--- · 9/12/2013 · 3627 i I I I 4. a2 u1-; = I 6rdnp 6i6n6rT6.r6i)[6oru lgSp+Cleneoru Glung$gr...

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3527

t.

(2) !z-!t:o(4) y2-2y-t-2y:o

(2) !z-!t:o(4) y2-2y-t-2y=0

Sl rqb+ni$p Gl u gLoLb Gl uSflgu rgein rsuGlug Gt rqgogr6 Gtu@fl grigLb.

St rq!+nrigp 619 Lo Lb G1 u f fl6u r9l ein r.6ri E g EE Lo g: 6 Gt u[fl6ri6Lb.rf u G u 19 Gl u 19 Lo Lb Gl u gfl 19u r-91 ein $r r_qD+rn$p Gt u g Logr Lb G u gfl gri;g6.rf eE gr Sl g Lo Lb Gl u gfl 19Lr Gl ein $t rqlenri$p 6t g Lo gr Lb 6 uf fl 9696.

(1) (a) LopgLb (b) (2) (a) LoggLb (c) (3) (c) tog96 (d) (4) (a), (c) LblD&'rb (d)Identify the corect statements,

For a continuous funcfion

(u) if it has local maximum, then it has absolute maximum.'(b) if it has local minimum, then it has absolute minimum.(c) if it has absolute maximum, then it has local maximum.(d) if it has absolute minimum, then it has local minimum.

3. eflun6dr a.fgeenar Gpri$Glp@ria

gg Glpnrri+E u-r nem enriq :

(u)

(b)

(")

(d)

(1) (a) and (b) (2) (a) and (c) (3) (c) and (d)

(2) StpainrnLb eneb u1g$l

(4) pneinsnd sneb u1g$

- z qgrehp nb s nel u 19$ u9lel 4aeoLo$pneb z elarLoqLi snei.rulq$ :

(1) gopeb aneb u6$(3) geinpnLb aneb ug$1

If - z lies in the third quadrant then z lies in the :

(1) first quadrant (2) second quadrant(3) third quadrant (4) fourth quadrant

2. y:ea(Acosir+ Bsinr) ereinp Glpnrrirlleir A ou.rqrb, B ou,q6 $ri6u Glupuu@tbotelrsGlsqgri eLoeiru n@ :

(1) yz+ yt: o

(3) yz*2y1+2y=0The differential equation for:rned by eliminating A and B from the relationy: ex(Acosjr + Bsirur) is :

(1) yz+ yt:o(3) y2-2y1+2y=0

(4) (a), (c) and (d)

HSE SEPTEMBER 2013 www.kalvisolai.com


3627

i

I

I

I

a2 u14. ; -; = I 6rdnp 6i6n6rT6.r6i)[6oru lgSp+Cleneoru Glung$gr +pfgLb G:ngt

Scrr6g6 $lruGunqafl ein s6ur<9{6n6q :

(1) 48 t (2) 64 r (3) 32r (4) 128 n

The volume of the solid oblained bv revolvins t * "

g G : I about the minor axis is :

(1) 48 tr (2) 64 tr (3) 32 r (4) 728 r

-, --) -+s. 4i- j +2k g$6inrf5l3i+ j-k$einoSpetu:

9 -9 81 -81(1) lfr (2) 6 (3) 7i (4) Jn-

rlre proir.:tior of :i + i - i* 4i - 1 + zi is :

9 -9 81 -81(1) A (2) A (3) A (4) A

6. (N, l.) grd x*y:maxlx, U) @cb Clurflug, x, y eN, oroflel (N, *) ereinugr :

(1) +)l6oL-ul1 or9lp Lo-@Lb Clu,rg$grb

(2) el6o[6(gob Lor@Lb "ry196

(3) e LD6rfl r.16o Lur .g6n[6(ga;ri LoL-@Lb 9196(4) B@ 15606

In (N, +), x*y:max|x, y\, x, y e N, then (N, *) is :

(1) only closed

(2) only semi group

(3) only monoid

(4) a group

[ $]Ouq* / Turn over



362\

dx7. m < 0, -%6 $gurSleh dy +rru=0 $ein $ne1 :

