t-r,...l-l=!ar @drp [email protected]'t,'qtioc6trllg{n eegldiglb rr&6nfie pomi ot...

t-r,mt[[il

(l)

(2)

+2-PART - III

3321

601S789

[ 6]6tr$p DdnclL6fi5.n : 200

lNlaximm Mark : 200

oom'flpd / MATHEMATICS

,,iJ jeg b+,n6o 6qi / lam'l& tiBl+ v.^on.,

sotr6gl dl@r&6@6 snurs !61@rd L(ilnptr 66itrupotrsnutriligdr Gsrdrmqb s6,6!u6l6Ed gtrpuqgnt5].ir sr@pB6aitr5trdin! rsriL6 LLdL+ur6E Clpfl€Aise Lil

6oLn q6Dog aonq dDugdd $LOC@ d(4grtupion !LlnIO6FGd@06 ' d,6.n a6.ogj9 G d4.n tru66p(6

Che.k rhe question paper ror laimess ol printing. tr ther. is any tack offaimss, inlom the tlal Superuisor imediat€ly.Use Blue or Bla.k ink b Mite and pencil to .Law diagram.

ugd - e! / PARI-A

€xod9gl 6ljl@tr66@6(9ln 66laLL6ndi6. 4or1=40

6arOd,6n!LL pr€itro 60@Ls6fld dl6qLir 6rirr@Lu 6r9@L!9@d

66s,CpgJ gp lsr o 6i .616 , 6a @qj Cs-pgr . usJ-.

0 Arl qlstims ap compulsory.

(i, ChooE lhe most suitabla answo.riom the Biven iour audmrives and wlire6a opiion cod€ ad the conesponding answer.

(21

(5dn,.t , (,(,)

HSC JUNE 2013 www.kalvisolai.com

3.

(1)

(3)

1. J /c)d' +i

Itr

trt Iet 'lrrl I

trt Iet 2lrrt I

1

;+,

r.r j

r.l J

(i)

6di DOnq :

(2) 1

(a) u

t2)(,r)

12)

(4)

(1

(3

{ { =, .-, *6---mLpi6i6 p,1) ddip ,rndiue6rtoEs dorunuo6

cprocstr@Eond G$r@prain l

(2) 9r+3!+77-014) 8x+9!+72=0

or c.nla.i or tangenh rrcm (2, 1) to th€ hypetnola ,i;

(1) 9r-8r-72=013) 3a 9! 72-A

The €quation of rhe ciord

(1) 9r 8y 72=0(3) 8x-9y-72=0

!-


l- l=!ar @drp [email protected]'t,'qtioc6trllG{n EEgldiglb rr&6n

fie pomi ot inrerse.iion of tangents

mppLir ti r6ilp q.ndn6an6]5loEg dorunuoLi

at tr' md t2' b tlE paDbola f =,1ar is

(2) (at1 t a(r]+rr)

(4) (atj t 4ir r,)

r)= i[r ,),r.

rr) ; D. ib

Y-o

'i'(' l,r.(1) i=o-

rrr i=o'

(r)

(3)

(4)

;-d

(:.;).. [;., I ""i,r,

tl_i

1zt i=i

O6dLtr(qb

(; ;).; l;

or; rn,r i aft pdarrer

lJ


6i#p srdfl&(6 CDtbro

0) (2)

io o r-l

e. loroll, o o]

li;:lro o rl

lr;:

t"-,,,tlo o,l

ll;31

l: ;:l

,"1o ol

forrr lo

L1

(4)

(2)

[o o r'llo ' u

l, o o

li:IL,rr]

t","L,oo]

(4)


\ -.n, 90 p@ir6tsd

S.f)@ I(x=0)=r6q;l'

X N a discrete random variable

P (r=1)=;A then P (r=2) is :

(1) l1r

3321

1, 2 r&p 04nltsmmri Cl6r.n4lpgl

(1=216€itr DdnLr .

14321ol ""

(2)

(4)

12)

(,1)

o)

(3) L

wh,.h takes th€ ralues 0, 1 2 and P l1=01=:;

e dl=;; n6i.n.

