I C.S.E-M nins 2008
MATHEMATICS

PAPER-I
SECTION-A
1. Attempt any FIVE of the following:

Show that the matrix A is invertible if and only If the adj is invenible, Hence find adJ(A)
LetS be 3 non-empty set nod let V denote the set of all functions from S into R. Show that V is a vector space with respect to the veector addition snd scalar multiplication
Find the value of hmfl l .l')cot r.;r
•• 2
12)
I
E1slua(c J d.< ..

Tbe plane 2y+3z =II is rotated through u nght angle aboul tts hne of 11lersecllon with the
plane 3y -4z.-5 fmd the equation of the plane tn tis new posttion.
12)
If) Find the equations (in symmetnc form) or tbe t1mgent liue to the sphere >.1 1 i Sx 7y
8 . .3x 2y +4J. 3 5. 4).
Itt)

Show that B 0.) is a basis of R3 Lei T . RJ be a linerutransformation
such thnt Tl 1. II, 1. 0. 0). (I. I IU1d Tfl. 1. (I. I.
Fmd T(xs. z).

Determine Ute maximwu and minimum distances of the on,gio from cun•e gtven by the
equsriou
3x! 14(1

A snhere S has pomts I. at opJ)ostte ends or a diameter, Find the-equahon of
sphere intersection sphereS " i lh the plane 4z 7 35 a £Tlat
ctrcle
Let a be a non-smgular mnlrtx.
Sho1 that If
I t A-t A1 .. . .... • A" O.
lhen A 1 A"
Evaluate double mtegral


4.
tel

II
.ll.x
by sing I he order of inlegntliun
20J
lf f"'lfOS011lQne uf a $CI Qf 1 2 3 -· Uu:ee ·,·.-..·.y,e udlotilnt of the "'l11"
5y:L 3xy !l.
rind the equaliofl!l. ofth" two,
Find lhe dinlotlSlon spnnnod b.Y Ill,. Set
1.2.0, i
H;,nce find a Iori he
(201

Ob1:1iu Lite volume bounded by ll1o p:orabo loide gil .:n bJ tltu equ:ttious
and z 18
Sliow ll10t ll1e eylindi:t'> of l11e ellipsoid nx= by1 r cz1
p"'l'<"dJcubr to z muct tho z 0 in parnboiOJi,
SECTION.a

with generators

I. Alt=pt :my ofll>e fotlowing:
Sol'e ll•e dill'<m:ntinl equation
ydx • ..ll Jdy o.




Use th.e method of vnnatum of paramet= to find lbc gener•l $olu1Jon ol .._ :y. 6y -11 )C.

12)
A lUbe i• plocc:d with vertex plano. A plllticlc <lidCII down the tube from
under the inllu.:nc.; i>f gravity, Prole th:ol in I he cif thu tube is e<1u:•l
to 2w h
1
" Whore ·w· 1s I he wctght of lhe pttrtlcle. ·r' the radius of ClllValme of the tube,
it.s I rectum and 1l•c 111iliRI Ve>1loal beighl of u,e porticlo abuV<: the >CJ'tes of tl10
lube,

A • tr•igbt uui lbnn beam loogth in lim iting cqutlfurium. In conllU:t wilb n rougll
veniCD! woD ofheight wllb one ood rough borlzon!Jl) plnoe Rn<l 11itl1 U1e olher end
projectmg heyond the wn I L lf bolh the wnll and tho plane he equa lly rough. prove lltoJ ·r the
angle of lriclion, is given by sin 21. ·a· being the inclin>lion of the beom lo the
hocizur1.

Jlind !he coml,linl 3 nnd b !hot the •mf nce by1. 2 will be ortltogon>llo the
•urfilce4x' y zl= the llOinl (l. 1.

Show Umt l j-.h z1k il a eonservntivo force field. Find the s<:alat
polcntllll tor F •nd lhe wn1 b. douo itt muvingllll objeo.t in lhi.•field from 2.1) to 4
12)
6. 1 p3rttele moVel in n plan.: such that il acted on two consmnl v<:lucltle$ u ond v
altm.S llte dircclilm 0X 3n<l lhc dircclion p¢11K'ttdicular lo OP. 0 is
7.
8
some fixed pom1 •. thai io origjn . Show lhnt the pud1 hy P 1s. o come seCLion wilh
ol 0 and 1:CcenlnciL)' urv.
Using L<!plru:c tr.rnsfunn. soh" initial valuo pr'ilblom
l• 2y 4t
with y!OJ I.
!ISl
I Solve I he dltl'crenhnl equation
f ·9' sinlin.'<J L


A part:ic'le of m move• under a fqree l(a1 l. u .!. • b. n. nnd
r
being consl.ant•. tl is projected from nn apse at a distance • b with
Ji o Sh01 jiJ; ls given the equntkm r a 1 1). where Ito are 1he
a-b
plnne pol :or CO()rd inale;< or A p Orn I •
OS)
A ollclllying in 11 •tr•igbt <mooU1 horizoniJ!l wbc 6udd.mly b•-o•ks into two pnrtiom of mas.• e.<
m1 and"'" lfs be the dist.tnee b•llweert lhe two ma.sse< inside the tube aft<>r thai
lhe walk dune by lhe expto.ion can he wrillen 01s I Ia



1 m1m.r t 1


A ladder of wefghl tO kg. re>ts on 1 •mooU1 hori7ontnJ ground le:ming agains1 •
,o,:rlicat walt nu inclination lau · • 2 with U1c horizon aud is prev.:ntcd from 5lippl'n,l by a
•Iring oltncbod ol ils lower cod, and to the junction of tho flour and the wall. A boy of weight
30 k.g, tu tl1e b dder, If t·he string can bear a ten< ion of 1U kg. wL, haw for along
I he ladder can I he boy ril;e with s.tety?
15)
A sol id right circular cone who•o buighl i• b and rndius of whase base .is i• phced un an
inclined plnne. It prevenl from sliding. Jf the inclination U1c plonc 0 (to l10nzonul)
be decre3sed. fiJld when the cone will tnpple ova. " hose
angle is 30°1
detcm1ine the critical value of tl which when excc.::Jetl, the cone will topple
UVt.T.

t)lut 71j rl r 2 c!f 1 • r I y I • 0 hnd f1 llllll 7"f1r)
dr rdr


Sho11 tML roo I he space curve
X • l 11
• 7. l
1
•
3
the oun·alure :and lonuop Dre n .mc Jt lWery poi:nL
Evalunle J alan& the cmve ll! 1. z 1 !Tom CO. IP L. if


Evulu3tc JJ F.ii tiS where 2yl J :•ir nnd S •• the of lhe cylinder hounded
• byx1+ 4. 0 :andz 3.

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