Veer Narmad South Gujarat University 2011-1st Year B.Sc Mathematics SB-0617 - 2 ( New ) - Question Paper
SB-0617
First Year B. Sc./B.A. Examination March / April - 2011 Mathematics : Paper - II
(Calculus & Differential Equations) (New Course)
[Total Marks : 105
Time : 3 Hours]
""'N Seat No.:
6silq<3i Puunkiufl [qaim SuwiA u* qsq d-onql. 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 (NEW)
Student's Signature
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-Subject Code No. |
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-Section No. (1,2,.....): NIL |
UMKl WH ?HLHl :
*1M
X
(*l) % y =-r Ah dl v
ax + b n
(0 /(x) = (x-1)2/3;xg[0,2] Hia H.HH *Url&l.
71
(3) v = sinx-fl <ttdl * = - *iWl.
r dx
0 (1 + x2 j
(m) feci \d-t-dx= +X-0. dlL HfcHlSl
J dxz
dx
(e) 6M : (l + xf = l + y2 dx
U) llm DsHd Hicfl.
xyoo
dx
(e) feci ~ + P'x = Q' Wi P';Q' I faMl d - dl
&HI-H (3K-1 <HHl.
(h) % _y = eax cos(bx + c) 6lH dl yn HlAl dLL y = ex cos5.x-cos3.x HIS yn ?lWl.
(h) % j = sin 1 x; |x| < 1 dl lOid *l
(l - x2)yn+2 - {2n +1) xyn+l - n2yn = 0
2x + 1
(i) * y = (x-l)(2x-l)
3 M Ac-i-i H.HH lOid d*U d. WLlPHlxti *i?had tt*m<il.
sinx
(H) lim
x>0
v y
-(I &Hd 5>llHl.
($) <HBUSHl UHHdl (3HH>L M lOid
/ \ia
= (e-A,) ; Wl O<a<rk<b-aa
(*i) iUtl'j, H.HH <HHl *id lOid S$L.
. . ,. tanx
(h) lim, :-{I &Hd Hl<Cl.
x>jr/2 tan3x
U) lOid : 100-/99 -(I &Hd 7- *id --{l &.
18 20
Y (*i) M H>SL Hi _y = /(x) Hia Hdl fcmi
3/2
(1 + >f) 2
*ufad wi y\=-; y2=TT-
ax axA
(h) x = (y-\)-(y-2)-(y-3)<\. HlrdL HRlH [H|*ildL HIH
a"
(<h) Hlr
rOT = aOT sinwG iCid 3RI Is P (-mj.-[\vm-\
(m + \) i
U) 'U }' =
HL8 *idd
M (*l) foh&i (ax + by + c) dx + (a1 x + b' y + c') dy = 0 d
Gk-tqHl &d ctstcfl.
(<h) (3&Hl : cosx -y-sinx + y2 =0 dx
U) 6M : (l + x) + (l + 2x)_y = (l + x)2
H*l<U
H M fohQ ?HLl dd Gk-tqHl &d %|M.
(&) (3ll : (2xy + y- tan y)dx + (x2 - xtan2 j + sec2 yjdy = 0 e M f(x,y,p) = 0, % j Hia (3<L-CIh Ah dl
dd Gk-tqHl &d
(h) (3r<yl : xp2 +(y-x)p-y = 0
(h) (36ll : y = 2px + y2 p3
H*l<U
e M M.-i fohQ C-tHl ?Hd dd Gk-tqHl &d
d (y-xp)(p-1) = p dl &IHI-H 6kl <HHl.
(H) 6M : * = + />
P
(&) (3r<yl : (px-y) (py +x) = h2 p
Instructions : (1) As per the instructions no. 1 of page no. 1.
(2) Answer all questions.
(3) Figures to the right indicate marks of the question.
(4) Follow the usual notations.
1 Answer the following questions : 15
(1) If y =-- then find y
ax + b n
(2) Verify Rolle's theorem for the function
/(x) = (x-l)2/3;xe[0, 2].
71
(3) Obtain Curvature of the curve _y = sinx at x = .
(4) Find the value of i
dx o (l + x2 )5
(5) Find order and degree of the differential equation f %dx =
d2y (dy2 'x
\2 dy
+ x
\dx j
(6) Solve : (l + -x) ~r = + y2
dx
dz dz
(7) If Z = x2 + 2xy + y2 then find and
(8) Evaluate : lim xX
V 7 X
(9) Write the general solution of differential equation
dx ,
+ P'x = Qwhere P' and Q' are functions of y.
(10) Solve : y2 p2 = x2
2 (a) Obtain yn for y = eax cos(6x + c). And find yn for 6
y = ex cos5.x-cos3.x
(b) If j = sin 1 jc ; |jc| < 1 then prove that 6
(l - x2)yn+2 - (in +1) xyn+l - n2yn = 0
2x + 1
(c) Obtain : y for y = -----6
(x-l) (2x-l)
OR
2 (a) Obtain yn for y = lx + j)) x a,> e ax + * . From 6
that find yn for y = log(ax + b);ax + b >0.
(b) Find yn for y = sin2 xcos4 x. 6
(c) Obtain v for v =-3x,+ 1--6
(x+1)2(x-2)
3 (a) State and prove Rolle's theorem and explain its 6
geometrical interpretation.
1
sinx
X2
6
(b) Evaluate : lim
x>0
V X y
(c) Use Lagrange's theorem to prove that 6
- = (e-X)b a; where 0<a<X<b-
aa
OR
3 (a) State and prove Cauchy Theorem. 6
tanx
(b) Evaluate : lim. 7 . 6
v 7 xjt/2 tan3x
(c) Prove that the value of /lOO - 99 lies between 6
1 1
and .
18 20
dy d2y
curvature is
dx dx2
(b) Find the point of inflexion for the curve 6
x = {y-i)-{y-2)-{y-3).
(c) Obtain J cosn xdx an(j hence evaluate J cos10x<ix. 6
OR
4 (a) Obtain the reduction formula for jsecn x dx and hence 6
evaluate |sec5xt/x.
(b) For the curve rm =am sinmQ > prove that 6
am
P =
(m + \)r
m1
x2-2x-8
x 1
5 (a) Explain the method to solve the differential 6
equation : (ax + by + c)dx + (ya'x + b'y + c')dy = 0
dy . ~
dx
(c) Solve : (l + x) + (l + 2x) y = (l + x)2 6
OR
5 (a) Define linear differential equation and explain the 6
method to solve it.
(b) Solve : x- = y + Jy2 +x2 6
dx
(c) Solve : (2xy + y- tan y)dx + [x2 - xtan2 j + sec2 yjdy = 0 6
6 (a) Explain the method to obtain general solution for the 6 differential equation / (x, y, p) = 0 which is solvable for y.
(b) Solve : xp2 +{y-x)p-y = 0 6
(c) Solve : y = 2px + y2-p3 6
OR
6 (a) State and explain the method to solve Clairaut's 6
differential equation and hence write the general
solution of [y-xp)[p-\) = p
1
(b) Solve : x = -+p 6
p
(c) Solve : (px y)- (py + x) = h2 p 6
SB-0617] 8 [ 2500 ]
W l-rt'xabeRax+b* 6lHdl V Hicfi. anvil e
(ax + &)
>> = log(ax + 6); ax + b > 0 HL8 yH H.M..
(h) fcl j; = sin2 xcos4 x 6lH dl yn ?iWl.
3x + l
U) * ,' = (x+l)2(x-2) A
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Attachment: |
| Earning: Approval pending. |