(r) 2.y+y:+.,=c(3) rr+yr 2xr =.

rf dr = r+, rh6

(+) t1t769

(r) 16e

(3) r6e

@mooryz=G,)G b)r, a, b >0 oigr6 a > b adgr ddrupr66 6u@rp lOF(1) .>" (2) x=b(3) b<r<a la) r=arhecuive!,=(r-a) (r b)2, a,b >o dda >bdoes norexisrfo.:

(1) x>a (2) r:b(3) b<x<a (1) i=a


3327

11. 2i +3

0) a-z b=3,

o r&!gr 1 6d', n +Lb uqu@6 6an6(1) 1+or+oa+....= d+b3+05+.. .

(2) on=o

II o i! rhe nih bor oI !n,t, rhen :

(1) 1+0,+oa+ ....= o +d3 +05 +.(2) oo=0

-5

j +4k, ai +b j +.l. q6lL Gdr,r 'i6rn

Gsingiigr 6tdal i6mtrs O(9d;6

(2) a=4, b:4,.=s(4) a= 2, b=3,.=l

ai +b j +ck ar popenJitule when

P) a-q rJ=1, c=5

(4) a= 1b=3. .=4

72. "4riL) f q6s . u5l@ gL(i4triEp 6t!(96Li sdlogr ,l0rD,n cuDg l.G)6l@d66oirt!]6t /1q=0" ngrLir &pptrdsr :

(1) oMLO 6y')nlr1i cpriD6

(3) gl@Lmdirq 6EEr

(1) oE extrene value rheoreh

'lhe staiement I -Irlhas a to.al erbemum (tuinum ormin,muh)

(2) "".C1!iO6L cpipLi)

(2)

(4)


(1) 1 t.2) 7

14. (1+r')2-!''? d@!r do6&Gs(9i. sD6n!rL'+@ dfl6s ob$6 uq (!)opcu

(r) 2, 1 12) t,2 (3) 2, 2 (4) 1, 1

Tlre ordera.d degree ordt diffeMtial equailon 0+fl'z=y''?are:(r) 2,1 12) 1,2 13) 2,2 (4) 1,1

rr+yr+u2 6r+3v roz+1=od6t9 G6mE46n oDUL! 6ipr'n +[ln goopc! :

(1) ( 3,4, 5),4e (2) ( 6,3,-10),1

(3) (3, r, 5), 7 (4) (6, 3, 10), 7

'flE centre and radins oI the sphft x':+y'?+22 6J+3y-102+1=0 are :

0) ( 3,4,-5),1e (2) (-6,3,-10),1t.3) 13, t,1),7 \4) 16,-3, 101,7

9r! 56,flud;6n Glungondr onselip DOnq 36D6ln$(n (P) Gpidl6lprj9{i)

6msqor6lpq. 699(9 srrp od666l5(4!. 6Ddur@ (k O@p 6air) :

3321

ctrn$lOg Gld6[r@r]ugldr

@%tunctionl(r)=x,+2:-1 ; a=0 ;

1r) *=llactive el€ment dislnteSlau at a rate FoPortiolal b n6

cotresp@dins ro rhe above statemenr is G is negative) :

?

a=0, b=] 6dd 6l6trdin0 J (r)-x'?+2r 1 ndtrp

taoLr6l!1]6 Gs,6Dli6ld! alt.nm '. ' g6n Ddnq l

(1) 1 (21 1 (31 0

'I1r valuc oI'c' of l.agrrngds ma,n vahe rhorem lor lhe

(r) $)%

lr) a:F

E=\

,,,4s=r, (4)

Tle asouni (f) present in a Dd,oamou.i. Thd dLff€rdntial o$.tion

(4)J, =" !P=re $=r(r) (2) (rl


l32l

17. I(A)=r 6an6 sdrdodd rg€n mgr &ji ?

(1) r oRoEuroLU €r@d6Ar dtDDni6 c6tr6d66tznOoe6rgr

(2)'A +@9l g@pEpllsLir eO rdd@j SddtLb,i, jipor; c6rotuLtrtugrOu6rio&(5o Digr6 r dnoa6(9 cDord c{adFgr JeEanrr c6ndds6ft6itrb6!qb $r$rors 6r9ijgL_1)

(3) 'A'+dgr OdpESULsLb e(9 (rlr) aflo6uroLL ;nlaE csnod!96;o, !" DdA\Jsob o. s, Gl.. l

{4) aod69 t+1) dfrmE bepro €r@p6gL s66otr-.Jnn, S6rdirL$E6tuoop ctipain6 c6trdd6.n grod(gd

II P(A) = r dren wh(h of iie tollowing n .o(..i ?

