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. 5. 6. 7.
Explain least square solution. Find the least square solution to the system Write down the stable formula for generalized inverse. + Find A for A = ⎡ 2 − 2 ⎤ ⎢1 ⎢ ⎢⎣ 2
⎥ ⎥ 2 ⎥⎦
1
8. Define singular value matrix. 9. State Singular value decomposition theorem. + 10. If A is a non singular matrix, then what is A ? Find the norms of 12. 13. 14. 15.
,
.Also verify that x and y are orthogonal .Find < x ,y >
Define orthogonal and orthonormal vectors. Write short note on Gram Schmidt Orthonormalisation process. What is the advantage in matrix factorization methods? 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
⎥ ⎥ 0 ⎦⎥
1
20. Check whether the given matrix is positive definite or not PART –B 1. (a) Determine the Cholesky decomposition of the matrix (b) Obtain the singular singular value decomposition decomposition of A= ⎛ ⎜
2
⎜−1 ⎜ ⎝ 4
2.
− 1 ⎞ ⎟ 1 ⎟ − 3 ⎠⎟
2i 2 ⎞ ⎛ 4 ⎜ ⎟ (a) Find the Cholesky decomposition of the matrix ⎜ − 2i 10 1 − i ⎟ ⎜ ⎟ ⎝ 2 1 + i 9 ⎠ ⎡0
(b) Find the QR factorization of A = ⎢ 1 ⎢ ⎢1 ⎢ ⎣1
1
1
1⎤
0
1
1
1
0
1
1
⎥ ⎥ 1⎥ ⎥ 0⎦
3. (a) Solve the following system of equations in the least square sense
⎡9
(b) Determine the Cholesky decomposition of the matrix ⎢ − 3 ⎢ ⎢0 ⎢ ⎣− 3
−3 6 6 0
− 3⎤ ⎥ 3 0 ⎥ 3 0 ⎥ ⎥ −3 6 ⎦ 0
4. (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) 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 +3 x3 = 2 ⎛ 1
2 ⎞
⎜1 ⎜1 ⎝
1⎟ 3 ⎠⎟
6. (a) Obtain the singular value decomposition of A = ⎜
⎟
⎡16 ⎢ (b) Determine the Cholesky decomposition of the matrix ⎢− 3 ⎢5 ⎢ ⎣− 8
−3
− 8⎤ ⎥ 16 − 5 − 8 ⎥ − 5 24 0 ⎥ ⎥ − 8 0 21⎦ 5
6. (a)Find the QR decomposition of 1 (b) Obtain the singular value decomposition of ⎛ ⎜
⎜1 ⎝
1 1
3 ⎞
⎟
3 ⎠⎟
7. (a) Construct QR decomposition for the matrix
⎡3 ⎢ (b) Find the canonical basis for A = ⎢ 0 ⎢0 ⎢ ⎣0
2
0
3
0
0
3
0
0
1 ⎤
⎥ ⎥ − 1⎥ ⎥ 3 ⎦ 0
− 1 ⎞ ⎟ ⎜1 0 0 ⎟ ⎜1 1 1 ⎟ ⎝ ⎠
1 8. (a) Find the QR factorization of A = ⎛ ⎜
1
(b) ) Solve the following system of equations in the least square sense x1 + x2 + x3 =1 ; x1 + x2 + x3 = 2 ; x1 + x2 + x3 = 3. 10. (a) Obtain the singular value decomposition of A = ⎛ ⎜ 1
⎜1 ⎜2 ⎝
1 (b) Find the QR factorization of A = ⎛ ⎜
⎜2
3 3 ⎞
⎟
3 0⎟ ⎜ 2 0 3⎟ ⎝ ⎠
1 ⎞
⎟
1⎟ 2 ⎠⎟
1. 2. 3. 4. 5. 6. 7. 8. 9.
UNIT – II CALCULUS OF VARIATION PART -A Define functional and extremal. Write Euler’s equation for functional. Define isoperimetric problems. Define several dependent variables. Write a formula for functional involving higher order derivatives. Define ring method. State Brachistochrone problem. Write other forms of Euler’s equation. Write the ostrogradsky equation for the functional
10. Write Euler-Poisson equation. 11. Define moving boundaries. 12. Define Geodesic. x1
13. State the necessary condition for the extremum of the functional I =
∫ F ( x, y, y' ) dx .
x0
14. Write short note on Rayleigh - Ritz method.
⎛ y '2 ⎞ Find the extremals of the functional ∫ ⎜⎜ 3 ⎟⎟ dx . x ⎠ x ⎝ x1
15.
