VALLIAMMAI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS MA7163 – APPLIED MATHEMATICS FOR ELECTRICAL ENGINEERS UNIT- I MATRIX THEORY PART – A 1. Define Hermitian Matrix. 2. Write the necessary conditions for Cholesky decomposition of a matrix. 3. Find the Cholesky decomposition of 4. Explain least square solution. 5. Find the least square solution to the system 6. Write down the stable formula for generalized inverse. 7. Find A+ for A = ⎡ 2 − 2 ⎤ ⎢1 1 ⎥ ⎢ ⎥ ⎢⎣ 2 2 ⎥⎦ 8. Define singular value matrix. 9. State Singular value decomposition theorem. 10. If A is a non singular matrix, then what is A+? 11. Find the norms of , .Also verify that x and y are orthogonal .Find < x ,y > 12. Define orthogonal and orthonormal vectors. 13. Write short note on Gram Schmidt Orthonormalisation process. 14. What is the advantage in matrix factorization methods? 15. Write short note on QR factorization. 16. Let .Compute A2 using QR algorithm. 17. Determine the canonical basis for the matrix A = . 18. Define the generalized Eigen vector, chain of rank m, for a square matrix. 19. Find the generalized inverse of ⎡ 1 1 ⎤ ⎢0 1 ⎥⎥ ⎢ ⎢⎣ 1 0 ⎥⎦ 20. Check whether the given matrix is positive definite or not PART –B 1. (a) Determine the Cholesky decomposition of the matrix −1⎞ (b) Obtain the singular value decomposition of A= ⎛⎜ 2 ⎟ − ⎜ 1 1 ⎟ ⎜ 4 − 3 ⎟⎠ ⎝ ⎛ 4 2i 2 ⎞ ⎜ ⎟ 2. (a) Find the Cholesky decomposition of the matrix ⎜ − 2i 10 1 − i ⎟ ⎜ 2 1+ i 9 ⎟ ⎝ ⎠ ⎡0 1 1 1⎤ (b) Find the QR factorization of A = ⎢1 0 1 1 ⎥⎥ ⎢ ⎢1 1 0 1⎥ ⎢ ⎥ ⎣1 1 1 0⎦ 3. (a) Solve the following system of equations in the least square sense (a) Fit a straight line in the least square sense to the following data X: -3 -2 -1 0 1 2 3 Y: 10 15 19 27 28 34 42 ⎡1 1 −1⎤ (b) Find the QR factorization of A = ⎢⎢ 1 0 0 ⎥⎥ ⎢1 0 − 2 ⎥ ⎢ ⎥ ⎣ 1 1 1 ⎦ ⎛ 2 2 − 2⎞ 5. (a)Find the QR decomposition of (b) Obtain the singular value decomposition of ⎛1 1 3 ⎞ ⎜⎜ ⎟⎟ ⎝1 1 3 ⎠ 7. (a) Find the QR factorization of A = ⎜ ⎟ ⎜1 0 0 ⎟ ⎜1 1 1 ⎟ ⎝ ⎠ (b) ) Solve the following system of equations in the least square sense x1 + x2 + x3 =1 . 10. x1 + x2 +3 x3 = 2 ⎛1 2⎞ 6. ⎡ 9 −3 0 − 3⎤ (b) Determine the Cholesky decomposition of the matrix ⎢− 3 6 3 0 ⎥⎥ ⎢ ⎢ 0 6 3 0 ⎥ ⎢ ⎥ ⎣− 3 0 −3 6 ⎦ 4. (a) Obtain the singular value decomposition of A = ⎛1 1⎞ ⎜ ⎟ ⎜1 1⎟ ⎜2 2 ⎟⎠ ⎝ ⎛ 1 3 3⎞ (b) Find the QR factorization of A = ⎜ ⎟ ⎜ 2 3 0⎟ ⎜ 2 0 3⎟ ⎝ ⎠ . x1 + x2 + x3 = 2 . (a) Construct the singular value decomposition for the matrix ⎜ ⎟ ⎜ 2 2 − 2⎟ ⎜− 2 − 2 6 ⎟ ⎝ ⎠ (b) Solve the following system of equations in the least square sense x1 + x2 +3 x3 =1 . x1 + x2 + x3 = 3. (a) Obtain the singular value decomposition of A = ⎜ ⎟ ⎜1 1⎟ ⎜1 3 ⎟⎠ ⎝ ⎡16 − 3 5 − 8⎤ ⎢ 16 − 5 − 8⎥⎥ (b) Determine the Cholesky decomposition of the matrix ⎢− 3 ⎢5 − 5 24 0 ⎥ ⎢ ⎥ ⎣− 8 − 8 0 21⎦ 6. (a) Construct QR decomposition for the matrix ⎡3 2 0 1 ⎤ ⎢ 0 ⎥⎥ (b) Find the canonical basis for A = ⎢ 0 3 0 ⎢0 0 3 − 1⎥ ⎢ ⎥ ⎣0 0 0 3 ⎦ ⎛1 1 − 1⎞ 8. Write Euler’s equation for functional. 9. State the necessary condition for the extremum of the functional I = ∫ F ( x. Show that geodesics on a plane are straight lines. Define moving boundaries. 