(1) all ihe mino:s oI order r whrh do nor vanish(2)'Ahasatleastonehinoroforderrnli.hdoesnoi\anrhind:0hisherorderninor

(3) 'A'hd ad€ ond t+1) order drnor whrh va.ishes(4) all t+1)and hisher or.der minoE srourJ n.r vannL

f,Ein!D ltiJL!E66'r5

(3) 10 n (l) 30 nrdius 5 intd.epred brh'een hro p:r.lht pla.es ot

13- +[L! 5 Lino c6rd6mp, pminsrn @DL6g65](9Ep

e,r 6f a ;lc'6i a q 6,dm -o pr.s. ,

(2) 40 rThe.!tued surf.ce aiea of a sphere .fdstan e2and 4 6om the cenhe in rhe

(2) 40 -

1e. exr+,!:=1Bo 66nr! 5.ndLLE6l6fi g6ltLE6(g6jloLCL 1nn €lFn6o6!:(r) 4 (2) 6(3) I (4) 2Tho disran.e betueer ihe f.ci of rhe.nipse 9rr+5d=lS0 r :

(1) I {2) 6

13) 3 $)2


9

6d dod6@69rn 6$ 4nutrdp{ o ?

(1) z1+4< \l+ lzll (2)

13) 1\-z,l> a - z2 (4)

Which ol the following is incoGt ?

0) zr+zJ< jzll + La \2)

t3) \21-2' > z1 '2 14)

3r2r

<lnl * lal

2r. OO p6l6oo sDdrunq odl :

(1) (pqdp!, @6indAeoau9ord odlL:rtrsddn GugdlpgJ' .

(2) sdl.OLL Bo @oLa@qflullgldm nddtr @6lnq6omrltb

1! 66i@la-dN D4ur$o6 n 6Jd6r$.OUBBp$.

(a) eO Ooqqop sroogJ d@r@,flL59&a 66lnll6@@u

A discrete Endom variable takes :

(1) only a linib nmbq or values

(2) all lo$ible valu6 berwean .dlain glven linits

(, irfilbnutrbdot'Le'

(4) a linite or couhble nmber of Yalues

r d.irp s6on nq Ddhil€a ufdplq a.G6grLb +ttran 2 d6fl

()2 (2) 4 (3) 6

varnn.€ of the random lanabb 1 is '1. Iis mean G 2 Thm E ($ is

,l - ) ,_r 4

Gl!gdpgJ. i:i i

.6nq


r;)

,;)

,,. ;=( i-,i..i)-,[ ,ir;

; = (,; -.;.,;)r.(i r,; -

(1) \2, r,1) (2) (1,2,.t) 7,2) (4) (1,1,1)

i..i).,[ ,r,i.i)."G) n,

Ths pqnt of DteBecrron of rhe trnes (-il

zt=4+sr,zr= 3+2 ' 66fld) aL

(1 1. 2) lr) (r r.1)

; = (.i..;..i)-,(i -,i..i)-,(1) t2_1,1) 12.) (,2,1) (3)

u,+sror son6,-i El6h2a. Cluo6r6.n dgou! Gt!rur5$ Ooordlu edtr6r6n promb

(1) 4 (2) 3 O)2The ordd ot - i in rhe mlripticarive group ot 46 tuois ol niq is :

(1) 4 Q)3 (3) 2 (4)

(4)

i ,, i*?'

A' @ tr*Zt

ol ..3-fit e)

If\=4+5i,4- 3+2i

2 22.13 13t

--223rr) li ii22

r4 ;i+1' 0)

13

(1)


26' ;;iF =

27. ltD)=lD 4 EtD), sG) +0 @6.fl.n do6dr6l6(4, +D€itrutrGl /(D)y=ee €itr 6Dnqp6n6t l

The pa*icular inteE,-.al oI

(3) s(a) e- (4) 8(d

eituation /(D)y-er where I(D) =(D-a) r(D)

(4)

(2)s(.)

