North Maharashtra University 2011 B.E Computer Science and Engineering .f.ecommon - Question Paper
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niRHIM*
Scat No.
v&m - 19
ENGINEERING MATHEMATICS
- II (New)
P. Pages 3 Time r 3 Hours
Max. Marks 100
Instructions ; I) Do not write anything on question paper except Seat No
2) AU questions are compulsory
3) Figures to the right indicate full marks
4) Use of non programmable electronic calculator it allowed
5) Neat diagram must be drawn wherever necessary.
20
1 Attempt any four ,
dJ*
a) If z3 - xz - y = 0, then prove that
- j -y x + y x-y
b) If u tan '*
c) If x * * tan v and y * e" mc v , then how that
du du '
*r-*yr
d* dy
dv dv dx dy
0
* v r \ du du du n
i.I.i Lthcnprowlh* + y
y i x ) y
d) If u * f
dx
- *
v
$L
e) If (cwx-Ciiny)* then find
Ii
|
x + y + r |
 |
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f) Verify EuJeri theorem for u
\\iy- -y:-' -
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2. Attempt any four :
20
Verit\ JJ I for x u cos (in v.
I
If u x + y + s, v x2 y2 *2; w x3 + y) + p lhcn f ind <*
dependent, Also find relation between u and v.
d) Prove that the stationary value of xmy*zp under the condition x + y + r. >au
If the sides and angles of a plane triangle vary in such a way that rs circum
j .da db dc radius remains constant. prove that +-+ 0, where da, db.
cos A co B om C
' <jkY+ email ifk'rnu*nlc in s K r
each
3. a) Trace any one curve with justification i) r * a sin 10 it) xy2 *a2(y2- x1)
b) Attempt any two ;
i) Obtain Fourier icnes expansion for f(x) x* in the interval 03 x 2*
ii) Find the Fourier enes expansion for
a
fll'lViH m 19
14
dxdy, over the im bounded by x + y e. h. * =0, y 0
oiWiinisiit
4 a) Attempt any two
0 Evaluate Jj xm ' y""'
%
*i) Evaluute II COS 2y/l-a2 sin2 x dx dy
o o
, . . . in) Evaluate j J J
a o
b) Attempt any one .
i) Find the area common to the circles r % a and r 2a co* 0
a) Using triple integral find the volume common to ihe cylinders x2 y2; and x2 > jt2 s* a2.
5 Attempt any four .
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a) Find the directional derivative of 4 xyl + yz* at (2, -1,1) along the Imc
iv r ----------
b> Find the tangential and normal components of acceleration of a partkk kMT' po*itJon (x, y) at time t la given by x * e* cos t. y e* wn t
c> Prove that (r> f#(r> - Hr) r
d) Show that F is irrotational an well as toknotdal vetlor field.
e) Find the constants a and b kuch that ihe surface ax* - 2by* * (a 4*x will be orthogonal to the surface 4x2y i* * 4, at (I, -1, 2).
Show that the necessary and sufficient condition lor Hi) > haw a constant magnitude 1* Fft)
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Attachment: |
| Earning: Approval pending. |