12. 8. x0 14. Find the extremals of the functional ∫ ⎜⎜ x 3 ⎟⎟⎠ dx . Define isoperimetric problems. 7. 3. x1 17. Solve the Euler equation for ∫ ( x + y ' ) y ' dx x0 18. Solve the Euler equation for y ' ) y ' dx x0 PART –B 1. Define functional and extremal. 2. ⎛ y '2 ⎞ x1 15. State Brachistochrone problem. UNIT – II CALCULUS OF VARIATION PART -A 1. 4. 3. 2. (a)Show that the straight line is the sharpest distance between two points in a plane. x1 ∫ 19. (a) Solve the extremals (b) Prove that the sphere is the solid figure of a revolution which for a given surface has maximum volume. 5. y' ) dx . x0 ⎝ 16. Find the curve on which the functional straight line [( y ' ) 2 + 12 xy ] dx with y(0) = 0 and x0 y(1) = 1can be extremised. Write a formula for functional involving higher order derivatives. Define several dependent variables. x1 ∫ (1 + x 2 20. Write the ostrogradsky equation for the functional 10. Write short note on Rayleigh . Write other forms of Euler’s equation. Define Geodesic. (b) Find the extremals of . (b) Solve the boundary value problem y” + y +x = 0 ( 0 ≤ x ≤ 1 ) y(0) = y(1) = 0 by Rayleigh Ritz method.Ritz method. x1 13. 11. y. (a) Find the extremals of (i) . Define ring method. Write Euler-Poisson equation. Prove that the shortest distance between two points in a plane is a straight line. 6. x1 ∫[y + ( y ' ) 2 − 2 y e x ] dx 2 (b) Find the extremals of x0 5. (a) Show that the curve which extremize given that (b) Using Ritz method find the approximate solution of y” . 2 (b) Find the extremals of 0 10. Find the shape of the curve. . so that the surface area generated is a minimum. 2 2 (b) Find the extremals of 0 .e.y(0) = 1 and y’(1) + y(1) = . (a) Determine the extremals of the functional that satisfies the boundary condition x1 ∫[y + ( y ' ) 2 + 2 y e x ] dx 2 (b) Find the extremals of x0 π 2 ∫[y + ( y ' ) 2 − 2 y sin x] dx y(0) = y(π/2) = 0. z(0)=0. (a) A curve c joining the points ( ) and ( ) is revolved about the x-axis. y(π/2) = 1 is stationary for x = . 6. π 2 ∫[y ' − y 2 + 2 xy ] dx with y (0) = 0 y ( π2 ) = 0 be 2 (b) On what curve the functional 0 extremised. given y’(0) . (a) Solve the boundary value problem by Rayleigh Ritz method.sin t y = sin t. . 9.y + 4x ex = 0 . 4. 7. 2 8. π 2 ∫[y − ( y ' ) 2 − 2 y sin x] dx y(0) = y(π/2) = 0. π ∫[y ' −y + 4 y cos x] dx y(0) = y(π) = 0. (a) Find the extremals of 0 π 2 2 ⎛ dx ⎞ ⎛ dy ⎞ 2 (b) Show that the functional ∫0 [2 xy + ⎜⎝ dt ⎟⎠ + ⎜⎝ dt ⎟⎠ ] dt such that x(0) = 0 x(π/2) = -1 y(0) = 0 . (a) 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 . (a) Find the extremals of the functional = . UNIT –III ONE DIMENSIONAL RANDOM VARIABLES PART-A 1. 2.10 and 45 respectively.Find the variance of X. 19. If 3% of the electric bulb s manufactured by a company are defective. f(x) =2x . find the probabilities that will take a value between 1 and 3. The random variable X has a Binomial distribution with n =20 .If he sells his product in boxes of 100 and guarantees hat not more than 10 will be defective . 2. Find the mean of the Poisson distribution which is equivalent to B(300. 