(r)

23. EDdtruuq 6tr 1 r6(n urdd"trirr F(x) eO :

0) OpEqLn sriq(2) sopur (6pd6r) afliq

(3) btr laSld atrnll

(9)pr6l.i, dgD l9{ntrr opEgrir smnlo)

(1)

(2)

(3)

(4)

dGhbuhon fnn.tion F0) of a random variable r is l

a non deorasing funchor

ln.,!asi!g first thcn de.reasing


3321

2e. I3l+u (sl+1116) 6n DSlnr.r :

(1) lot (2) trtrhe varue or t3l+rr (sl+ I6t) i5l(1) tol (2) I1l

dr O. 66lnq:

(r) t2t

121

(1)

(1)

t3l

I3t

+i aclin8 on a parti.le, dr paft.le is displa@d

(4)

(3)

30. y=r' d6np domdofugld o@m6] btrEgrn Lt.n6n

(1) x=0 (2) r=3(3) r=12 (4) d6rol6l6@@Ihe point ol i.i.ction of rhe cune , = tr4 n at :

(1) 1= 0 (2) x=3(3) x=12 (a) no where

n.n0 6El@s. A19 g16@o A (3, 3, 3) 6€!ri 6oou965loEgJ B (4 a,9

Fet96ldtr$ .9rcn60@e OELirqd Go@@un6rt :

(2) 3 sd196dr(4) 7 e'do6d

=i+j(4,4,4)G

cr(.{)

%32 I

tr, 'l\1) -l

erlo) "l

(ll

h

(r) (4)


66i, '-' F(D@iLt+ Glau6&utrsLir

(1) N (2) 0-{o}' ' is a bin ry opdation on :

(1) N (2) a {01

(3) R- {0}

p) R l0l

(4)

(4)

(2)

-1 1(3)lt$i

The slop€ of dE n(lml to lh€ @e y=3r2 ai thb point rhose a coddinate is a is

rI(1) ii (2) ra

bdlnlgL Gdair+u (9d@ orrl&6fl@ s@66p U.ii8r Gpnu 3sr!+6prd,D.ir!trLOEj Apr6nl96n A=0 @Egb

^x=0, ^y+0, A,=0 a#li, GlptrOnLrG6trdE

6;q ,

(1' 6rcr eO 6-6! (2r Ooaho 6id6o

{3) n€iroa,iim6uirp 6n6ts{ft (41 6iq odotrmbIn a slslem ot thEe lined non nodogEneous equationi, with lhree unktuwtu, il A=0 and

Ar=0, Ar+0, A? =0, thm the sysrern has :

y=31d€irp domdorli;(q r rir +urbGg'rooq 2 d@6i G6rditrO(ild qdrdfl!96!

c&dcsLq<ii erudrdgl :

1

i,,t ,rl

$)+G]u

(3) inrinitely runy solutiore

E


(rl

(r)

(1) (r)

c)

o)

(3) (4)

(,r)

q n.itrp str69Dg@ osLnLLL G6trtrLir

;i

t;i ;

;; = q 6.onn{hd b\ the

lil3

ljl

3r. i =; rt;

H

(2)

.i',', = 3.I;

;ir(2)

(.r) ;r(


15 3321

1=4 d6' : st,i@6 eri!:sr6 dd65r38. y= Ji;7 6.hp d@ndo, ,=o 66r66toEsr

epDDn!016 SlLn6l!trOdfd @sm6! :

Gl r0o n 100 100 (4) -.(2)

(2) (r)

(3)

100

3e. eb@flqdL! c!6r6ooLil OdDdpdo s4idl ceuu cdai,Tu 66r6urdsr ,

(1) sroLnlr 6{]0 (2) CEnlLt 606

(3) cDal) 6061 (4) n6tji u6p6sld

A mndd bMes a Ercup if n abo satisEes itre :

(l) .l(Hre arbr (2) Nociahv. dDm

(3) rdahty d<iom (4) nNffi dj@

40. r, 2!+3r-23=O a.iiD ![domui6]dr

Th€ volune vn.n rhe dfley= J3+,, froml=0t6r-4jsrraledabolr,_an!is:

(1) r00 i

(1) y= -1(3) x=3

(1) y= -1

(3) ,-3

(2)

(4)

{4) Y=1

100 (4) -t


{gdn'l ,

16

UOS-ES/PART'B

n6dcugr6 !651 6$6iri5@6(9 661@Lu6i6i66l6.(L)

(ii) 651dtr n6ir s5 t;6 dffqn!16 6OoLLqiaa6l6, Ap

661dtr;d6n60O.ieJ 6/cpg!'n eih!51 6s1trtr66€DaO 66loLU6id'56rtir'

(i) An$ver.ny ten qu.stio'rs.

