Download AKTU B-Tech 1st Sem 2015-2016 AS 101 Engineering 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 AS 101 Engineering Mathematics I Question Paper

Printed Pages: 5 1327 AS-101
(Following Paper ID and Roll No. to be ?lled in your
Answer Book)
Roll No.
B.Tach.
(SEM. I) THEORY EXAMINATION, 2015-16
ENGINEERING MATHEMATICS-l
[Time : 3 hours] [Tolal Marks : 100]
SECI?ION-A
l. Attempt all parts. All parts carry equal marks. Write
answer ofeach part in shots. (10X2=20)
. . . au 9
(a) Ifl1=10?.3,(x2/y)the:n value of X? + y?? = s
(3an
62
(b) lfz=xyf(x/y) then value of xii + Y 5; = 22
(c) Apply Taylor?s series ?nd expansion of f (x, y) =
x3 + xy2 about point(2, 1) upto ?rst degree term.
d It ? ? 2 2['dth l I?M
() x u-v,y u-v,1n evaueo 6(x,y)
(e) Find all the asymptotes of the curve xy2=4a2(2a-x)
4000 (1) P.T.O.

Attempt any ?ve from this section.
. (1?) Find the inverse of the matrix by using elementaly
1 2
row operation.A=[5 7]
?100
_ 2 -
(g) lfA? l 43 02 ,?nd the eigen value of/\2
(h) Evaluate
0?...
~v?.~
3
I 1ij dx dy dz.
2 t
(i) If ?(x,y,z)=x2y+y2x+zz, ?nd W) at the point
(1,1, 1).
.\
r(8/3)
r(2 /3) -
(j) Evaluate
SECTION-B
(l 0x5=50)
? - n l . -1
2. It x - 51%; 51? y} ?nd the value of y" at x=0
if u, v, w are the roots of the equation
a
(xi?x):+(2,?y)3+(?1_z)3 =0 in/l?nd (u,v,w)l
a(x,yJ)
(2) AS-lOl
4000
If r the distance of a point on conic ax2 + by2 + cz?=l,
(x + my + nz = 0 from origin, then that the
stationary values of r are given by the equation
[2 m2 n2
, + ,+?
l?ar' 1?br? l-cr
Find the Eigen values and corresponding Eigen
6 ~-2 2
vectorsA= _2 3 ?1
2?13
The plane g+?+? =1 meets the axes in'A, B, and C.
Apply Dirichlet?s integral' to ?nd- the volume of the
tetraheadron OABC. Also ?nd its mass if the density at
any point is kXyz. ' ' ' H
, _ l 24x
Change the order of Integrauon 1n I=, Io?IxZ xydxdy and
hence evaluate the same,
Verify gauss?s divergence theorem for the ?mction
F = 321? + z] + yz/E, taken over the cube bounded by x=0,
x=1, y=0, y=1 and z=0, z=l.
?2?
F ZVW is irrotational as well
Show that the vector ?eld
as solenoida]. F ind the scalar potential.
(73) P.T.'O.

Attempt any two question from this section.
10. (a)
(b)
(C)
11. (a)
1
12. (a) Evaluate I
4000
(b)
(C)
SECTION?C
(15x2=30)
Expand e?cos by in power of in powers of x and y
as terms of third degree.
Determine the constant a and b such that the curl
of vector.
A
21- : (2xy+3yz)f +(x2 +axz?4zz)j ?(3xy+byz)l?
is zero.
Examine the following vector for linearly
dependent and ?nd the relation between them. If
Possible.
xl=(1, 1,31, 1),x2=(1, [,2,-l), x3=(3, 1, o, 1).
De?ne Beta andGamma functioh and Evaluate
j dx
0 V 1 + x4 .
Find the area between the parabola y = 4ax and
x2 = 4ay. ,
sz3 x15"! x1?12 3(y2,y2,y3)
y = , = ? = _
If ' I X1 Y2 x2 Y3 X3 ?nd a(x1,x2?x3)
_???__
0 (0? ? x" W
(4) AS-l 01
4000
(b) Reduce the matrix in to normalform and hence
I 2 l 0
. ?2 4 3 0
?ndltsrank
0 2 ?8
132
(c) It u =u[yx-yx ,z?x} show that :
(5) ' AS-lOI