University of Mumbai 2007-2nd Sem B.E FE Applied Maths II - exam paper
(3 Hours) [Total Marks : 100
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N.B.: (1) Question No. 1 is compulsory. M/F
(2) Attempt any four questions from remaining six.
1. (a) Prove that
| + x = (-x2j* sec Ttx provided -1<2x<1. 5
3
x 2
(b) Solve (x2 + y2 + 1) dx - 2xydy = 0. 5
* log(l + ax2)
(c) Show that J -;--dx = <a > ) - 5
o x
(d) Find the total length of the curve y% + + y% = a% 5
"-1 ,n-1
f Sin x 2"" Jn
2. (a) Prove that J (a+bcosx)r, (a2_b2 PU" 2 J
(b) Change the order of integration and evaluate 6
r r7i-y cos-1 x
J J I 2 I 2 2* y.
(c) Solve (D3+l)y = e2 sin
2
3. (a) Find by double integration the area enclosed by the curve 9xy = 4 and the line 2x + y = 2. 6
ff dxdy
(b) Evaluate JJ v4 . v2 where R is the region x > 1 and y > x2. 6
r * +y
(c) Solve (D2 - D)y = ex sin x by method of undetermined coefficient. 8
4. (a) Find the volume bounded by y2 = x, x2 = y and the planes z = 0andx*y + z = 1. 6
00 -x8 * -x4
Evaluate Jx e dx jx2 e dx.
(b)
-1
(c) Solve (d2 -t)y = 2 (l - e"2x) 2* by variation of parameter.
[TURN OVER
5. (a) Find the mass of the lamina bounded by the curve ay2 = x3 and the line by = x. If 6 the density at a point varies as the distance of the point from the x axis.
CO
(b) Verify the rule of D.U.I.S. for Je sin btdt. 6
d n2 1
(c> So,veU+xJy = 7- 8
6. (a) Evaluate ]\\ x2yz dxdydz throughout the volume bounded by x = 0 y = 0 z = 0 and 6
x y z .
(b) Find the length of the upper arc of one loop of lemniscate r2 = a2cos20. 6
(c) Solve (1 + X)2 + (1 + x) + y = 4coslog (1+x) . 8
dx dx
7. (a) Change to polar co-ordinates and evaluate II "xy dxdy where R is the region of 6
Integration bounded by x2 + y2 - x = 0.
dv Q q
(b) Solve -p = xJ yJ ~xy. 6
dx
1
n +-
(c) Prove that pfn + n + -1 = -J- -==?- -Jk
n + | = 1.3. 5....{2n - 1)Vjt.
deduce that 2n
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Attachment: |
| Earning: Approval pending. |