(ii) Question No. 55 n .omP!150ry and choose anv nine qnestios rrom tie

.9reild;Gsrdd (lDtrpuq6n Si65 :

S.lve by determina.t nethod :

ls il'or"=fi,

ll """'=[-', ,'] *,+ *,1,0 u1''=" 'o-'

,'l .". ,^ Br =3 | \ lddup@d sR

t;

Gld6Lt grop@! uutr!GF6. C5rd6$l6h 66lLL6 GDp!,n6@ 6/GpglLir

BO Lrd6n!5ld q9!6FAD C6r@Lb Gl6EG5tr@6 dddr 6rL66

Show tlut diametei ol a sPha'e stbtdnds ! d8ht angle at a loint on the suifacc bv $itr


a5. 66ir dO6 sb6iiLrLoL 6t@D6! 6t&uqLir r bugo i !9d 6lDU D9llr6dmd

{1 ,)r +i +r) r/: r-r

t, .- __ _rl, o o ..-"1J=' t.b=l [ !=l

l------,1rr;=i,; i=j-r;=l -;.

Fi,rd the rcal ralues ofr dnd(1-i) x+(1+i) r=1 3 i.

!/ for whi.h dre lollowinA cquaLior is sariltied.

(,r)+rsk, d@ 6l(}6166

For any Mr comple! tumtras zr and:.,prove 21zr = zr zr.nd €kr:r)= s(zr)+arg

47. @ouLb ( 2, %) bbs. lt, %)*@y,,1,;,"t1dd6 Gs6s6 dLLi G6dd5

ar6!rd6mu59.n 666itr!nO 5tra{i6

Find trre equaiion ot thc standard rc ansular lypclboLa wh(xe Gnft c {_r, -%) md

\hrh pa\Ys rln.ugtr ih. pn (t -%)


!13,

(,)

(,)

13

(,) l(r)=h, 1<r<1 m.t9 6r;qirg c[r6!.il cFipEmg Enltri66

dmnddtrdltr L5c.rr g.6t.rr FrrL6s.a65 str..r5

vejiy Rolht theoien rorl(r)= r,, t<x<l

Determi.e rhe aomai'r or.on.iljiy (.onrcxtr) of dE.nnp q=2 rl

@dlt!6tGl5: \/J

rr ,r. I

E,otu"te. rtul ,r I

DdL6Gr6: .),'

(Dr+ r)! =0 d.itrp do66ctsed,

cDe6 i=; ndnd, v= -2

6Dd!trL6Lp 6trse FFrs r=u. -nd r=r

i'

solve the dilrcrenrial dquarjon (Dr+1)L/-0 r,trer x=0, {=r an!r rhn r_


52. i DLi, sr-Lio@6,1!j atjrii,arr (r, ^ t-q) v ((_r,) r q) rnp &6!1 G)D!@DUf

eroagl rar-ritr!tur.rdd s,.nrAu5e dn, rrlrrr trti e !) lrerer nrnle $nclher fie sritencnr (p ^ (-q)) v (_p) v q) is a tautotosy

(i) r1.\e rhl " \.nriil etefrert of a Aronp1L') Pue ih rlic inld\e or ea.h ol€mpnr

anLirEFA" 611519;4

dEriD6D6Ln 6iuEdio6ELir"

" o. . p. og D qo j6,!J.fl parpEd ,t

.m6dr@o tr 6Ci ,.Ap@e,

IhPFrcLrhl,lr.i{..Bsoi.rnerenrispandthalofltrituroEq frnd thc expe.te.t number

oIh,als to 3.r i hrstsu..ess.

9!! 6lprltjatro dtd {p!E6LLtr6Lb Ftr+a!r.n66i.n 20%(9@rr!rL6;r(ilmtr 1() 9trpn!tr.i6.n ebdtruujr (!16pu96! dGi6n!6Lb cutrgreflLrs 2 Frin!tren6.ir g69ql.h 6t(9l;6 FOprn!! lrdd D6pJ6!trLEr.i !ra.1! ilrdorg C5rp4q 6r6itr5.h ,=013531

3r.noaa66dos-i! !u.n!66d (1e7f 6@ cprrtru b6n@! 6r@i,@$bdFrd93D(9 6trGd06lu " d 1rr b,nG pro.iuF.i r i h.r.,), .u! unLl ro Lrr !1erd.vd rnrJ the p,.brb iq.l :"' "' ' " r r" ' J'."-^'rb"Jac' " .

ri: or" t. ou

OR(I.) Irftl .n .$r.\nnarp rrlue.r (1916, usrngd,arereniiats ro ra,o

I


u6!;l- 6r / PARr,c

Odlnq: (, d@dcugl6 liii5 66l6rt6@drg 6$dLU6nii56!rb. lorr0=100

(r) 66ldr @ai, 70 69 6ahqnur6 6$oL@6fl6s6trn jll9

A6irugl 6$dr&6(E;Ci60dtri66i60o6sr ocggl6

AEwe! any tm questions.