11. then what is the standard deviation? 4. 9. 0. Obtain the moment generating function of Geometric distribution. How many times must he fire so that the probability of his hitting the target at least once is more than 2/3. 10. (a) A manufacturer of certain product knows that 5 % of his product is defective .4 Find P (X = 3). If a RV has the pdf . find the mean variance of RV X.2).what is the approximate probability that a box will fail to meet the guaranteed quality? (b) The probability density function of a random variable X is given by .4.05 that a certain kind of measuring device will show excessive drift . 6. A RV X has the p. Show that the mean is 5 and variance is 3. If the probability is 0.2. 8.f. find the probability that he waits 1) less than 5 minutes for a bus and 2) atleast 12 minutes for a bus . If the mean of a Poisson variate is 2. The first four moments of a distribution about 4 are 1 . 12. If X is a continuous RV with p.d.p =0. The mean variances of binomial distribution are 4 and 3 respectively. The probability that a man shooting a target is ¼. PART – B 1. State the memoryless property of an exponential distribution.then find the value of K 17.0 <x< 1 .f.If a passenger arrives at a random time that is uniformly distributed between 7 and 7. (a) Buses arrive at a specific stop at 15 minutes intervals starting at 7a. 14.30 am . (b) Derive the MGF of Poisson distribution and hence deduce its mean and variance. Find P(X=0).3 and 4 such that 2P(X = 1)=3P( X =2 ) =P ( X =3 ) = 5 P (X =4 ) find the probability distribution. If X and Y are independent RVs with variances 2 and 3 . Find the moment generating function of Exponential distribution.m. . If a RV has the probability density . Write two characteristics of the Normal Distribution.Find the variance of 3X + 4Y.Find the CDF of X 20.d. then find the pdf of the RV Y = X3 3. 5. what is the probability that the sixth measuring device tested will be the first to show excessive drift? 18. 7. 16. If X is a Poisson variate such that P ( X =2) = 9P(X =4 ) + 90 P(X = 6) . Find the value of X . 15. find the probability that in a sample of 100 bulbs exactly 5 bulbs are defective. 13. If the RV X takes the values 1 . 6. (i)Find the value of k (ii) P (0.What is the probability that at least 5 components are to be examined in order to get 3 defectives? (b) The number of monthly breakdown of a computer is a random variable having a poisson distribution with mean equal to 1. (b) Let X be a continuous random variable with p. 3. 5. (a) If X is a discrete random variable with probability function p(x) = .000 gallons. how many would you expect to have i) at least 1 boy ii) 2 boys iii) 1 or 2 girls iv) no girls.000 gallons .Find the mean and variance of the distribution.2) (iii) What is P[0. Find the probability that this computer will function for a month (i)Without breakdown(ii) With only one breakdown and(iii)With atleast one breakdown. .f of X. On average one in 500 is defective. (a) Let the random variable X has the p. What is the probability that the stock is insufficient on a particular day? (b) Find the MGF .f.000 sets? 8. what number of lamps might be expected to fail in the first 700 burning hours? 