Gujarat University 2007 B.E Computer Engineering-Advance Mathematics - Question Paper
3/1264 Candidate's Seat No
GUJARAT UNIVERSITY B. E. Sem Til (C.E. (New) / I.T.) (New) Examination Advance Mathematics-11
Saturday, 5th January', 2008] [Time : 3 Hours
Max. Marks : 100
Instructions : (1) Attempt all questions.
(2) Answer to the two sections must be written in separate answer books.
(3) Assume suitable data if required.
(4) Figures to the right indicate full marks.
SECTION I
1 Attempt any three: 18
1
( a ) Express f(x) = (71 - x) in a fourier series in the interval 0<x<2rc.
( b ) Find the fourier series expansion for periodic function f(x), if f(x) = - 71 ; - rc<x<0
= x ; 0<x< 7T
CD A 2
rn 1 TT
state the value of the series at x = 0 and hence deduce that
n=1(2n-1>2 8
{ c ) Obtain the founer scries to represent the function f(x) = n2 x2, jt < x < 71.
till n1
Hence deduce that -z----r + r ~r.................= 7 .
I2 2 3 12
( d ) If f(x) = x; 0< x <2 then find half range cosine series.
( a ) Solve the following differential equations (any two):
(i) (D2 - 4D + 4)y - e2* + x3 + cos 2x
(ii) (D2 + 5D + 6)y = e 2x sin 2x
(iii) (D2 - 4t> + 3)y = sin 3x cos 2x
d2y _ 2
( b ) By using the method of variation of parameters solve : 5 V--
dx 1 + ex
OR
( b ) Solve the simultaneous equations .
( c ) The charge Q on die plate of a condenser of capacity C charged through a resistance 4 R by a steady voltage V satisfies the differentia! equation.
dQ Q
R 1 = V If Q 0 at t = 0. show that Q = CV [l~et,KC]. Find die current flowing into the plate.
3 ( a ) Solve in series the differential equation ; 6
d2y dy (1 - x3) 7-2x ~r + 2y = 0 dx2 d*
OR
d2y dy x ~~ + ~ + x2y = 0 dx dx
( b ) Attempt any two parts : 10
(i ) State cayley - Hamilton theorem and verify cayley-Hamiltou theorem for the matrix.
1 1 2
[: ; :] i_ 1 1 -
A
2 3
( ii ) Using cayley -Hamilton relation obtain the inverse of the matrix.
1 1
r: ; ;
_/i _A
-2 -4 -4
(iii) Reduce the quadratic form 6x- + 3V2 + 3z2 - 4xy - 2yz + 4 zx into canonical form.
SECTION II
4 ( a ) Form the partial differential equation from the following (any one) : 4
( i ) z = (x2 + a) (y2 + b ) ( ii ) F (x5 + y2 + z2, xyz) = 0
32z Bz
( b ) Solve = sin x sin y, given that = -2 sin y when x = 0 and z - 0 when y is an 4
odd multiple of %/2
( c ) Attempt any two : 8
dz 02.
< i ) x (/- yJ) -1- y (x" -7r) ~Z = z (v: - x:)
ex. <7y
( ii ) p - q = x2 + y2
(iii) z2 (p2x2 * q2) ~ 1
5 ( a ) Using the method of separation of variables solve : 4
du du -
4 + zr -3u and u = e~Sy when x = 0. dx dy
( b ) Attempt any three : 12
( i ) Prove that the function sinhz is analytic and find its derivative.
(ii ) Find the image of |z 3ij =3 under the mapping (iii ) Show that the function u =e2xy sin (x2 - y2) is hannonic.
(iv ) Construct the analytic ftinction f(z) of which the real part is e'cos y.
6 Attempt any three parts : 18
(i) Find bilinear transformation which maps the points z = 1, i, -1 on to the points cn~ 0,1, oo.
1+i
(ii) Evaluate J (xJ -iy)dz along the paths ( a ) y = x ( b ) y = x2.
0
rC e'2Z
(iii) Use cauchys where integral formula to evaluate 9 , , .3 dz, where C is the circle I z| =2.
c
r COSTTZ
(iv ) Evaluate J 2 * around a rectangle with vertices : c
( a ) 2 + i, -2 i ( b ) - i, 2 -i, 2 + i, i.
GUJARAT UNIVERSITY B.E. Semester IU ( CE (Old) / IT (Old))
Advance Mathematics 3X
Saturday 5th January, 2008] [Time: 3 Hours
[Total Marks: 100
Instructions: ()) All questions arc compulsory.
