C.S.£-Jmains 2001
MATHEMATICS paper 1
Time followed: 3 hours
marks 300
Candidates anempl QuesLion Nos. and 5 11 hlch arc compulsory, and any three of the remaining.
ques!ions selecli.ng at least one queslton from each section.
PAPER-I
SECTION-A
1. Attempt any fivc of the following:
Show that the vector form a basis for the vector space R3 (R).12
If lambda is a characteristic root of a non-songular mairix A then prove chat |A|/lambda. is a chracteristic
root of Adj. A 12
Let f be defined on IR by selling iF x is rational, and 1-X if xis irracional.
Show that f is continuous at N=1/2. but ls continuous at every other point.12
Test the convergence of integral 1 to n sin(1/x)/root x dx 12
Show that the equation x2 -5xy y2+ 8X- 20y 15
represents a hyperbola. Find the coordinates of its centre and the length of its real semi-axes. 12
Find the shortest distance between the axis of z and the Line ax by +cz d 0. 0 12

2.If

I
Show lhat for every integer n 3 •
• 4" 1
Hence, delennme
12)

When is a square matrili: A said co be congruent to a square macrix Prove !hat e1•ery matrix
congruent to a skew-symmetric matrix IS skew-symmetric.
15)
tel Delennine an orthogonal matnx P such lhm P-l AP is a dmgoJJal where
4 8
IS)
3.
Show lltat the real from
9 xi: .... •·· •xJ
Inn variable is positNcscmi·dcfinit.c.
Find the equation oflhe cubic cuiVe which h•• the nsyrnptot<"S
2x(
And which x·axis ullbe uri gin pMscs tlorough tho point I.
Find tbc maximum and minimum radii vectors of the section of lite surlllce
• .:1 tl i J b1:i t c1=2
Ry the plane f.y my • 112 0
Evaluate
JJf y d,<dydz
Over the region dt:tinetl by
x O. U,..t: • y t


15
IS)
Find tlte volume of the solicl generated by tl1e cnrdiotd r a(l co• about the
illi(i;llline.


Find lht c:<fU3tion or the circ.fe cm:umscribing triangle tumc:d by the point
o.O,U O.b.O O.O.c) Obtain also the coord tll3tes oftl1e centre ofthe circle.
Find the lt)Cus of equal co1li ug.1te of the eWpsQid

-· 1
li fl
Prove that
r 8:1 1 8y: S.:x 12.t l2y • 0
Reptesenls a cylinder whose cross-section is an ellipse of ecc<."ntrlcity 11../2
lfTP. TQ. and aU lie on • conic.
SI!CTION · B
liS)


IS)
5. tIJempt any five of Ute following:
A continuous func.tion sn(j,jfi e.s the difft>rential equ•t.ion
d)•f dt= . .

l O t l
2t 3f, 5
I lind

Solve:
r 3y- log.
ch.' dx

Flnd the of force to Ute pole " 'lten the path of o particle the c.trdioid r I cos
and prove that ifF be the lbree at the nnd u the veloQit"y there, then 4oF
12)

1.

The middle points of the opposite side. joinood quadrilateral nre c<)nnected by light rod'
of lengths. 1fT. be tettsions is these rods, prove tltal
1 r·
I I

A soHd rgbt ctme with sc:mi·Vc.t1iofil •n&Jc e1. is just imm.,•·•cJ in " li4uitl whh •
g.;n.,..!lng Hnu on the u· 0 be tllu of the wiUt Ute NSult.ont UmtsL
on the curved sur13c-o. pro'c tbnt
U>n0 . 3smaco•a
!121
en find the length or the a:re IJ! the lwi.<tetl curve r 31, from the point l n to the
point I. Find olso the unit tangent t. uniluonn•l nand Ute unit binom ial b nt l J.

Svlve:
rll' los v
-· •
d.; X • ••

Find the general sohll lun ur
ll.tl 0

Solve:

. y 24xcns.<
Given y Dy 0 1 r 0 •ntl 0 1 12 whatx II.
15)

A comet descdhing a r•rabob lln<ler inven;e S<tn•re litw •bout the sun. when nearest to it
sudd<:uly breaks up. witlt out g•in or loss of kinetic eueoogy. into two equal pootions. ooe of
which describes A oirclc. Prove Utot other will dc5crib" a hyperbolo ofi:<lccntridty 2.
15)
A pankle a mass M is at rest and begins to move under tile a cHon of a force In •
dircctioo.lf the of • sl>cnm of fin., dust moving in the opposite
dir..x:tion with n>lodty V. which dcposilll rontter on it ot 3 colliltJJnt o·at" p. Show Utot tloe oua•s
of the particle will be m when 1t hu tnwelled a distance
JJ
k F •

OA. OB und are edges of• of side n. and AA'. ond CC" are its diogonals.
'long 08'. O'A. BC ond nc.tlo rces P. 2P, and 4P rcspectil<>ly. RcJucc the
system to a at 0 " 'iUo a couple.

A rigbt circulur inder Ooating with nxis hori'tonlol untl in tht: suo:face is displac-ed on tht."
vt:t'fiCJ t pl:mt: lbrvugh the aKi.i.. Oiscw;• $1llbmty of c-qullihrium


a r a 3r
c11rl 3
a.r)
r r r
Vhere a is a consl3nt vector,
15)
Fiudtltedir<'Ctionalde•·ivativeof f >long .r e·' . y l 2sin l oosl ol 0

Show thotlhe vector tield defined by
F j 3x1
is irrototionoL Find olso lbe u such the F u.
15)