Q."s-or No. 70 ..ompJl.oD r'd Jr q rn) nine qua.,6 ftm $e

57.

r bDgrin p 6(itr dLi,Ddnq6gt(g x+r+u =6, x+ 2r+3,= 10, r+2y+[=E 6diD

(i) u lptrO 6i6r6 GllDrlrrgr

(i)

(rtt

9G[ eO 6i@dn C]!0d1o6(q6.

(D

(ir)

66iflmi6@6ubp 6n6,r5ddn GuDdtO66Li.

n€irlp@d g[ (!!6Du.9l6i +[tru&Invesrigate lor what valuos of I and p rhe simultaneousr+ 2y+3? -10. r +2y+ \.= p have

6) a unique sorurion and

(ni) an inlinits nun$e! ol $rudons, by usins Enk merhod

(A-B)*(E a cos B+3in A snr B d@ 5lgJ6ts.thal cos ( B)-cos .os B+sin A sin R.

53. ( - 1, 3, 2l d6np Lr{n6i ag5li Cts.ndgrb x + 2v + 2u = s 6iso 3r +i +2-r ql0upmrn6(€966t Gsdlgiiprdgrbr@ Sn95d Ctu;Li 6tig6 itj*x5 sD.nur056md srair6

Iind the vector and .ariesian dquarions to thz praR thman dE Fi ( r, j, 2) andperyendicular to the planes r+2y+22=5 ad 3r+!+2:=8.


5e. P dgrb Ltdrdi @loueir Drli z e6 odiitprd ? 6{n 6ubirlr6pou sh(22+ t\I r-r I ' " oD jrJrtPooigr u OE 6rd-r

-repiEerlthev db'ecomprerrLlba, ErdrFproft.or * [i:;+)=,

60. 6yO dtrrn 606h166h +d5) efiLodii eEd u@omun lr@pugl€i 6t"66lp$.big6 6flL6ir ur@odu5$€itr (960!Fd{n smodtpsr. dr{n 606i',56tgrflu611dloEgJ 80 L5l6)dud 6.6 Cpr6@6d16 c{6oti5l 6rOdjOL! cur5l dr.n

66lai 6md-E &lu@trq@ @emaoo c5,@ u 6p €rssLd IG5tr@Ep6d 6ri!665l6trtu6.

(i) drd) 606h,-66idr !trdplad tu&lrLoL& gtraifla.

,,i' drd 6$,imi.ir Of Jg4O ndr@d6r €OCa d-qr+, b

Gr6i.r6. (!r@O odsrLre6 dDuu6uptr6 6t&trfi5).

A comet is tuvin8 in a parabolic o.bit arcund ine in wljch is at rheWhen OE.o@t is 30 million Is 6om the sur! tlE line se8ment 6om

hat"" r. .rB.cor I rddift Mdl dr.Eof 1Eorb,l. f nd

(i) the equation or the comet's oftii

(ii) how close does the conet cm. neE io the sh ? Crake the orbjr as open righl

61. l-y+r=0 d6irp Gpncdfo 6drdLLL! f+3f=12 ag Gt5tr6r c6rlr5 Ldrns6o 90360 GDgi JnE(n Gpio Ldef@r b 6.ai'6

Showrharrhelinex-r+4=0i5arangenttorheeltiprerr+3f=u. Find also the coodirar€soI the Poinr df ontact.


52. a: 4u2+6i+16y 1r=o n6np 3rFu[ mdu6$d 6DUF C]6roofp66,r.

