MODEL QP- CAT-I
School of Advanced Sciences
Department of Mathematics
CAT – I, AUG, 2017
Class Number Slot-A1+TA1
Course Code; MAT5005 Time : 90 minutes
Course Title: Advanced Mathematical Methods Max. Marks : 100
ANSWER ALL QUESTIONS (5X 10=50)
5 −2 1
1. Find all the Eigen values of 𝐴 = [7 1 −5] by Rutishauser method
3 7 4
2. Use Gerschgorin Circles theorem to locate all the Eigen values and hence by
5 −2 4
Power method to find the least Eigen values of (−2 1 1)
4 1 0
2 −1 −1
3. Tridiagolaize the matrix 𝐴 = [−1 2 −1] using House Holder Method,
−1 −1 2
Hence finding the strum sequence and so find the Eigen value of A.
1
1 √2
√2
3
4. Using Jacobi’s method find all the Eigen values of 𝐴 = 1 √2
1
1
[√2 1
√2 ]
7 −4 2
5. Deflate the matrix (9 −6 5) if one of the Eigen value is 4 and the
6 −6 7
corresponding eigen vector is (0 0.5 1 )𝑇 and find the reaming Eigen value
of A.
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MODEL CAT-II QP
SCHOOL OF ADVANCED SCIENCES
DEPARTMENT OF MATHEMATICS
Continuous Assessment Test – II- October 2017
(OPEN BOOK EXAMINATION)
Course Code: MAT5005 Advanced Mathematical Mathematics Slot : A1+TA1
Faculty G. Murugusundaramoorthy
Max. Marks : 50 Answer ALL the Questions Duration: 90 Minutes
______________________________________________________________________
1. a) Determine the shape an absolutely flexible, in extensible homogeneous and heavy
rope of given length L suspended at the points A and B. (8)
1
b) Find the extremal of the functional∫0 (1 + 𝑦 ′′2 )𝑑𝑥 subject to
𝑦(0) = 0; 𝑦(1) = 1, 𝑦 ′ (0) = 1 𝑎𝑛𝑑 𝑦′(1) = 1 (7)
2. An uncharged condensor of of capacity 1 farad is charged by applying an e.m.f
𝐸 (𝑡) = 2𝑡 through the leads of self inductance of 2 henry and neglible resistance. Find
the charge q by Galerkins method up to two approximation if 𝑞 (0) = 𝑞(1) = 0. (10)
3. The differential 2𝑦 ′′ + 5𝑦 ′ − 3𝑦 = 0 represents the damped harmonic oscillations of the
particle. At time 𝑡 = 0 the particle at a distance -4 unit from the origin and its speed away
from the origin 9 units. Find the displacement at any time t by converting in to system of
homogeneous differential equation . (10)
4. A body excutes forced oscillations without damping is , 𝑦 ′′ + 4𝑦 = cos 2 𝑥.If the particle
starts from rest from the origin initially, find the displacement of the particle by by
converting in to system of non-homogeneous differential equation. (15)
MODEL FAT QP
2 3 1
1. a. Find all the eigen values of the matrix (3 2 2) by Jacobis method. (10)
1 2 1
2 −1 −1
b. Find the srum sequence for the given matrix (−1 2 −1) (10)
−1 −1 2
23 36
2. a. Find the eigen value of a matrix ( ) by deflation one of the eigen value is 50 and
36 2
4
the eigen vector ( ). (5)
3
1 1 1
b. Find all the eigen values of (2 1 2) using Rutishauser method (Iterate till the
1 3 2
elements of lower triangular part are less than 0.05 in magnitude) (15)
3. a.Determine the shape of a solid of revolution moving in a flow of gas with least
temperature (10)
1 2
b. Find the extremals of the isoperimetric problems 𝐼(𝑣) = ∫0 (𝑦 ′ + 𝑥 2 )𝑑𝑥 given that
1
∫0 𝑦 2 𝑑𝑥 = 2 given that 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(1) = 0. (10)
4. a.The differential 2𝑦 ′′ + 5𝑦 ′ − 3𝑦 = 0 represents the damped harmonic oscillations of the
particle. At time 𝑡 = 0 the particle at a distance -4 unit from the origin and its speed away
from the origin 9 units. Find the displacement at any time t by converting in to system of
homogeneous differential equation . (10)
𝑑𝑥1 𝑑𝑥2
b. Solve = −5𝑥1 + 𝑥2 + 6𝑒 2𝑡 𝑎𝑛𝑑 = 4𝑥1 − 2𝑥2 − 𝑒 2𝑡 (10)
𝑑𝑡 𝑑𝑡
5. a. Reduce 𝑥𝑈𝑥𝑥 + (𝑥 + 𝑦)𝑈𝑥𝑦 + 𝑦𝑈𝑦𝑦 = 0 into a canonical form in the region where it is
hyperbolic. (15)
b. Solve by method of seperation of variables 𝑢𝑥𝑦 = 𝑒 𝑐𝑜𝑠𝑥 given
−𝑡
𝑢(0) = 0𝑎𝑛𝑑 𝑢𝑡 (𝑥 = 0) = 0. (5)
6. a. Classify the partial differential equation
(1 + 𝑥 2 )𝑢𝑥𝑥 + (1 + 𝑦 2 )𝑢𝑦𝑦 + 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 0 (5)
1
b. Solve the wave equation 𝑢𝑥𝑥 = 𝑐 2 𝑢𝑡𝑡 − cos 𝑤𝑡, 0 ≤ 𝑥 < ∞, 0 ≤ 𝑡 < ∞ subject to the
boundary conditions𝑢(0, 𝑡) = 0, 𝑢 is bounded as 𝑥 → ∞ and the initial conditions 𝑢(𝑥, 0) =
0; 𝑢𝑡 (𝑥, 0) = 0 using Laplace transform. (15)
7. a. An uncharged condensor of of capacity 1 farad is charged by applying an e.m.f 𝐸(𝑡) = 𝑡 2
through the leads of self inductance of 1 henry and neglible resistance. Find the charge q
by Galerkins method up to two approximation if 𝑞(0) = 𝑞(1) = 0.
(15)
b. Solve the Srum-Liouville 𝑦 + 𝜆𝑦 = 0; 𝑦(0) = 0, 𝑦(𝑙) = 0 𝑎𝑛𝑑 𝜆 > 0.
′′
(5)
