# 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 R^{3} into R defined by 7^{T}( , x_{2}, x_{3}) = x*, x_{2}^{2}, x^{2}. Show that T is not a linear transformation.

(b)

(c)

(d)

(e)

(0

(g)

00

Obtain the half range sine series for e^{x} in 0 < x < 1.

_{(} . [1 for \x <1 rsinx ,

Find the Fourier transform of t (X1 = < . Hence evaluate - ax.

10 for x > 1 * x

s^{2} +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 x^{2} + y^{2} + z^{2} and V -xi + yj + zk, show that div (wvj = 5m .

Find the work done when a force F = (x^{2} y^{2} + x) i (2xy + y) j moves a particle in the xy-plane from (0,0)to(l,l) along the parabola y^{2} =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:R^{3} > R^{3} defined by

<b)

| ||

y |
* |
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 A^{8} 5 A* +7 A^{6} - 3 A^{5} + A^{4} 5 A^{3} + 8 A^{2} 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 \ + x^{2} 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 y^{m} + 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 = 3n_{9} 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 = (lx^{2} ~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.

x^{2} 3v^{2}

IfF = grad, show that = - + z^{2} + 2xy + Axz - yz.

2 2

Verify Stokes theorem for F = (x^{2} + y^{2} j / 2xy j taken round the rectangle bounded by the lines x = 0_{>(}y = 0, = b. . 7*

Attachment: |

Earning: Approval pending. |