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) (b) Find the MGF .d.2<x<1. (b) Find the MGF . 31 % of the items are under 45. 7. 4. mean and variance of the Gamma distribution.. (a) The slum clearance authorities in a city installed 2000 electric lamps in a newly constructed township. (b) Find the MGF .mean and variance.d. (a) In a company . mean and variance of the Uniform distribution. (a) If X is Uniformly distributed over (0 . If the lamps have an average life of 1000 hours with a std deviation of 200 hrs .8.10) . (a) A discrete RV X has the probability function given below X : 0 1 2 3 4 5 6 7 2 2 P(x) : 0 a 2a 2a 3a a 2 a 7a2 + a Find (i) Value of a (ii) p (X <6) .in excess of 20.5 < x < 1. Find the mean and variance. mean and variance of the Exponential distribution 9.find the probability (i) X < 2 (ii)X > 8 (iii) 3 <X <9? (b) A wireless set is manufactured with 25 soldered joints. P ( 0 < X < 4 ) (iii) Distribution function.5/x ≥1 (iv) Find the distribution function of f(x).f (i)Find a (ii) Compute P(X (iii) Find the c. (a) The daily consumption of milk in a city .5% defective components are produced . x =1.2 . (b) Out of 2000 families with 4 children each . mean and variance of the Binomial distribution. 10. 8% are over 64.in approximately distributed as a Gamma variate with the parameters k = 2 and The city has a daily stock of 30. How many sets can be expected to be free from defective joints in a consignment of 10. (a) In a normal distribution. (K constant ) then find the moment generating function ..d. P ( X ≥ 6 ) . 10. 18. What is a travelling sales man problem? 14.P. Define Assignment Problem. List any two basic differences between a transportation and assignment problem 4. Explain optimal solution in L. UNIT – IV LINEAR PROGRAMMING PART –A 1. Write down the mathematical formulation of L. Define transshipment problem.P by using graphical method . 16. What do you mean by degeneracy? 5.P. Explain Row. Find the initial solution to the following TP using Vogel’s approximation method D1 D2 D3 D4 Supply F1 3 3 4 1 100 F2 4 2 4 2 125 F3 1 5 3 2 75 Demand 120 80 75 25 300 . What is the difference between feasible solution and basic feasible solution? 9. 15. 6. Obtain an initial basic feasible solution to the following transportation problem by using Matrix minima method D1 D2 D3 D4 capacity O1 1 2 3 4 6 O2 4 3 2 0 8 O3 0 2 2 1 10 Demand 4 6 8 6 8. Subject to 19.P. Solve by graphical method . Subject to 7. Differentiate between balanced and unbalanced cases in Assignment model 3.P. Column. minima methods to find initial solution of transportation problem . Solve the following L.P. 17. What is an assignment problem? Give two applications. When will you say a transportation problem is said to be unbalanced? 12. 13. 