(1) x=cen! (2) r=c e-ml/ (3) x=my+c (4) .r:c

dxThe solution of 4, +rnt=O, where m < 0, is :

(1) ;r:c em/ (2) x:c e-mv (3) x:my+c (4) x=c

- -) -+ l---r --+l

8. p, q Lopgtlb p + q gdluaoar tr a-ainourenoq Glsneubu Gleurirrisornu9ldr lP - qltt.2t6r151 :

(1) 2 \ (2) .,6 r, (3) Jz r (4) L

--) -+ --+ -+ i-+ -+lIf p, qand p + q are vectors of magrritude ), then

ln - 11 is,

(1) 2I (2) J5 r (3) .,8 x e) 1

drrf . i; -y tanr=cosx areinp areosliGlaqg$ +Loeinu nr-rqeiir Glpnarats 5n[6r0fl :

(1) sec .x (2) cos r (3) etanx (4) cot .r

The integrating factor of the differential equation f{ -/ tu*:.ou i, ,

(1) sec r (2) cos r (3) etanx (4) cot r



362L

10. jrir,l, .*3, dr $rein Lo$)uq :

0

(1) n (2) % (3) % (4) o

The value of Jsin2x cos3r dr is:0

(1) ,,, (2) % (3) % (q o

11. g1g saru Glu ermad ein eS&o Tf erafila:9$g arenr :

(1) CpDdgttb afuoar crain (2) Cp$dfgtO Glou: eraiur

(3) 0 (a) GLoutqLoeieu, ssuatemqLoeieu

If tlre amplitude of a cornplex numAer ts /2, then the number is :

(1) purely imaginary (Z) purely real(3) 0 (e) neither real nor imaginary

72. +Lourq$pna'r Gprfru* +Loeinun@aofleiir Glpngurgreb p(A) < o nfle oflein erainaufr&aos,er.euflai Glpnlgu u rradTgl :

(1)

(2)

CloLrafl u ueor$ $noq tor@GLo GugflgrigLb.Cl euafl u u ar r$ $ nq topg Lb erainmfl 6eneu pp Gl erafl u u ao rur f p $rieqocjrClrfflgrigLb,

(3) ClouafluuearruSp $ireqsdr Lor@GLo GlugflgGgLb.(4) $fqseir GLgflg6en51.In the homogeneous system p(A) < the number of unknowns, then the system has :(1) only trivial solution

(2) trivial solution and infinitely many non_trivial solutions(3) only non - trivial solutions ,.

(4) no solution

[ $6uqs / Turn over



3521 6

13. a=0, b=1 6r66r{i Glsr6iinGt ,f (r) =x2+2x-7 a'einp enri6l$g Gleuri g nq$flu9lein

StarrLo$ Lr q$ Gppp$$einu q p-eiror'c' ein Lo$ u q :

(1) *1 (2) 1 (3) o (4) %

The value ol'c' of Lagrangds Mean Value theorem for the function f(x)=*+X-t; a:0,b:1is:

(1) -1 (2) 1 (3) o (4) %

i

l-

74. E eineu qgou eorer.rpg 6ir 619] erflu: neurpoo?

(1) Z erarugt Gltou.r "9&616i z $ein rfll$ue6luq.

(2) z $eii grqo arrAeuri (r, -0)

(3) -z eranugJ gparu-ru Gungpgl z rfr1g eto&#qna erartosp qeirafl

(4) -, g6itr gqgar ourqorLb (-t, -0)

Which o{ the following is incorrect ?

(1) 7 is the mirror image of z on the real axis.

(2) The polar form of Z is (r, -0)

(3) -z is the poin! symmetrical to z about the origin

(4) The polar form of -z is (-r, -0)



3627

-) / --t -, -+\ls. +Sr9l"0O.ogt r'lsi +4i+1'2kl=26 6t6'6rp pdrp66(g 6J6DITTJUULLr.i

Sersrgr5dtin $orLb :

(1) 26 (2) # (3) 2 (4) ;

The length of the perpendicular from the origin to the plane ? f3i + +l + rzr.'l=zo,\i

is:

1, (1) 26 (2) # (3) 2 (4) ,

L_ r-3 v-l _z- 5 . . x-l _y-2 _z-3I ,6. 4

:-, : _t rDpg6 _ :T:

= 6r6i,rp 6t-*

i Csn@sqg68aouGu: e-airen rJlsd genp$p Glpnoreuq:

(1) 3 (2) 2 (3) 1 (4) 0

The shortest distance between the parallel 1i.,". f = +: $ ,"a

,-L:y-2:r-3 r..4 2 -3 '"'

(1) 3 (2) 2 (3) 1 (4) 0

I

77. 9x2+5f :780 ereinp $eira:L-r-$$leiir gergu:rusqg66larL-Gu,: reiror Glpneoeuq :

(1) 4 (2) 6 (3) 8 (4) 2

The distance between the foci of the ellipse 9P+Sf =Bg it '

(1) 4 (2) 6 (3) 8 (4) 2

tr [ $lqguqe / Turn over



18. L9leinorgrb eu6o6nari6otr5(6r6eir ergr EpGpned <gdlu,l Glu$geirorg ?

(1) y= -x2 Q) !:x2 (3) !=e, (4) y:r2+2r_ZWhich of the following curves is concave down ?

(1) y= -x2 e) !=x2 (3) !:ex (4) !:x2+2x_j

79. X gg parflpeoeu eLoeunuruq tonfl areofleb, +rflunen a,!g :

(1) 0 < F(:r) < 1 (2) F(- -;=9 Lofgr6 F("") < 1

(3) n[X: r,.,1: F(rn)- F(rn-r) @ F(r) gCD ronslerlls +nriq

If X is a discrete random variable then the correct statement is :

(1) 0 <F(:r)<1 (2) F(--;=gand F(co) <1(3) n[X=r,.,]:F(r.,)-F(rn_r) (4) F(r) is a constant function

20. pu$pamrri e,.pgl p -+ q &qgri eouon60Tgl :

(1) pvq (2) pv(-q) (3) -pvq (4) p^qThe conditional statement p --> q is equivalent to :

(1) pvq (2) pv(-q) (3) -pvq (4) p^q

2T. !2=4ax ereinp ulouarorurpSl6i,T 6+ruqg$5t$ 6lpn@Gon@oeir Gleur@Lb qeiraflugleiirpu:Louuneop:

(1) Clsou6uooLb

(2) Stu6lgouarT

(3) garemu9eb areoTuuu@Lb Glpn@Gsn@

(4) uToueooru$$ein eroa

The locus of the point of intersection of perpendicular tangents to the parabola f :4ax is:(1) latus rectum

(2) directrix

(3) tangent at the vertex

(4) axis of the parabola



3521

22. r=O 60OEgj ,= f a: ar pr!}uu 116r.r y=sin1. LoDg Lb y=cosx 6.r6inp eueoeneuaogaaflein

@anuuurr_ uquq :

(1) "E+t e) JI -t (3) 2nE*2 (4) zJi+zThe area of the region bounded by the graph of y:siro and y=66r, between .r=0 and

ii:r:7 is:

(1) Ji+t (2) A-t (3) 2Jr_2 $) zu1+z

23. r-9leinou geu elrot f 0l6ir 6rgJ GlLcu:eoLo u nlgLb?

(1) pvq (2) p^q (3) p v (-p) (1) p ^

(-p)

Which of the following is a tautologv ?

(1) pvq (2) p^q (3) pv(_p) (4) p^(_p)

24' 4t+2y=a ereinp Gsn$t f :16x eremrp u ryorrarenu$$ein Gpn@Gsr@ e,.afleb c Lo$uq :

(1) -1 (2) _2 (3) 4 $) _4

The line 4x+2y:g is a tangent to the parabola y2:16:r then c rs :

0) -7 (2) _2 (3) 4 e) _4

2s. E(X+C)=8 oDgb E(x-C):12, erofleL C ein o$luq:

(1) -2 (2) 4 (3) _4 (4) 2

Given E(X + C) : 8 and E(X - C) :12, then the value of C is :

(1) -2 (2) 4 (3) _4 (4) 2

[ $lguqe / Turn over



I

10

26. y:e^'LDEgb !=e ^', m > 1 ereingt Lb ari6i)6noi6i)r5@riqg @a:uuurr GsnermLb :

