Kurukshetra University 2008 B.A Mathematics Vector and Geometry - Question Paper
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Roll No. SKCTION-I I
Tula! Pages : -1
33
j A.:3t> ' (.Sc :45
(.Maximum Marks
3(4)
N'oic : Attempt fir,- que%tmus in .ill. selecting am- question Itoin each scction.
SKCTION-I
I. (a) Show lieu
(l> > V t/t < . /i X ((: y ,J) + f XliiXM = o.
(ii) ;i r - citswOi + viu {ntij.
lb) Prove ihul V/(.i)xr = ()
where r.rf + vi+;ft. 3<4Vj)
2. (a) lf/ .rV~. find div (grad/). 3(4*j)
(h) For what value of A. \x and v ihe vector
/s(2.r + 3,v+A:)i + (/i.t + 2y + 3;)j,+<2.v*vy + 3z)k
3(416)
RAK/A08 MATHEMATICS Papcr-DM-103 ( Vector Analysis ami Geometry)
. ... tfr *
show that r x = nk. dt
Tunc : Three llmrs|
is irrotational?
3J/HOWKD/2M)
t
X (a) livaluate J/sir from (0. 0. 0) to (3. 9. 0) along the
curve >* .*. z = 0 for / = j*2/ 4(x *)ii/+<2a - z)J*.
3(4W)
(b) Evaluate JJJ45a*> r/l' where V is the region bounded y
by the planes 4x + 2y + z s 8 and .t = 0. y - 0. z - 0.
3(4W)
4. (a) IKing Gauss Divergence theorem, evaluate
Jjtjrfyy t dy) over the surface
tHHtnded by the co-onhnutc planes ami the planes jr * y = t =* 2 3(4*/*)
(b) fcvaluatc J-dr whca* f * y'i +x:J-<.t :)i and C c
is the boundary of triangle with vertices on ((). 0. 0). <l.O.O>antl(l. 1.0). 3(4 Vi)
SKCTION-III
5 (a) Show tliat r + 12xy - 4\J - (iv + 4y + 9sl) represents a hyperbola. Kiiul its centrc and lengths of axes and eccentricity. 3(4'/)
(h) Find the polar equation ol a conic with locus as pole. 3(4i6)
(t. m) 1-itul the equation <f the sphere through the circle t2 + \~ + z~ + 2.V + 3y + tf* = l). ,i - 2v + 4c - *) = 0 and the centre of the sphere
t* + >- + r 1* * 4v - <*: + 5=0. 3(4'/,)
VXntKVKIV 2
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(h) Find the equation of (he right circular whose vertex is x y z at origin, axis the line = = - and has scmivertieal 12 3 angle 30. 3(4'.*) 7. (a) Prove that 4.V2 - y3 + 2z2 + 2xy - 'Syz + lit - 1 ly + 6z + 4 = 0 represents a cone. Find its vertex. 3(4'/i) (b) Find the equation of the right circular cylinder of radius 3 and axis as the line |
10. (a) Show that two con focal paraloloids cut everywhere at right angles. (b) Prove that the surface given by l(uJ+4y2+422+4<y2-r+ 8.ry+4.r+4y- I6z- 24=0 isan elliptic paraboloid. Find its vertex. 3(4'/i) |
3(4W)
SECTION-IV
8. fa) Find the equations of (he tangent planes to the surface .r - 2y + iz2 * 2 which are parallel to the plane
* - 2y + 3 = 0. 3(414)
(b) Find the equation of the enveloping cylinder of the conicoid or + by + cz~ - 1. whose generators arc
X y *
parallel to the line =- = -.
3(4!4)
/ m n
9. (a) Find the equation of the plane which cuts the paraboloid :r - ly2 = z in a conic with its centre at the point
( 3
2. 4j. 3(4)
(b) Find the equation of the generators of the hyperboloid
2 2 2 X V
+ ---s| which pass through the point a' b c
(a cos ft b sin 9. 0).
3(4 /*) IP.T.O.
33/S()l XVKI)/:6J
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Attachment: |
| Earning: Approval pending. |
