Cochin University of Science and Techology (CUST) 2007 B.Tech Civil Engineering Engineering Mathematics II - Question Paper
IT/CS/EC/CE/ME/SE/EB/EI/EE 301 ENGINEERING MATHEMATICS H
BTS(C)-ni- 07- 065(B)
B, Tech Degree III Semester Examination, November 2007
(2006 Admissions)
Time: 3 Hours
Maximum Marks: 100
PART A
(Answer ALL questions)
(All questions carry FIVE marks)
(8x5 = 40)
Define rank of a matrix. Find the values of / and m .such that the rank of the matrix
(a)
2 1-13 1-12 4 7 -1 / m
is 2.
Let T be a linear transformation from R3 into R defined by 7T( , x2, x3) = x*, x22, x2. Show that T is not a linear transformation.
(b)
(c)
(d)
(e)
(0
(g)
00
Obtain the half range sine series for ex in 0 < x < 1.
( . [1 for \x <1 rsinx ,
Find the Fourier transform of t (X1 = < . Hence evaluate - ax.
10 for x > 1 * x
s2 +s
Find the inverse Laplace transform of log
Find the Laplace transform of the saw toothed wave of period T, given /(0 = p</<7\
If u x2 + y2 + z2 and V -xi + yj + zk, show that div (wvj = 5m .
Find the work done when a force F = (x2 y2 + x) i (2xy + y) j moves a particle in the xy-plane from (0,0)to(l,l) along the parabola y2 =x.
PART B
(All questions carry FIFTEEN marks)
-2 2 -3
(4x15=60) and find the eigen vector corresponding
(a) Find the eigen values of A -
2 1 -6
w-l -2 0 to the largest eigen value.
Find ker(r) and ran(T} and their dimensions where T:R3 > R3 defined by
<b)
rx + y | ||
y |
* |
z |
<z) |
i H |
OR
1 1 1 0 2
2
0 1 1 1
and hence find the matrix
II. (a) Find the characteristic equation of the matrix A =
represented by A8 5 A* +7 A6 - 3 A5 + A4 5 A3 + 8 A2 2 A +1. (b) Test for consistency and solve the following
systemx - y + z = 1,2x + y - z = 2,5x - 2y + 2z = 5.
Find the Fourier sine transform of Hence show that
x sin mx n __m _
I _ ox =e ,m > U. i \ + x2 2
OR
, si \ . / ri \ fOfor ~2<x<0
Expand / I x) in Fourier series in the interval I 2,2) when / (JC1
W ' W [1 for 0<x<2
, fl Oxctt Express the function J I asa Fourier sine integral and hence
[0 X>7t
f(l-COSl)
evaluate I--- sm x X a a .
j X .
Find the Laplace transforms e~a) _
(i) --(ii) sin/M(/-;r.)
Solve by the method of Laplace transforms ym + 2 y* y' 2y = 0 given
(0) = /(0) = 0 and y(0) = 6.
t
Evaluate jt e~2t s'mt dt. o
OR
Define a unit impulse function and find its Laplace transform.
r-l f 1 1
Apply Convolution theorem to evaluate L \ , x ,.
Find the Laplace transform of the periodic function of period 2a defined by ( . J 1 for 0t<a
11 for a<t<2a
Prove that jj = 0.
Show that F.dr = 3n9 given that F =* zi + xj + yk and C being the arc of the
curve r = cos// + siiUj + tk from t = 0 to t = n.
If F = (lx2 ~3zi 2xy j - 4xk, then evaluate jJJ V.JFdv where v bounded
V
by the planes Jt = 0,_y = 0,z = 0 and 2x + 2y + z = 4.
OR
Find the constants a,b,c so that
F (x + 2y + az)i + [bx-3y-z) j + (4x + cy + 2z)k is irrotational.
x2 3v2
IfF = grad, show that = - + z2 + 2xy + Axz - yz.
2 2
Verify Stokes theorem for F = (x2 + y2 j / 2xy j taken round the rectangle bounded by the lines x = 0>(y = 0, = b. . 7*
Attachment: |
Earning: Approval pending. |