11. What is degeneracy in a transportation model 2. Write short note on North west corner rule. Enumerate the methods to find the initial basic feasible solution for transportation problem. Solve the assignment problem P Q R S A 5 7 11 6 B 8 5 9 6 C 4 7 10 7 D 10 4 8 3 20. B .B 1.z to destinations A . PART . (b) Find the initial feasible solution for the following transportation problem D1 D2 D3 D4 Supply 11 13 17 14 250 16 18 14 10 300 21 24 13 10 400 Demand 200 225 275 250 950 5. (b) A military equipment is to be transportated from origins x. . Maximize . Find the least cost route To City A B C D E A ∞ 4 10 14 2 B 12 ∞ 6 10 4 C 16 14 ∞ 8 14 D 24 8 12 ∞ 10 E 2 6 4 16 ∞ 4. Maximize .C. Maximize . (b) Solve the following transportation problem: A B C αi F1 10 8 8 8 F2 10 7 10 7 F3 11 9 7 9 F4 12 14 10 4 bi 10 10 8 3. (a) Solve by Simplex method. Subject to . (a)Solve by Graphical method .P. (a) Solve by Simplex method. (b) A travelling sales man has to visit 5 cities .y. D. visit each city once and then returns to his starting point. (a) Solve the L. Subject to . Subject to .He wishes to start from a particular city. Subject to .Maximize .P by Simplex method Subject to (b) Solve the assignment problem P Q R S A 18 26 17 11 B 13 28 14 26 C 38 19 18 15 D 19 26 24 10 2. Cost of going from one city to another is shown below. (a) Solve by Simplex method. (a) Solve by two phase Simplex method Minimize Z = x1 .x4 . Subject to . x1 + 2 x2 + x3 + x4 = 10 and x1.x3 ≥ 0 (b) Solve the transportation problem Supply ⎡ 15 51 42 33 23 ⎤ ⎢ 80 42 26 81 44 ⎥⎥ ⎢ ⎢ 90 40 66 60 33 ⎥ ⎢ ⎥ ⎣ Demand 23 31 16 30 100⎦ 10.2x2 . x2 . 8x1 . x2 .3x3. 2x1 + x2 + 5x3 = 20 . 6x1 + 5x2 + 10x3 ≤ 76 .x2 . (a) Solve by Simplex method Maximize Z = x1 + x2 + 3x3.x3 . 2x1 + x2 + 2x3 ≤ 2 and x1.4x2 + 3x3 . x3 ≥ 0 .3x2 + 6x3 ≤ 50 and x1. x2 ≥ 0 7. Find transportation plan so that the time required for shipment is the minimum Destinations A B C D Supply Origin X 10 22 0 22 8 Y 15 20 12 8 13 Z 21 12 10 15 11 Demand 5 11 8 8 32 6. (a) Solve by Graphical method Maximize . Subject to 2x1 + x2 ≤ 1 . (a) Using Simplex method Maximize Z = x1 + 2 x2 + 3x3 .x3 ≥ 0 (b) Solve the assignment problem Job ⎡ 13 8 16 18 19 ⎤ ⎢ 9 15 24 9 12 ⎥ ⎢ ⎥ Machine ⎢ 12 9 4 4 4 ⎥ ⎢ ⎥ ⎢ 6 12 10 8 13 ⎥ ⎢⎣ 15 17 18 12 20⎥⎦ 9. the demand at the destinations and time of shipment is sown in the table. x1 + 4x2 ≥ 6 and x1. Subject to 3x1 + 2x2 + x3 ≤ 2 .6x3 = 20 . 2x1 + 3x2 + 4x3 = 1 and x1. Subject to 2x1 + x2 . (a) Solve by two phase Simplex method Maximize Z = 5x1 . Subject to -2x1 + x2 +3x3 = 2 . x2. Subject to x1 + 2 x2 + 3x3 = 15 .x4 ≥ 0 (b) Solve the assignment problem for optimal job assignment Job ⎡ 10 3 3 2 8⎤ ⎢ 9 7 8 2 7 ⎥⎥ Machine ⎢ ⎢ 7 5 6 2 4⎥ ⎢ ⎥ ⎢ 3 5 8 2 4⎥ ⎢⎣ 9 10 9 6 10 ⎥⎦ 8. The supply at the origins. (b) Use two phase Simplex method solve Maximize Z = 5x1 + 3x2 . f(t) = 2cos 5πt + sin6πt using time domain analysis. 10. with example. 18. Find the Fourier constants bn for x sinx in ( −π . 12. 9. in with periodicity . Find the root mean square value of the function f(x) = x in (0. 0 < x < π. Write short note on Eigen value problem and orthogonal functions. 17.x) 1 ≤ x ≤ 2 . Find the value of in the Fourier series expansion of 11. find the assignment of machines to the job that will result in maximum profit. If the periodic function f(t) ={ where f( t + 2π) = f(t) is expanded as a Fourier series . 