(2) Figure to the right indicate full marks.
(3) Attempt all questions from each section.
(4) Answer to the two sections must be written in separate answer sheet
1 Attempt any three 18 ( a ) Find the Fourier Series of the function, F(x) ~ x + x2;-x <x < n
l-lence deduce that,
1111 nl
l2 21 32 42 ..............~~{2
( b ) Find Fourier Series for F(x)= x, 0 < x < 7t
= 2 7T-x, x < x < 2n .
( c ) Find the Fourier series for F(t)=l -t2, -!</<!.
( d ) Expand F(x)= x as a half range sine series in 0< x <2.
( c) Obtain the Fourier series of sinli ax in - rf'x< ir.
2 (a) Solve any two of the following Differential Equations: (
(i) * (Dz-O.DH-L3)y=8e3l< Sin4x +2*
(ii) (D2 H3D+2) y=xSin2x
(iii) (D2 -i 6D +9)y~=e3x/x.:i +cos2x
l b ) Apply the method of variation of parameters lo solve Differential Equation 5
(D2+4)y=4 sec22x
OR
( b ) Solve the following Simultaneous Differential Equation 5
d2x . - o d2y ci,
- + 4a- + 5 v = r f + 5.v + 4y = I. + I
dr - dt
(c) Solve the Differential Equation 5
- x + 4y cos(logx))- xsm(logx) dx" dx
OR
( c ) The Di fferential Equation of a circuit is L- ~ 4- R---\- - - 0
. %
dt* dt C
Solve the equation with initial conditions that q=qp and dq/dt =-0 when t=(j and CRI<4L.
(a) Verify Cayley Hamilton theorem for A
2 1 1
0 I 0
1 1 2
, and hence find A'1
1 + 2/ J + 2/ 0
unitary matrix, where i is an identity nuilrix.
o
( i) Define a unitary matrix. If A -
then show that (1-A)('1+A)" is a
( ii ) Reduce the Quadratic form 6x|1 +3x:?2+3x}"-4xix2-2x2x;!+4x.|x.3 to canonical form and litid the corresponding linear transformation.
8 - 6 2
(iii) Dmgonalize the matrix
-6 7 "4 2-4 3
, by orthogonal transformation.
( a) Construct any one Partial Differential Equation 4
(i) z= (x2+a) (y'b) (ii) z= xy +F(x2+y2)
( b ) Solve any two Partial Differential Equation 8
(i) x2(y-z)pt y2(7.-x )q=/2(x-y)
( n ) (1 -x.)p+ (2-y)q-3-z
(iii) py +qx = pq
(c) Using the method of separation of variable, Solve = 2+m .given u(x,0)6e"Jx 4
dx dt
Attempt any three: 18
( a ) Define Analytic function, llF(z) is Analytic Function then prove that
f)-
( ) I
dx vv
( u ) lj F(z)= u(x, y) + i v(x, y) prove that u and v are orthogonal to each other.
( b ) Show that u(x, y)= e"2x:' sin-y2) is a harmonic function. Find the harmonic conjugate and an analytic function F(z). t, c ) Derive C'-R Equation in Polar form. If u=r2 Cos28, find Analytic function F(z).
( d ) An eleirli'ustaiic field in the XY- plane is given by the Potential Function <f=3.x2y-y'}.
Find stream line function.
( e ) Find the Analytic function F(z)= u l iv, if u-v ex( cos y sm y). .
d~ Z 2 3z
( a ) Solve--= a z. given that when x=0, a sin y and =0 4
d\dv ' dx
t b ) Attempt any three 12
( i ) Show that the transformation w=z+l/z maps the circle r=c of z-plane into a family ol ellipse in w-plane discuss the case when r=l.
( ii) Determine the region of the w-plane into which the region < x < I and < y < I
mapped by Iran:.formation wit.
(iii ) lfw-1/z (A) Find the image of square whose vertices are lb, 4H, 4+4i, 114i.
(B) Find the image of tine y=2x and x-fy=6.
(iv ) Define bilinear transformation, find bilinear transformation which maps the point z-0, I, ocinto point w5, -1, 3, respectively VVbal are the invariant points of the transformation?
( v > Show that the transformation w-(2M !)/(y-4) maps the circle x"+y-4x=0 onto the straight line 4u I 3~U.
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| Earning: Approval pending. |