0
16. Prove that the shortest distance between two points in a plane is a straight line. x1
17. Solve the Euler equation for
∫ ( x + y ' ) y ' dx
x0
18. Show that geodesics on a plane are straight lines. x1
∫
19. Find the curve on which the functional straight line [( y ' )
+ 12 xy] dx with y(0) = 0 and
2
x0
y(1) = 1can be extremised. x1
20. Solve the Euler equation for
∫ ( 1 + x
2
y ' ) y ' dx
x0
PART –B 1. (a) Find the extremals of (i) (b) Solve the boundary value problem y” + y +x = 0 ( 0 Ritz method. 2.
. x
1 ) y(0) = y(1) = 0 by Rayleigh
(a) Solve the extremals
(b) Prove that the sphere is the solid figure of a revolution which for a given surface has maximum volume. 3. (a)Show that the straight line is the sharpest distance between two points in a plane. (b) Find the extremals of
4. (a) 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. x1
(b) Find the extremals of
∫ [ y
2
+ ( y ' ) 2 − 2 y e x ] dx
x0
5. (a) Find the extremals of the functional = z(0)=0,
,
,
. π
2
(b) On what curve the functional
∫ [ y '
2
− y 2 + 2 xy ] dx with y (0) = 0 y ( 2 ) = 0 π
be
0
extremised. 6. (a) Show that the curve which extremize
given that
(b) Using Ritz method find the approximate solution of y” - y + 4x ex = 0 , given y’(0) - y(0) = 1 and y’(1) + y(1) = - e. 7. (a) Determine the extremals of the functional
that
satisfies the boundary condition x1
(b) Find the extremals of
∫ [ y
2
+ ( y ' ) 2 + 2 y e x ] dx
2
+ ( y ' ) 2 − 2 y sin x] dx y(0) = y(π/2) = 0.
x0 π
2
8. (a) Find the extremals of
∫ [ y 0
π
2
(b) Show that the functional
2
⎛ dx ⎞ ⎛ dy ⎞ ∫0 [2 xy + ⎜⎝ dt ⎠⎟ + ⎜⎝ dt ⎠⎟ ] dt such that x(0) = 0 x(π/2) = -1 2
y(0) = 0 , y(π/2) = 1 is stationary for x = - sin t y = sin t. 9. (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
.
π
2
(b) Find the extremals of
∫ [ y
2
− ( y ' ) 2 − 2 y sin x ] dx y(0) = y(π/2) = 0.
0
10. (a) Solve the boundary value problem by Rayleigh Ritz method. π
(b) Find the extremals of
∫ [ y ' 0
2
− y
2
+ 4 y cos x] dx y(0) = y(π) = 0.
UNIT –III ONE DIMENSIONAL RANDOM VARIABLES PART-A 1.If X is a Poisson variate such that P ( X =2) = 9P(X =4 ) + 90 P(X = 6) .Find the variance of X. 3 2. If X is a continuous RV with p.d.f. f(x) =2x ,0
, find the mean variance of RV X.
15. 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 3. 16.
Find the value of X ,then find the value of K
17. 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? 18. If a RV has the probability density
, find the probabilities that
will take a value between 1 and 3. 19. A RV X has the p.d.f.
.Find the CDF of X
20. Write two characteristics of the Normal Distribution. PART – B 1. (a) 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 , f ind the probability that he waits 1) less than 5 minutes for a bus and 2) atleast 12 minutes for a bus . (b) Derive the MGF of Poisson distribution and hence deduce its mean and variance. 2. (a) 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? (b) The probability density function of a random variable X is given by
(i)Find the value of k (ii) P (0.2
, x =1,2 ... (K constant )
then find the moment generating function ,mean and variance. (b) Find the MGF , mean and variance of the Gamma distribution. 4. (a) 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? (b) The number of monthly breakdown of a computer is a random variable having a poisson distribution with mean equal to 1.8. 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. 5. (a) Let the random variable X has the p.d.f. Find the mean and variance. (b) Out of 2000 families with 4 children each , how many would you expect to have i) at least 1 boy ii) 2 boys iii) 1 or 2 girls iv) no girls. 6. (a) In a normal distribution, 31 % of the items are under 45, 8% are over 64.Find the mean and variance of the distribution. (b) Let X be a continuous random variable with p.d.f (i)Find a (ii) Compute P(X
(iii) Find the c.d.f of X.
