Download AKTU B-Tech 1st Sem 2016-2017 RAS 103 Engineering Mathematics 1 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) 2016-2017 RAS 103 Engineering Mathematics 1 Question Paper

Printed Pages: 8 , IRAS- 103
(Following Paper ID and Roll No. to be ?lled in your,
' Answer Books)
Paper ll) : 2012439 Roll No.
B.TECH.
Regular Theory Examination (Odd Sem - I),2016-17
ENGINEERING MATHEMATICS?I '
Time .' 3 Houis Max. Marks : 70
Note .' The question paper contains three sections -'A, B & C.
Read the instructions carefully in each section.
[SECTION -A
Attempt all questions of this section. Each?p'art
carries 2 marks.
'1. ? a) For what vaer of ?x?, th? eigen values of the given
matrix A are real
10 5 +i 4
A = x 20 2 ' ' (2)
4 2 ?10 '
0 1
prove that A3 = 19A+ 301. ? (2)
' - ?5 -3 1 0
b) For the glven matrlx A =[ 2 0 J and I =[ ]
103/12/2016/47100 ? (1) [1:10.

C)
d)
g)
RAS -103
Find the maximum value of the function
? f(xyz) = (z? 2x2 4 2y2) where 3xy?z+ 7 = o. (2)
If the volume of an object expressed in spherical
coordinates as following :
27:
?t
F ind the condition for the contour on x ? y plane
?'?ua
I .
Jr 2 Sin (15 dr 61? ?19 Evaante the value of V.(2)
0
where the partial derivative of (x2 +y2) with respect
to y is equal to the partial derivative of (6y + 4x)
with respect to x. ? (2)
The parabolic arc y=x/; , 15x32 is resolved
around x - axis. Find the volume of solid of
revolution. (2)
' > 2 2
For the scal?r ?eld u = ?2? + y? , F ind the magnitude
ofgradient at the point (1, 3). (2)
103/12/2016/47100 (2)
2.
RAS- 103
SECTION - B
Attempt any three parts of the following. Each part
carries 7 marks.
,a)
b)
0
Express 2A5?3A4+A2?4I as_a linear
. . A? 3 1 (3)
polynomlalmAwhere ? _1 2 .
1 2 ?2
Reduce the matrix P: 1 2 1_ to
-l ?l 0
diagonalfdrm. - i (4)
V V
If u =sin?1 M then evaluate the value
x% + y%
(3)
103/12/2016/47100 (3)
[P.T.O.

d)
RAS -103
Trace ~ the curve x : d(B?sin 6) ,
y=a(1?c086). (4)
Find the relation between u, v, w for the values
u=x+2y+z;v=-x?2y+3z;
w = 2x32 ? 2x + 4yz ? 222. (3)
Divide a number into three parts such that the
product of ?rst, square of the second and cube
of third is maXimum. ? (4) ?
Change the order of integration for
l 2?.r
I = J. I xydxaj? and hence evaluate the same.
0 r2
(3)
Evaluate the triple integral
I l I [:5 I JET; (HZ) dx dy dz 0 '(4)
0 0 0
103/12/2016/47100 (4)
RAS- 103
e) i) If ?=(x3+y3)f?2xy]_ then evaluate the
value of C! Pdf. (3)
1 .
ii) Find the directional derivative of (7] 1n the
direction of ,7 where F = ix + }y f k} . (4)
SECTION-C
Attempt all questions (if this section, selecting any
two parts from each question. All questions carry
equal marks. (5X7=35)
?
= :x" (er logx), show that [n = "In-1+ n_1.
a) If 1,,
b) If? e_:/(X2'_yz)l=x?y then show that
E + xgzt = x2 ? y2
y ax ay
103/12/2016/47100 (5) [n.0,

b)
b)
RAS -103
H W: x2 +y2 +z2 &x= cosv,y= usinv,z=uv,
V?w ?} "
then prove that 6" av -\/1?+?vz.
If x=v2+w2, y=wZ-1-u2,z=u2+v2thensh0w
a(xyz) 6(uvw)_'
t t???l
6(uvw)' 6(xyz)
Express the function?xy) = x2 + 3y2 ? 9x ? 9y + 26
as Taylor?s Series expansion about the point (1 , 2).
F ind the percentage error in measuring the volume
of a rectangular box when the error of 1% is made
in measuring the each side.
IfA=[
1
2
0] then evaluate the value of the
expression (A + 51 + 2A?).
1 4 2
Find the eigen value of the matrix [2 4]
101
corresponding to the eigen vector [10]].
103/12/2016/47100 (6)
RAS- 103
1 l 1
c) Showthat A=i 1 -w W2 is aunitary matrix;
J3 2 -
1 w _ w
where w is complex cube root of unity.
6 a) Changing the order of integration in the double
integral 1:]- ! f(WHWk leads to the value
0 x/4
S q
1: I I f(xy)dm3? . What is the value ofq?
r p ?
b) Evaluate ?I xzyzdxdydz throughout the volume
. ? x y 5?1
bondedbyplanesx=0,y=0,z=0& Z+Z+c ? .
c) For the Gamma function, show that
(1] ?]
/3 ?
3 __6 =(2)1 J7:
2)
3
103/12/2016/47100 (7)

RAS -103
7. a) Verify Stokes theorem F_=(2y+z,x?z,y?x)?
taken over the triangle ABC cut from the plane '
x + y + z = 1 by the?coordinate planes.
b) Verify Gauss Divergence theorem for
j?)? ? yz)? ? 2x2yj?+ 213]r?zd;v where S denotes the?
- surface of cube bounded by the planes x = 0, x = a;
y=Qy=mz=Qz=av
c) If}; : (xzzf+ 2y}?3le?)and r3 = (3xzf+ 2yzj-ZZIE)
Find the value of [AXWXBU & [(AXV)XB]. _
++++
103/12/2016/47100 '(8)