Biju Patnaik University of Technology 2008 B.Tech (B Tech) , 2nd semester mathematics-II . - Question Paper
BPUT(B Tech) , second semester mathematics-II ques. paper.
Total number of printed pages - 7 B. Tech/B. Arch
BSCM 2102 / SCM 2002
Second Semester Examination - 2008 MATHEMATICS - II Full Marks 70
Time: 3 Hours
IWL Answer Question No. 1 which is compulsory
and any five from the rest.
Figures in the right hand margin indicate marks.
1. Answer the following questions : 2x10
(a) How many solutions does the following (inear system have and why :
x + 2y = 2
j 2x + 4y = 1
- P.T. O.
1
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(b) If a - 0 is an eigenvalue of the matrix A, then what one can say about \A\. (cj Find the eigenvalues of the matrix '1 01 .2 4 (d) If A is any 5 x 5 matrix, and a polynomial associated with the matrix A is defined1 by n \Xl~A\- X , then what is the deter- n=0  minant of the matrix A. {e)> Find the period of the following periodic functions f(x) = sin(6x) and g(x) = cos(3x). (f) Identify which of the functions are even, in the specified period ii |
(g) Change the order of the integration * x j Je-*yydyctx o o (h) Find the parameter u such that the vectors = (1. 1. a), u2 = (1,*0, a) and u = o (0, a, 0) are linearly dependent. (t) Find the dimension of the vector space V= {u = (vUa,D2)|u1+U2-D3 =0}. Q) If the set of vectors = (1,1,1), = (1,0,1), and = (0,1,0) are dependent, then find their dependency. (a) Solve the following linear system by Gauss elimination method: 5 |
and
12-x, 2<x<3
9{x)=
x + z = 1 2x + z = 0 x + y + z = t
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(b) Find the inverse of the following matrix by Gauss-Jordan elimination method: 5 |
(b) FindUhe matrix Psuch that P~'AP= D where D is a diagonal matrix containing eigenvalues of the matrix A along the diagonal |
( 1 11
1 1 1 1 1 1
(b) Find the directional derivative of the function o (x, y) = xy3 - Ay at the point (2, -1) in the direction of the vector v = (2,5). 5
7. (a) Find the volume of the sphere + y* + z*
= a2 using multiple integral, 5
(b) Evaluate the line integral j jfdx + xy dy
c
by using Greens theorem where c is the triangular curve consisting of the line segments from (0,0) to (1.0), from (1,0) to (0, 1) and from (0, 1) to (0,0). 5
8. (a) Show that F(x, yt z) = (yz3, 2xyz3t
3xy z2 ) is a conservative vector field, i tail
and evaluate J (ytfdx + 2xydy +
3 xfi? dz). 5
(b) Evaluate JJ F.dS where F{xf yt z) = (xy s
y2 + e*, sin (xy)) where S is the surface of the region bounded by the parabolic cylinder z - 1 - x2 and the planes z - 0, y-0 and y + z = 2. 5
IWL
BSCM2102/SCM20D2 7 -C
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Attachment: |
| Earning: Approval pending. |
