Download AKTU B-Tech 1st Sem 2015-2016 NAS 301 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 NAS 301 Engineering Mathematics I Question Paper

\
Printed Pages: 189 NAS-301
(Following Paper ID and Roll No. to be ?lled in your
Answer Book)
Roll No.
B.Tech.
(SEM. I) THEORY EXAMINATION, 2015-16
[Time:3 hours]
ENGINEERING MATHEMATICS?l
[Total Marks: 100]
Section-A
Q.l Attempt all parts. All parts carry equal marks. Write
answer of each part in shorts. (10 X2=20)
(a)
(b)
(C)
(d)
[f Y=esin?1x, ?nd the value of (l ?x2 )y2 ? chl -? azy .
62
lfV?( 2+ 2 2)""2 then?ndxg+y?+z?
? x y +Z a ax 5y ax-
?xayazaw
If f(x,y,z,w)=0, then ?nd a X 5; X 6?w? X a .
If pvz : k and the relative errors in p and v are
respetively 0.05 and 0.025, show that the error in
k is 10%.
? (1) NAS-301

(e)
(f)
(g)
(h)
(i)
0)
Examine whether the vectors x1={3,l,1], x2=[2,0,-
l}, x3=[4,2, 1] are linearly independent.
-I 0 O
A = 2 ? ~
H? l 43 02 , ?nd the etgen values of A2.
I dub,
Evaluate L E m
Find the valueofintegral fe '6 x.""dx
Find the curl of F = xyz?+ylj+leg at (3,4,1)
State Stoke?s theorem.
Section-B
Attempt any ?ve Questions from this section:
0.2. If
(5x10=50)
cos.l x =10g(y)?- , then Show (I ?lev?+: ?
(2n + llryml ? (n2 +mz}y, = 0 and hence Calculate
X. when xzo.
(2)
Q3
Q4
Q5
Q.6
Q.7
lf u,v,w are the roots of the equation
6(_u___ W)
(A_gwy?HA?szin 1 ?nd y,z)a(x,
Using the Lagrange?s method ?nd the dimension of
rectangular box of maximum capacity whose smface area
is given when (a) box is open at the top (b) box is closed.
Find the characteristic equation of the matrix
1 and verify Cayley Hamilton theorem.
Also evaluate A6 ?6A5 +9A? ?2A3 ?12A2 +23A?9I?
dxdydz n
inethalIHW= 8 theintcgralbeing
extended to all positive values of the variables for which
the expression is real.
Verify the Green?s theorem to evaluate the line integral
L(Zyzdx + 3xdy), where c is the boundary ofthe closd
region bounded by y = x and y = x2 .
? (3) RIO.

Q.8 Determine the values ?a? and ?b? for which the following
system of equation has.
x+y+z=6
x+2y+3z=lQ
x+2y+az=b
(i) No solution
(ii) A unique solution
(iii) In?nite no of solutions.
. . x \I? y q (er
' ? ?' ? ? :1
Q9 Fmd the mass ofa sol1d (ab) +(bj + c , the
density at any pbint being p = kx"? y?"?z"? Where x, y,z
are all positive.
Section-C
Attempt any two questions from this section: (2X15=30)
Q10. a) lfu=f(r) where r2=f+?showthat
62L! 62u ,, l ,
?+6y2 "f (r)+rf(r).
(4)
13)
Q.11a)
b)
Q.12 a)
b)
NAS-301
Change the order
of Integration in
1 2?
l = I0 L2 xxy dxdy and hence evalute.
Find the rank of the matrix by reducing to normal
?1
6
5
\l-D-U)
$NN
A fluid
Vz(y+z)f+(z+x)j+(x+y)l?. Show that the
motion is given by
motion is irrotational and hence ?nd the velocity
potential.
[f x+y+z:u, y+Z=uV,Z=uVWY then find
6(x, y, 2)
6(14, v, w)?
Prove that, for every ?eld 17 ; div curl 7 =0.
Evaluate ?IR ()6 + y + Z)dx dy dz where
R:0$xSl;lSySZ;25233.
Trace the curve y2(20 - x) = x3.
. (5) P.T.O

c) Verify Euler?s theorem for the function
)1
bolt?
4.
?<
wl?n
H
Nl"
+
V
NIB
(6) NAS-30l/ 71700