@Ddr] (q6!iuri,6d GD8rrb L,+46@n5 6trad6 G6Brro tu@fl@6r@L

Find rhe e..enhi.ity, .ent.e l.ci and vertices ol thc htp€rLola r': i1y:+ 6r-r.15r.'11=o

63. 6l6r66ia!!LL 90 &Oo@69@tr6 6l6r6irL Gs.no6|;l5@fl Esr6 ULOGD

6!(q6 !rnln@ddj Grar@qo6(5irb ntri snLG

shou, thai of all the p.ianglas with a glven perimeter, ihe .ne nrh ihe Sreai.n ar.r is a

5a. Gd66FoLou Cla956u A6hd,i eg dtrBda, 1' c0drq66i.n 6l&ngl6 gnfLi)

i s. \=tu Ze n6np eD6filrLL(li) grnL66l!gr.66.io

(i) cd66p@L CrsgrEpnlLL dnsd56ltr GosD (61 6i D6i)

(r) sr.ndtrsd6 Ca66 5l6a.ij(q d([email protected] s5l sLpp gn[Li]

(i) the speed or the vehi.le (in kn/rnj atihe in*ant Lh. brakes anapplled and

+6:lLdp@ps sr6ir6

lhe disian.p r meie$ tuvelled bya vehicle in tnn.'t seconds .fter the biakes .rd aFplied is

siven b-v r:2ot 1,,. o*"--g

( I ih, lLtan.e thecar b^elled bern'e riops


1.+, 1 ar"=( n l- dafL{n r- +

\!,+nyl [=+J""[++J**,*1

i l;+l-.1='r1" ',.t;i+lvfu=

65- r=a(r sint),y=a(1-cos, d€irp doddo[u9.n S@666d t=0 CA'n t=?

Find the lefgrh or the curue r=a (t- sin t), y=a (1-.osi) b€tween t=0 and t'

67. eO 651flu5&n Gllrodr rl@pul6 Dtrg6]spDtr@gl. rrp.n n6DLri6 66ClpDtr5

sroop$(nngr. srp€itr 66L 10 r&.6)rrL, +5 go{i;66 GLrg Adgq'n otrgl

69ELir Erclntr.i(8ng oo51 dl.drtrb nald slgid 6oL 10 61dlltrdla,lopgJ 5

dl.6rror66 O@pu d@Fglri Gl5r.ng6 6nd snoda 5rait6 0osJ =rl6931)

A ra.tioactive substdce disEtegrates ar . rai. PioPoriioml to ils mass' wher ils ma$ is

10 m 3m, the ratd of disinieCHhon is 0.051 m.8m Pei day Hot l.ng {ill ir ia*' lor the mass

Lo bo reduced tuom 1o6.gn1 lo 5 m.sh. (loger = 0.6931)

,,' ll l,'.- lor d6tD sroDn.ll'n L'ntr srd'n6d !trqLi srLriJcr! 6md c

+asr el6,i.Gl!O6i66,r@ tF eO l5otn 6@dr 6trL05

show tha*heseiGof a, ma.r,.", ., *" t- li l] ;-.n-rorr".e",o-.a**"t'*

li=a


3321 24

6e. p6S6 6iEr.6gr66a!n Gl!trOFpnLOLir &it6[niJ66aalbijs E@tunq G]@pu9€ncpnE6p@*6nuoL! sri6r66.n 6tr!tp(95p6 Oud' doo! Lrd@oOgE(D66lp5l. Jlpd 6trip(9Ep eftrefl 31psj, 6LL6e@i6Lb 02 psi o6!Ri)4bdruuq @@pu96(, 30.5 bDgJtr 31.5 Psi @6LCu(ii) 30 bDg6 32 lsi 6@LCu(1D 30.s psi ,n(q Gb@r6

(b)

G)

&ie[6d]d a@peiip6 g066 616&g6q6mn 6tr{nr& *ir6P l0 < z < 25 )-0.4933 oliprnl

P [0 < z <s ] =0.5000TIE air pBsue i! a ra.lomty selered qae Ptt ddislriblted with rean 31 psi rd stanildd devialionpressft fdr a radonny dect€d t ae.

(i) ter{een 305 dd 31.5 Psi(0 netween 30 and 32 psi

Here P ( 0 < z < 2.5 l=0.4YJB a.d

(a) y=sinx bpp6 y=.osx d&p d6dd@[56ir, '=0,

oigLi r=. qiDG4rOs& +dlu@Eg&s @oLcu sdrn g{rlils66ld Lrnoui {(i&

srddOJ

.onput€ rhe dea letween the tures r=situ and y=rcsr o.l dE lim r=0, x= ?.

a erhi. dE e i malry0.2 Fi. Eld oE Fr'taDiu9 that tE

@)


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