(1) '''_,[#) Q)'-'-'[;;1

_.,( -2m \ ./ 2m _)(3) t'"-'[ffi,j (4) t*'l;*,,1

The angle between the curves y=e*'ard !=e-^' for m > 1 is :

(1) *-'[#) Qt '*-'[;tr)

(3) *.-'[#) e),-'-'[#])

27. -2x+y+z=l ; x-2y*z:rn; x+y-22:n, creiirgr eoeinun@seir l+m+n:O, ergl Longl

eleoLoq ronu9ein er$Glpnlgurflein $rioq :

('l \ eaGr rarm r-Lri8 u orir m Eriot (2) Glouafluueor$ $ner1

(3) creonemfLieosu-rpp $rioqscir (4) $rieL1 $ebeuneom Glr$flgri;gLb

If the equations -2x+y+z=l; x-2y *z=m; x*y -22--n, such that I+m+n:0, then the

system has :

(1) a non - zero unique solution (2\ kivial solution,.

(3) infinitely many solutions (4) no solution



.t 1 3621

^.2 _.2

28. + - l- =1 6r6inm -srL6 g - -'-",p :,$luTeueooru$$lein Glpneoeu$Glpn@ Gon@aqg6genrGu zeiror

Gsrorr6 :

(1) .-2tan-,(%) @ ,-2tan-t(%)

(3) zt*,4(!l) @ zt^^n(/r)

The angle between rhe as)rmprotes to the hyperbola *

_ * =, O ,

(1) - - ?Iar,-'(%) (2) r' - ztan-l(/r)

(4) zt ^n(/r)

2s' "=w(* ;f )*"na '# . ,# ereirusr :

(1) 0 (z) u (3) 2u (4) u-lu u=6r(x2 + y2'l *,",, ,3 + y4 r ,( *y ) d, "dy''(1) 0 e) u (3) 2u @ u_1

30. i*rl*i a.grrb u,glar* i+ I crgLir qenafl orglGu$Glauebr@dlpg, l* icrglLb q#ofleou:u Glunp$gr .srpein $l19uqp $pefleir .96nq .,6, ei-edeu ,a,

einm$tuq :

(1) 1 e) 2 (3) 3 (a) 4

If the magnirude of moment about the point I + iof a fo.ce i * ,l _ i ".ti.,g

through

the point ? * I t J6, then the value of ,a, is :

(1) 1 e) 2 e) 3 (a) 4



362L L2

31. Glulryriaareuu Glrng$gr lgeuLo ndlu geinflein n.5Lb uq (,orfus;afleu ,k Stein er$lriLoeogr

(k<n):

(1) ^/u

In the multiplicative group of nth roots of unity, the inverse of ok is ft < n) :

(1) ./o (3) on-k (+) -''1

92. 6D(D unu6on6itr urJeleflei-r P(X:0):k erefleb urJouf uqu9lein Lo$1uq :

(3) to^- k (4) -/u

(2) logk (3) etr (4)

(2) iogk (3) etr

(2) b +6119l c &19 $eomr

,-r-+-+(4) a+b+c=0

If in a Poisson distribution, P(X: 0) : k, then the variance is :

1(1) tog t

.1(1) to8 r.

1

k

L(4) k

e r -) (- '\ t '\

33. a.b,,c ereorrp p6nLD.Dp Glouri;risq9615 , ><lu ".1=la x u lx c erefleb:l. l( /

(1)

(3)

a .9L60'r5r tr 6rg Glaoelrr

-,c +6rrgl a Bg $aram

a parallel to b (2) b parallel to c,.