16. Job ⎡ 62 78 50 111 82⎤ ⎢ 71 84 61 73 59⎥ ⎢ ⎥ Machine ⎢ 87 92 111 71 81⎥ ⎢ ⎥ ⎢ 48 64 87 77 80⎥ ⎢⎣ ⎥⎦ UNIT – V FOURIER SERIES PART –A 1. find the value of an. 20. (b) Find the Fourier series of the function = . -1 < t < 1. 7. 13. 6. Put the following DE in self – ad joint form x2 y” + 3xy’ + λy = 0. Define energy signals and power signals? 3. Distinguish Periodic and Non Periodic functions. State Parseval’s theorem. Define self ad joint operator. 4. State convergence of the series. (a) Find the fourier series of the sawtooth function f(t) = t. Define Periodic Sturm. 19. If in then deduce the value of 15. ⎧πx 0 ≤ x ≤1 (b) Obtain the Fourier series for the function f(x) = ⎨ ⎩π (2 . (b) Solve the assignment problem .Liouville System. Find a Fourier sine series for the function f(x) = 1 . (a) Find the Fourier series of the periodic Ramp function . Define a periodic function as power signals. 2. 5. 2.Liouville Systems. Write note on cosine and sine series. 8. Calculate average power of period T =2. Write note on Singular Sturm. Define generalized Fourier series? 14. Write the power signals Exponential Fourier Series. l). π ). f(t+2π) = f(t). f(t+2) = f(t). PART – B 1. y (-π) = y( π) . (a) Calculate the averge power of the periodic signal.. π )...... 0 < x < 1.3 3. (b) Find the Fourier series for f(x ) = x2 in (...5 5.. y(0) = 0. 0 < x < 1. (b) Obtain the Fourier series for f(x) = 1 + x + x2 in (.. 5.. (b) Find the complex form of Fourier series of f(x) = ex in -π < x < π. y(1) + y’(1) = 0. 0 < x < p. Hence find 1 1 1 1 1 1 1 i) 2 + 2 + 2 + .y(p) = 0.∞ 1 2 3 1 2 3 44 7.. (a) Find the eigen values and eigen functions of y’’ + λy = 0.. y(0) = 0. (a) Find the eigen values and eigen functions of y’’ + λy = 0. (b) Find the Fourier series of f(x) = x sinx in -π < x < π... 4). ∞ π4 ∑ (2n − 1) 1 (b) Obtain the cosine series for f(x) = x in 0 < x < π and deduce that 4 = n =1 96 8. Deduce that ∞ π2 ∑ (2n − 1) 1 2 = 1 8 4. 1 1 1 π −2 Hence deduce the sum of the series − + − .1)2 .Hence deduce 6... 0 < t < 1. -π < x < π. 0 < x < 1. 0 < x < 3. y(0) = 0. (a) Find a fourier series representation of f(t) = t2. 0 < x < 3. (b) Find the half range sine series of f(x) = 4x. (a) Find the eigen values and eigen functions of y’’ + λy = 0.π . y(0) = 0. 3. y(3) = 0..π . ***** ALL THE BEST ***** .∞ = 1. ii) iii) 4 + 4 + 4 + . y(3) = 0.. (a) Find the generalized Fourier Series expansion of the function f(x) = 1 . in terms of the eign functions of y’’ + y’ + λy = 0... (a) Find an expression for the Fourier coefficients associated with the generalised fourier series arising from the eigen functions of y’’ + y’ + λy = 0.. y’(-π) =y’(π).7 4 10.. y(1) + y’(1) = 0. 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 (b) Obtain the half range cosine series for f(x)= (x . 9.. (a) Find the eigen values and eigen functions of y’’ + y’ + λy = 0. period T =2 f(t) = 2 cos 6πt + sin 5πt Using (i) time domain analysis and (ii) Frequency domain analysis. (b) State and prove the Parseval’s theorem on Fourier coefficients.. π ). y(0) = 0.x2 in the interval (0.
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