Cochin University of Science and Techology (CUST) 2007 B.Tech Electronics and Communications Engineering Engineering Mathematics2 - Question Paper

ques. paper is in attachment beneath..

BTS(C)-in- 07 - 065(B)

B. Tech Degree III Semester Examination, November 2007

mCS/EC/CE/ME/SE/EB/EI/EE 301 ENGINEERING MATHEMATICS U

(2006 Admissions)

Time: 3 Hoots    Maximum Marks: 100

PART A (Answer ALL questions)

(All questions carry FIVE marks)

(8x5 = 40)

I. (a) Define rank of a matrix. Find the values of I and m jsuch that the rank of the matrix

2 1-13"!

I -I 2 4 is 2.

4

7 -1 / m_m

Let T be a linear transformation from R* into R defined by 7*(x,,x₂,xj) = x_l²,x₂²,x₃².

Show that T is not a linear transformation.

Obtain the half range sine series for e* in 0 < x <1. _rt . fl for Jxi <1    _rsinx

Find the Fourier transform of/(x)=i _    . Hence evaluate I -ax.

⁹ |0 for x >1    * x

Find the inverse Laplace transform of log

Find the Laplace transform of the saw toothed wave of period T, given

f(t)-j,0<T.

If h = x^J +y* + z³ and V *=xi + yj + zk, show that div (vj = 5u.

Find the work done when a force F = {x* y² +x}i (2xy + y)j moves a particle in the xy-plane from (0,0)to(l,l) along the parabola y^l = x.

PART B

(All questions carry FIFTEEN marks)

-2 2 -3

2 1 -6

-2 0 to the largest eigen value.

Find ker{r)asd ran(r) and their dimensions where T: Rⁱ * R³ defined by

/ \ X		/ \ x + y
y	m	t
		s-y.

OR

11. (a) Find the characteristic equation of the matrix A =

represented by A* -5A⁷ + 7A*- 3A* + A⁴ - 5A³ + 8A² -2A + /. (b) Test for consistency and solve the following

system x - >+z 1,2x + .y - z 2,5x - 2>>+2r 5.

Obtain the Fourier series for /(*) = |sin x| In the interval ft < X < K.

Find the Fourier sine transform of Hence show that xsinmx , n ..

Jxsmmx , n __

*-r

VR

Expand /(x) in Fourier series in Ac interval f-2,2)wheo f (x) = J^r 2<x< ^{W    V} 1 J W [lfor 0<x<

. [l 0x<x

Express the ftmction y(x} =    as a Fourier sine integral and hence

' [0 x>x Xl-cos) . . .. evaluate I----- sin x A dX.

6 *

Find the Laplace transforms

(j) ----(ii) sin/u(/-?r.)

Solve by the method of Laplace transforms y^m + 2y* - y - 2y = 0 given

>(0) = (O) = 0 and y(0) = 6.

/

Evaluate jt e~² sin t dt,

o

OR

Define a unit impulse function and find its Laplace transform.

Apply Convolution theorem to evaluate Z."¹ i -    I.

l(*+a)(s + 6)J

Find the Laplace transform of the periodic function of period 2 a defined by , . f 1 for OS/<a 1-1 for a<l<2a

f

Prove that div

Show that FMr = 3tt, given that F = zi + xj + yk and C being the arc of the

e

curve r cos/i + sin/J + tk from /=0 to t = rr.

If F = {2x⁷ ~3zi~2xy j - 4xk, then evaluate JJJ V.Fdv where v bounded

by the planes x * 0,.y = 0,z = 0 and 2x + 2y + z = 4.

OR

Find the constants a,b_tc so that

F = (x + 2y+02)i+(bx-3y-z)j+(4x+cy + 2z)fc is irrotational.

   x⁷ 3 v*

IfF - grad show that ---+ z* + 2xy + 4xz-yz.

2 2

Verify Stakes theorem for F = X² + y²i-2xy j taken round the rectangle bounded by the lines x a,y 0,y~b.

Attachment: cochin-university-of-science-and-techology--cust--2007-b-tech-electronics-and-communications-engineering-37111-1.JPG cochin-university-of-science-and-techology--cust--2007-b-tech-electronics-and-communications-engineering-37111-2.JPG

Earning:   Approval pending.