University of Mumbai 2007-1st Sem B.E FE Applied Maths - Question Paper
N.B.: (1) Question No. 1 is compulsory.
(2) Attempt any four questions out of remaining six questions.
(3) Figures to' the right indicates full marks. \
(4) Answers should be grouped and written pne below the other.
1. (a) If y = eax cos(bx + c) proved that yn = I a +b e cos bx+-c+ntan (b/a))and 5
hence find nlh derivative of e5x cosx cos3x.
(b) Prove that
(bxc) x(axd) + (c x a) x (b x d) + (a x b) x (c x d) = - 2 [a b cj d
du d -u
(c) If u = A e9x sin(nt - gx) satisfies the equation - Mj
dx
Prove that 9 - J
(d) Prove that the statements Rez > O and Iz - 11 < I z + 11 are equivalent 5
where z = x + iy.
2. (a) Show that the roots of z5 = t can be written as 1, u, u2 u3, u4. Hence prove that 6 (1 - u) (1 - u2) (1 - u3) (1 - u4) = 5
(b) Prove that -
= sinh 1x
-1
cosh
(0
f 2) /1 + X
-1
sin (x + iy) sin (x - iy)
= 2itan [cotxtanhy]
[ii) Log
-1
prove that
(c) If us cosec
Vx + Vy
~f3 w
xb + y
2 d2U 0 32u 2 d2U 1 .
tan u 13 12 + 12
x = + 2xy - + y 5- = tanu
3x2 3xoy ay2 12
3. (a) Find all stationary values of x3 + 3xy2 - 15x2 - 15y2 + 72x
(b) If y = sinrx + cosrx, prove that
yn = rn[j + (-1)n sin2rxj find y8 (it) where r =
(c) For the curve x = tcost, y = tsint, z = at, Find radius of curvature and torsion at t = 0. 7
4. (a) Expand gx _ upto x4 and hence show that -
4
X + .
[TURN OVR
} 4' V'} 4
-I -wwpn
t'vVVoV\ A'v-vS -~\ '.4.. n
v2 z2 . i ' T
~ 1, where u is a homogeneous
2 2 2 a+u b + u c+u
function of degree n in x, y, z prove that
2 2 2 0 ux + uy + u2 = 2nu
(c) If X = Vvw, y = Vwu, z = Vuv prove that
54) 56 56 5$ 5<j> 56 X+y57+Z=U+Va7+W' where <Hs a funciton of x, y, z
( \ n
5. (a) If xn = cos--j + isin
Then show that
l on J \fc J
(0 xt x2 x3...........xw = -1
xoo = 1
x0 X1 x2
(b) If z and are two complex numbers such that lz,+z2l = lz1-Zjl. Then prove that arg z1 - argz2 = ~
(c) If F, if, \j/ are continuous functions in [a, b] and derivable in (a, b). Then show that there is a value c lying between 'a* and 'b'
f(a) f(b) f'(c) *(a) <Mb) f(c) v(a) v(b) v'(c)
= 0
such that
(a-b) + i(a+b)
2ab
-1
2n?c + tan
6. (a) Prove that Log
2 .2 a - b
(a+b) + i(a-b)_
.j x(a + bcosx)-csinx
(b) Find the constants a, b, c so that llm 5 1
x-0 x
1 2(-1)n n! . , <W1
(c) If y =-g prove that yn = 3 ~n+T sin(n+*) where
1 + x + x -if 2x +1
f 2x + . [~ 4 I j- I and r = -1+ x + x
0 = cot
7. (a) Find the approximate value of |(0 -98)2 + (2 01)2 + (1-94)2
5z 5z . ~ 0 Then prove that x + V - 3z
_L Z
3' v
(b) If *
\x
(c) A tangent to a curve makes a constant angle $ with a Fixed line. Show that -= constant. Where k is curvature and t is torsion of space vector.
|
Attachment: |
| Earning: Approval pending. |