7. (a) If X is Uniformly distributed over (0 ,10) ,find the probability (i) X < 2 (ii)X > 8 (iii) 3
The city has a
daily stock of 30,000 gallons. What is the probability that the stock is insufficient on a particular day? (b) Find the MGF , mean and variance of the Binomial distribution.
1. 2. 3. 4. 5. 6.
UNIT – IV LINEAR PROGRAMMING PART –A What is degeneracy in a transportation model Differentiate between balanced and unbalanced cases in Assignment model List any two basic differences between a transportation and assignment problem What do you mean by degeneracy? Explain optimal solution in L.P.P. 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 u sing 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. 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. Write short note on North west corner rule. 13. What is a travelling sales man problem? 14. Enumerate the methods to find the initial basic feasible solution for transportation problem. 15. Define transshipment problem. 16. Define Assignment Problem. 17. Explain Row, Column, minima methods to find initial solution of transportation problem . 18. Solve by graphical method , Subject to
19. 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. 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
PART - B 1. (a) Solve the L.P.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. (a) Solve by Simplex method.Maximize
, Subject to .
(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. Maximize
, Subject to .
(b) 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 A
B
C
D
E
4
10
14
2
6
10
4
8
14
B
12
C
16
14
D
24
8
12
E
2
6
4
10 16
4. (a) Solve by Simplex method. Maximize
, Subject to .
(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. (a)Solve by Graphical method , Maximize , Subject to . (b) A military equipment is to be transportated from origins x,y,z to destinations A ,B , C, D.
The supply at the origins, the demand at the destinations and time of shipment is sown in the table. 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 . (b) Use two phase Simplex method solve Maximize Z = 5x1 + 3x2 , Subject to 2x1 + x2 1 ; x1 + 4x2 ≥ 6 and x1, x2 ≥ 0 7. (a) Using Simplex method Maximize Z = x1 + 2 x2 + 3x3 - x4 , Subject to x1 + 2 x2 + 3x3 = 15 ; 2x1 + x2 + 5x3 = 20 ; x1 + 2 x2 + x3 + x4 = 10 and x1,x2 ,x3 ,x4 ≥ 0 (b) Solve the assignment problem for optimal job assignment Job ⎡ 10 3 3 2 8 ⎤ Machine
⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
9 7 3 9
⎥ ⎥ 5 6 2 4⎥ ⎥ 5 8 2 4⎥ 10 9 6 10 ⎥⎦ 7
8
2
7
8. (a) Solve by two phase Simplex method Minimize Z = x1 - 2x2 - 3x3, Subject to -2x1 + x2 +3x3 = 2 ; 2x1 + 3x2 + 4x3 = 1 and x1, x2 ,x3 ≥ 0 (b) Solve the assignment problem Job
Machine
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
13
8
9
15
16 18 19 ⎤
⎥ ⎥ 12 9 4 4 4⎥ ⎥ 6 12 10 8 13 ⎥ 15 17 18 12 20⎥⎦ 24
9
12
9. (a) Solve by Simplex method Maximize Z = x1 + x2 + 3x3, Subject to 3x1 + 2x2 + x3 2 ; 2x1 + x2 + 2x3 2 and x1, x2 ,x3 ≥ 0 (b) Solve the transportation problem Supply
15 ⎡ ⎢ 80 ⎢ ⎢ 90 ⎢ ⎣ Demand 23
51 42 33
23 ⎤
42
26
81
44 ⎥
40
66
60
33 ⎥
31 16
30
100⎦
⎥ ⎥
10. (a) Solve by two phase Simplex method Maximize Z = 5x1 - 4x2 + 3x3 , Subject to 2x1 + x2 - 6x3 = 20 ; 6x1 + 5x2 + 10x3
76 ; 8x1 - 3x2 + 6x3
50 and x1, x2, x3 ≥ 0
(b) Solve the assignment problem , find the assignment of machines to the job that will result in maximum profit. Job
Machine
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
62
78
50
71 84
61
111 82⎤
⎥ ⎥ 87 92 111 71 81⎥ ⎥ 48 64 87 77 80⎥ ⎥⎦ 73
59
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 an. 2. Define energy signals and power signals? 3. Find a Fourier sine series for the function f(x) = 1 , 0 < x < π. 4. Define a periodic function as power signals. 5. Calculate average power of period T =2, f(t) = 2cos 5 t + sin6 t using time domain analysis. 6. State Parseval’s theorem. 7. Find the Fourier constants b n for x sinx in ( −π , π ). 8. State convergence of the series. 9. Write note on Singular Sturm- Liouville System. 10. Find the value of
in the Fourier series expansion of
11. Define self ad joint operator. 12. Write the power signals Exponential Fourier Series. 13. Define generalized Fourier series? 14. If
in
then deduce the value of 2
15. 16. 17. 18. 19. 20.
