Download AKTU B-Tech 1st Sem 2015-2016 EAS 103 Mathematics I Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU)) B-Tech 1st Semester (First Semester) 2015-2016 EAS 103 Mathematics I Question Paper

Printed Pages: 5 175 EAS-103
(FolloWing Paper ID and Roll No. to be ?lled in your
Answer Book)
Roll No.
B.Teich.'
(SEM. I) THEORY EXAMINATION, 2015-16
MATHEM?ATICS-l
[Time:3 hours] [Total Marks:100]
Section-A?
1.7 Attempt all parts. All parts carry equal marks. Write
answer of each part in shorts. . (10X2=20)
2 7 > au au
(a) Ifu = log(x /y) thenvalue Of x?+y?? = ? .
6x 6y
(b)
(C)
(d)
lf 2 = xyf[?) show that xz+y? = 22.
y 'ax 6y
Apply Tayldr?s'ls?ries find expansion of
f (x, y)=x3 +xy2 about point (2,1), upto ?rst
degree term. I
6(u, v)
5(xay) '
lfx;u?v, y=u2 _v2,?ndthe valueof
(1) 12.10

(e) Find all the asymptotes of the curve
2=4ar2 2a? . ',
xy ( x) 3. If u,v,w are the roots of the equatlon
(f) Find the inverse of the matrix by using elementary
6(u,v,w)
- 1? +1? +A?z =0inl?nd
row operations. A=[; ?72] ( x)? ( y )3 ( )3 a(xd??)
4. If r is the distance of a point on Conic
_1 0 0 ax2+by2+czz%1,lx+my+nz=0fromorigin,then
(g) [f A = 2V _3 0 ?nd the eigen values of A2 that the stationary values of r are given by the equation
14?2} V i2+mg+n2 _0
l?ar2 1?br2 1?ch
(h) Evaluate J: L2 Lsxyz dx dy dz. . _ 5. Find the Eigen values and corresponding Eigen vectors
; a
. 2 2 2 . I 6 ? ?2 2
(l) If?(x,y,z)=xy+y x+z ?nd V? atthepomt A? _ 3 ?1
(1,1,1). 2 _1{ 3
(j) Evaluate M.
r(2/3) x y z .
6. Theplane?+?+??=1meetstheaxesmA,B,andC.
Section-B a b c " .
' . V . . Apply Dirichlet?s integral to ?nd the volume of the
N :Att t f : . .
0? emp any we Qu?snons from ??3 sec?? 10 50) tetrahedron OABC. Also ?nd its mass 1fthe denSIty at
x =
. any point is kxyz.
2, Ifx = Sln{?sm?" y} ?nd the value of y? at x= 0' 7. Change the order of Integratlon In
In
I 2-x
I =J;J;2 xy dxa?i and hence evaluate the same.
(2) EAS-103 .
' ' (3) P.T.O.

Verify gauss?s divergence theorem for the function
_.
F
= x2; + z] + yzkA , taken over the cube bounded by
x=0,x=1;y=0,y=1andz=1,'z=1.
A
-> r . . .
Show that the Vector field F = ??3 IS 1rrotat10nal as
M '
well as solenoidal. Find the scalar potential.
Section-C
Attempt any two questions from this section: (2X15=30)
10.
a)
b)
\
Expand em cos by in powers of the powers of
x and y as terms of third degree.
Determine the constant a and b such that the curl
of vector.
A
A =(2xy+3xz) 15+(x2 +axz?422)j?(3xy+byz)le
is zero.
Examine the following vectors for linearly
dependent and ?nd the relation between them, if
possible, le=(1,1?1,1), X2 =(1,?1,2?1).
X3 = (3,1,o,1).
(4) EAS-103
11.
12.
'b)
0)
Vb)
De?ne Beta and Gamma function and Eval-aute
1 dx
LT?
Find the area between the parabola y2 = 4ax and
x2 =4ay.
[fy] = x2x3 yz =??, 3H; ?nd a(yl?yZ?y3)'
x, x2 x3 a(xl?x23x3)
E 1 t [1 dx
vauae 0W
Reduce the matrix in to nomial form and hence
I 21 0
?nditsrank ?2 4 3 0
102?8
lf u=u[y?x?z?x] show that
$31 xz
(5) EAS-103/8600