallons. What is the probability that the stock is insufficient on a particular day?

7 UNIT IV LINEAR PROGRAMMING PART A 1. What is degeneracy in a transportation model 2. Differentiate between balanced and unbalanced cases in Assignment model 3. List any two basic differences between a transportation and assignment problem 4. What do you mean by degeneracy? 5. Explain optimal solution in L.P.P. 6. Solve the following L.P.P by using graphical method, Subject to 7. Obtain an initial basic feasible solution to the following transportation problem by using Matrix minima method D 1 D 2 D 3 D 4 capacity O O O Demand What is the difference between feasible solution and basic feasible solution? 9. What is an assignment problem? Give two applications. 10. Write down the mathematical formulation of L.P.P. 11. When will you say a transportation problem is said to be unbalanced? 12. What is a travelling sales man problem? 13. Enumerate the methods to find the initial basic feasible solution for transportation problem 14. Define transshipment problem. 15. Define Assignment Problem. PART - B 1.Solve the L.P.P by Simplex method Subject to

8 2. Solve by Simplex method.maximize, Subject to. 3. Solve by Simplex method.maximize, Subject to. 4. Solve by Simplex method.maximize, Subject to. 5. Solve by Graphical method, Maximize, Subject to. 6. Solve by Graphical method Maximize, Subject to 7. Find the initial feasible solution for the following transportation problem D1 D2 D3 D4 Supply Demand A military equipment is to be transportated from origins x,y,z to the destinations A,B,C and D The supply at the origins, the demand at te destinations and time of shipment is sown in the table. Work out the transportation plan so that the time required for shipment is the minimum Destinations A B C D Supply Origin X Y Z Demand

9 9. Solve the assignment problem P Q R S A B C D A travelling sales man has to visit 5 cities.he wishes to start from a particular city, visit each city once and then returns to his starting point. Cost of going from one city to another is shown below. Find the least cost route To City A B C D E A B C D E

10 UNIT V FOURIER SERIES PART A 1. If the periodic function f(t) ={ where f( t + 2π) = f(t) is expanded as a Fourier series, find the value of a n. 2. Define energy signals and power signals? 3. Define a periodic function as power signals. 4. Calculate average power of period T =2, f(t) = 2cos 5πt + sin6πt using time domain analysis. 5. State Parseval s theorem. 6. State convergence of the series. 7. Write note on Singular Sturm- Liouville System. 8. Define self ad joint operator. 9. Write the power signals Exponential Fourier Series. 10. Define generalized Fourier series? 11. Put the following DE in self ad joint form x 2 y + 3xy + λy = Distinguish Periodic and Non Periodic functions, with example. 13. Write short note on Eigen value problem and orthogonal functions. 14. Define Periodic Sturm- Liouville Systems. 15. Write note on cosine and sine series. PART B 1. Find the Fourier series of the periodic ramp function, f(t+2π) = f(t). 2. Find the fourier series of the sawtooth function f(t) = t, -1 < t < 1, f(t+2) = f(t). 3. Find a fourier series representation of f(t) = t 2, 0 < t < 1. i) as a sine series with period T =2 ii) as a cosine series with period T =2 iii) as a full trigonometric series with period T =1 4. Calculate the averge power of the periodic signal, period T =2 f(t) = 2 cos 6πt + sin 5πt Using (i) time domain analysis and (ii) Frequency domain analysis. 5. Find the eigen values and eigen functions of y + λy = 0, 0 < x < p, y(0) = 0,y(p) = 0.

11 6. Find an expression for the Fourier coefficients associated with the generalised fourier series arising from the eigen functions of y + y + λy = 0, 0 < x < 3, y(0) = 0, y(3) = Find the eigen values and eigen functions of y + λy = 0, -π < x < π, y(-π) = y( π), y (-π) =y (π).. 8. Find the generalized Fourier Series expansion of the function f(x) = 1, 0 < x < 1, in terms of the eign functions of y + y + λy = 0, 0 < x < 1, y(0) = 0, y(1) + y (1) = Find the eigen values and eigen functions of y + λy = 0, 0 < x < 1, y(0) = 0, y(1) + y (1) = Find the eigen values and eigen functions of y + y + λy = 0, 0 < x < 3, y(0) = 0, y(3) = 0. ***** ALL THE BEST *****

HW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100]

HW7 Solutions. f(x) = 0 otherwise. 0 otherwise. The density function looks like this: = 20 if x [10, 90) if x [90, 100] HW7 Solutions. 5 pts.) James Bond James Bond, my favorite hero, has again jumped off a plane. The plane is traveling from from base A to base B, distance km apart. Now suppose the plane takes off from