Put the following DE in self – ad joint form x y” + 3xy’ + λ y = 0. Distinguish Periodic and Non Periodic functions, with example. Write short note on Eigen value problem and orthogonal functions. Define Periodic Sturm- Liouville Systems. Write note on cosine and sine series. Find the root mean square value of the function f(x) = x in (0, l ). PART – B 1. (a) Find the Fourier series of the periodic Ramp function ,
f(t+2π) = f(t).
(b) Find the Fourier series of the function
=
, in
with periodicity
2. (a) Find the fourier series of the sawtooth function f(t) = t, -1 < t < 1, f(t+2) = f(t).
⎧π x
(b) Obtain the Fourier series for the function f(x) = ⎨
⎩π (2 - x)
0 ≤ x ≤1 1≤ x ≤ 2
.
2
3. (a) Find a fourier series representation of f(t) = t , 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 2
(b) Obtain the half range cosine series for f(x)= (x - 1) ∞
∑ (2n − 1) 1
2
=
1
π
. Deduce that
2
8
4. (a) 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. 2 (b) Obtain the Fourier series for f(x) = 1 + x + x in (- π , π ). ’’
5. (a) Find the eigen values and eigen functions of y + λ y = 0, 0 < x < p, y(0) = 0,y(p) = 0. 2 (b) Find the half range sine series of f(x) = 4x- x in the interval (0, 4).Hence deduce
6. (a) Find an expression for the Fourier coefficients associated with the generalised fourier series ’’ arising from the eigen functions of y + y’ + λ y = 0, 0 < x < 3, y(0) = 0, y(3) = 0. 2 (b) Find the Fourier series for f(x ) = x in (- π , π ). Hence find i)
1 12
+
1
+
22
1 32
+ .......... ii)
1
iii)
14
+
1 24
1
+ 4 + 3
1 44
........∞
’’
7. (a) Find the eigen values and eigen functions of y + λ y = 0, - < x < , y (- ) = y( ) , y’(- ) =y’( ). ∞
(b) Obtain the cosine series for f(x) = x in 0 < x < π and deduce that
∑ (2n − 1) n =1
1
4
=
π
4
96
8. (a) Find the generalized Fourier Series expansion of the function f(x) = 1 , 0 < x < 1, in terms of ’’ the eign functions of y + y’ + λ y = 0, 0 < x < 1, y(0) = 0, y(1) + y’(1) = 0. (b) State and prove the Parseval’s theorem on Fourier coefficients. ’’ 9. (a) Find the eigen values and eigen functions of y + λ y = 0, 0 < x < 1, y(0) = 0, y(1) + y’(1) = 0. (b) Find the Fourier series of f(x) = x sinx in -π < x < π. Hence deduce the sum of the series
1 1.3
−
1 3.5
+
1 5.7
− ..........∞ =
π
−2 4
’’
10. (a) Find the eigen values and eigen functions of y + y’ + λ y = 0, 0 < x < 3, y(0) = 0, y(3) = 0. x
(b) Find the complex form of Fourier series of f(x) = e in - < x < .
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