Veer Narmad South Gujarat University 2010 B.Sc Nursing Mathematics : - 2 F.Y../B.A . - Question Paper
RR-0631 First Year B. Sc./B.A. Examination March / April - 2010 Mathematics : Paper - II
(Calculus & Differentiate Equations)
(Old Course)
Hours] [Total Marks : 105
Time
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Seat No.:
Fillup strictly the details of signs on your answer book.
Name of the Examination :
F. Y. B. Sc./ B. A.
Name of the Subject:
Mathematics 2 (Old)
Student's Signature
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-Subject Code No.: |
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-Section No. (1,2,.....): Nil |
(0 HHl UMHl (3tR ?HlHl.
(3) WiO. d UMdl %|R d.
M &IHI-H did *lL.
*1 -Mdl Wldi WUH ?HLHl :
(<l) y = x\ogx dl ?iWl.
1-x2
U) f(x) = e , xe[-l,l] HLa nm\l.
(3) lim/9 X &Hd x>71/2 secx
(y) 3/= x2 (;r + 2) <W-(l <ttdl (0,0) hM..
(h) lOid j' = logx mm ?HH:?HdH d.
k/2
(?) J sin3 xcos2 x dx-fl Old %iM.
0
dx
U) z = x +3xy +3x y + y 6lH dl
dy
3 2
a z dxdy
{<>) sinx + 4,ycosx = cotxdl *U<jh&r& HHHH ?iWl.
ax
dy
dy
(*lo) (3&Hl : y-x = edx .
dx
\ (*l) % y = log (ax + b) ; ax + beR+ Ah dl y *iWl Hd
d hM = log 2x + 5x + 3 Hia y_ *iWl.
m
Ah dl lOid i
(h) J'
2+2) +2+(2+!) x+i+(n2-m2)yn= 0
x
_2x
U) % = e x cos xsinx A.H dl %iM.
(?h) % 3; = eax sin (6x + c) Ah dl yn HlHl ?Hd d HVfl. y = ex sin (4x +1) HIS yn ?iWl.
(*) y = emsm~lx Ah dl *lP>ld *>U
1 - *2) yn+2 ~ (2n+!) x+i - {n2+m2) yn =0
/ \ -v x2 + 4x +1 . . . .
U) % >= >-2- LH dl J' *IM.
x + 2x -x-2
3 (?H) 611311 HHHK H.HH <HHl Hd RilOid
, v ,. tanx-x
(*0 h 2- &Hd SM.
x0 x tanx
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(&) ifcld 3Rl sinx=x---1-----1-....
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3 (?h) H.HH 5Hd lOid 6
(H) lOid S$L I e
a-6 -i _i a-b , ,x - < tan a-tan b<--;[0<a<b)
1 + a 1 + bz
xex - log (l + x)
xO &Hd ?iWl. 6
X (*l) J sin xdxi 181 Hiql 5Hd d HVtt J sin5 x dx 6
(H) j' = x4 - 6x3 + 12x2 + 5x + 7 dl-Uit IM|*iL 6
U) y = X ~5X + Idl ?Hdd 6
x - 3
H*l<U
X (*l) M H51 y = f(x) Hia lOid S$L I 6
7 2 <ttdl k =-
2\3/2
1 + 1
(h) j> = 3x5-40x3 + 3x-20 *id*KlHl (M *idH e ?HH:?HdH Hd d d
7 dx
U) i L 2\4 -fl &Hd
0 l + x
H (*l) ?Hd ddl (3K1 HLSdl &d <lsWL. 6
(<h) (3&Hl : (2xj' + j'-tanj')dx + |x2-xtan2 j- +sec2 1 = 0 . ?
U) 6M : - + = xy2
dx x
M (*l) Mdx + Ndy = 0 Hd d HlMl 5Hiq.%HS e
5Hd HHLkl %Rd <HHl 5Hd lOid S$L.
(h) 6M : (x2+j'2l (xdx +j'<ij') + (x(ij'-j'(ix) = 0 . e
U) 6M : (1 + X2]j + y = etan . M
e (*i) f(x,y,p) = 0, % p Hia 6ki-{h Ah dl dd e
Gk-tqHl ld ctstcfl.
