University of Mumbai 2008-1st Sem B.E FE Applied Maths I - Question Paper
c/A
[Total Marks : 100
(REVISED COURSE) (3 Hours)
N.B. : (1) Question No. 1 is Compulsory.
(2) Attempt anyjfour auestions out of remaining six questions. *a
- J- UAT /Hx fir. rpp& cA
K _0
4 2
then P.T.
1. (a) If u = log
20
tan
(i) coshu = Sec 0
(ii) sinhu - tan 9.
(b) Find the complex number 4z* if
arg (z + 1) = -?-and arg (z - 1)
2k
3 '
du
(c) Ifu = (l - 2xy-H y2)-,/2, P.T. x
(d) If u = a cost i + a sint j + at tana k
du d2u d3u
= a3 tana.
then S.T.
dt dt2
dt:
. -i
2. (a) If y = emsin x then P.T.
0 -x?)yn+2-(2n +1)xy>iH-(n2 + m2>yn = *
(b) Find the maximum and minimum values of
x3 + 3xy2 - 3x2 - 3y2 + 4.
(c) If z = f(x, y), x = eucosv, y = eusinv
OZ 07. _ 2u OZ
+ 8y
0)
P.T.
' dz + f dz V
dz' du
- 2u
+
= e
(ii)
dv
lyj
OX
3. (a) Prove that V( f(r)) = fr(r)y and hence find f if Vf = 2rr. 6
(b) Find the values of a, b, c so that 6
lim ae*-bcosx + ce~x _ x-0 xsinx
(c) If cosa + cosp + cosy = 0 and sin a + sinp + siny = 0 then P.T. : 8
3
(i) sin2a + sin2p + sin2y = cos2a + cos2p + cos2y - y.
(ii) cos(2a) + cos(2P) + cos(2y) = 0
(iii) cos(a + P) + cos (P + y) + cos (y + a) = 0
(iv) sin(a + p) sin(p + y) + sin(y + a) = 0.
(a) If z = f(x, y), x = r cosG, y = r sinG, P.T.
f*f+fi5f=ff+4f-i-)2
ax
4. (a) If z = f(x, y), x = r cosG, y = r sinG, P.T.
(b) If x + iy = c cot(u + iv), Show that x _ -y c
sin(2u) sinh(2v) cosh(2v) - cos(2u) (c) If is a constant vector and r = xi + yj + zk P.T. (i) div(axr) = 0
(ii) div(a*r)a = a2
(iii) (axrxa) = 2a2
(iv) curl(axr) = 2a.
7
x3 x5
5. (a) P.T. tanyx=xj~ + ~$---7
r fex2
59
dz 'I dr
'_dz
.5y J
x
b - a , f b 1
r.l<Jli.forO<a<b a a
(b) P-T. 5 J
1 4 1 Hence deduce that -<log-y<-j.
(c) Separate into real and imaginary parts tan"1 (cos 0 + sin 0)
6. (a) Test the convergence of
T2+J + 5 + 7g +...... (x > 0 and x * 1)
(b) If u = f(y/x) + ,/xV
then P.T. * J + yJpV*2+y2
(c) If y = 2* cos9 x then find yn.
7. (a) Find all roots of (x + I)7 = (x - l)7.
then P.T.
(b) If u = cosec
tanu
IT
txy
<5x
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- 1 |
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(c) If z = x log(x + r) - r where r2 = x2 + y2 &Z.&Z 1
13 , tan2 u
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Attachment: |
| Earning: Approval pending. |
