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# Cochin University of Science and Techology (CUST) 2007 B.Tech Civil Engineering Engineering Mathematics II - Question Paper

Saturday, 25 May 2013 11:50Web

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

Time: 3 Hours

Maximum Marks: 100

PART A

(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 < x1.

( . [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

U+4

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

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 ,.

l(J + fl)(J + *)J

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*

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