(h) 6M : 3; = 2px-ip2. e
o
(&) (3&Hl : sinpx cos j' = cospxsin y + p . 6
H*l<U
e (*l) C-lL5lL [ciM C-tHl ?Hd ddl (3K1 HLSdl &d 6
(h) (3&Hl : j' = /?tan/? + logcos/?. 6
U) (3&Hl : x2, (y - px) = yp2,.
Instructions : (1) As per the instruction No. 1 of page no. 1.
(2) Answer all questions.
(3) Figures to the right indicate marks.
(4) Follow usual notations.
1 Answer the following questions : 15
(1) If j' = xlogx then find yn.
(2) Verify Rolles Theorem for the function
f(x) = e1~x , x e [-1, l] tanx
(3) Evaluate : im/0 .
v ' x>7i/2 secx
(4) Obtain curvature of the curve y = x2(x + 2) at (0,0).
(5) Show that ,y = logx is always concave downwards.
k/2
f 3 2
(6) Evaluate : J sin xcos xdx.
0
(7) Solve : - = ex+y.
d2z
(8) If z = x3 + 3xy2 + 3x1 y + y3 , then find
dxdy ' dy
(9) Find the integrating factor of smx-+ 4,ycosx = cotx
ax
(10) Solve -.y-x- = edx .
ax
2 (a) If y = log (ax + b) ; ax + be R+ then find yn and 6
hence find yn if y = log|2x2 + 5x + 3
c i-\m
2 2 1
v /
+2 yn+2 + (2n+1)xyn+i+[n2-m2)yn=-
X
(c) If y = e2x cos2 x sin x, then find yn. 6
2 (a) If y = eax sin (bx + c), then find yn. Using it obtain 6
yn for y = e3x sin (4x +1).
-l
(b) If y = emsin x, then prove that 6
1 - *2) yn+2 - (2n+!) xyn+1 -(n2+m2)yn = o
X2 + 4x +1
x +2x -x-2
3 (a) State and prove Lagrange's Mean-Value theorem. 6
tanx-x
*0 x tanX
X X x}
3! 5! 7!
OR
3 (a) State and prove Cauchy's theorem. 6
(b) Prove that : 6
d-b _i _i a-b ,
- < tan a-tan b<--;[0<a<b)
1 + a 1 + bz
xex -log (l + x)
(c) Evaluate : j.11 2 . 6
4 (a) Obtain the reduction formula for J sin x dx and 6
f 5
hence evaluate I sin x dx.
(b) Find the point of inflexion for the curve 6 y = x4 - 6x3 + 12x2 + 5x + 7
i-10
x-3
4 (a) For any curve y = f{x), prove that 6
*2
curvature k
(b) Obtain the intervals in which the curve 6
y = 3x5 -40x3 + 3x -20 is concave upward or concave downward.
7 dx
(c) Evaluate : I L 2\4 . 6
0 \1 + X 1
5 (a) State Linear Differential Equation and explain the 6 method to solve it.
(b) Solve: (2xy + y-tan y)dx + |x2-xtan2 j' + sec2 yjdy = 0. 6
dy y 2
(c) Solve : -jL + - = xyji.
dx x
5 (a) State and prove the necessary and sufficient 6
condition for the differential equation Mdx + Ndy = 0 to be exact.
(b) Solve : (x2 + y2 J (xdx + ydy) + (xdy- ydx) = 0 . 6
(c) Solve : (l + x2) + y = etan~lx. 5
6 (a) Explain the method to obtain general solution 6
of the differential equation f (x, y, p) = 0, if it is solvable for p.
1 9
O
(c) Solve : sinpx cosy = cospxsin y + p. 6
OR
6 (a) State Lagrange's differential equation and explain 6 the method to solve it.
(b) Solve : j' = /?tan/? + logcos/?. 6
(c) Solve x2 (y-px) = yp2. 6
RR-0631] 7 [ 500 ]
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Attachment: |
| Earning: Approval pending. |