-) -+ -, -)(4) aab*c:0

rr I x [u. " ;,)= [;, ;l * ? ro.non-coptanarvectors 7, u',?tn"r'I /\

c parallel to a



13 2621

2aas4. I f O\a, :2Jf @ dx eret Srgrie Garain@onugein:

00

(1) fe^-x)=f(,) (2) f(^_,):f(t) (3) f(*)= _f(_*) (4) f(_4=f@)

a

I f on* : zlf tx) ax it,00

(i) f (za-x):f (t) (2) f ("_,):f (*) (3) f(4: _f _4 (4) f(_x):f(*)

35. or areinugr 1 eir gruurq qgoeu Lb arafleb (1_or) (1_o2) (1_t,r4) (1_t,l8) @eiir Lo$uq:(1) e e) - s (3) 76 (4) 32

If ro is the cube root of r:nity then the value of (1 _ to) (1 _ ro2) (1 _ ro4) (1 _ os) is :

(1) e Q) -s (3) 16 @) 32

i-r s 2j36'

I I I ;'] .-, er*flri(s GpriLong zarr@ ereufreb k @ein Logluq :

(1) k 6rGpg16 99 Gou:Gluain (z) k= _4

(3) k+-4 (4) k+4

f-", : ,ltf the matrix |

1 | --3

| f.,", an inverse then the vatue of k is :L1 4 sl

(1) k is any real number e) k: - 4(3) k+-4 $) k*4

[ $guqe / Turn over



3621 14

f (a + 24 = x2 Pa - x) er eory oueoorar eo qu9 ein Glpnaro$ Gpn@Gan@ :

(i) x=3a (2) .=; (3) ,:; (4) r=o

An as),mptote to the curve y2(a+b):*(3a- x) is :

(1) x:3a (2),:; (3) -:i @) r:o

38. y'+ (y")2:(x+y")2 oeiirp ouaostsGlslg& e r-ooinu nr-rqein arrfieoa Lo$ELb urq 6peopGu :

(1) 7,7 (2) 7,2 (3) 2,7 (4) 2,2

The order and degree of the differential equation y' + (g")2: (t+y")2 arc'

(1) 1,1, (2) t,2 (3) 2,7 (4) 2,2

39. .el6D(q elamfl I $ein au rfleo* n, k*0 g19 LonfierSl eroflet adi GI):(1) kn(adj I) (2) k(adj I) (3) k2 (adi D (4) k"-1 (adj D

If I is the unit matrix of order o where k+o is a constant, then adj ftf;:(1) kn(adj r) (2) k(adj r) (3) k2 Gdj r) (4) k-1 (adi D

40. f rn = I\.' ' o

-' 1'1 ^ ereiirugl 6"t!prq eir:tg$l+ +nriq creofleu, k eir o$luq :

IU IDMGITDEJ6I6DITDt'

1111(1) 5(2)a(3)i@i

rt f (x):{tl' o" 'a is a probability density tunction then the value of kis:' I0 , elsewhere

1111(1) z@a(3) ,(9r2



I

15 362'1.

u(961-e$/PARr - Bgfluq : (t) eianeuGu.rglLb u6gl orglewntssqg&g arglarru-Lafl6sqLb

--+ l+ .)\ t-l, (u' * i) n u' " (;' ;) * 7 "

(i " ;) : 0' erenri onr@s.

x2+y2+22+4x-By+22=S ereinp Gsnen$$lein eoou:Lb LoSg@ 976 snains.

Provethat, ;" (u,, ?) * i,. (;, ;) n i* (1, o,) : o'

Find the centre and radius of the sphere

t2+y2+ z2 + 4x- 8y +22=S

10x6=60(ii) efleur n eranr 55 6qg eainrguune orglar L_u afl riserl Lb, rglgt

eflenndserfleSlgpsr s.Gpgx 6 geinug e6larn6sqg61q o6len ru.l afl6e er1Lb.

Note : (, Answer any ten questions.

(i, Question No. 55 is compulsory and choose any nine questions from theremaining.

41. alenflBGsneoeu grarpugel $ri6e :

2x+2y+z:5x.-y+z:73x+y+22:4Solve by using determinant method :

2x-l2y-lz=5x-y+z=73x+t1+22:4

system by using rank method :

42' r9leinorgLb +Loeiirunr$t$ Gpnguq g(Drir66ooq L6DLurpr 616r'r ptr (lD6Dpugreb gpnurett4y+72=143x+8y-22:137t-8y+262=5Examine the consistency of the followingx-4y+72:\43x+8y-22=137x*8y+262:5

43. (,

(n)

(i)

(i,

[ $6uqo / Turn over



:l=l;; ?l

-+ -+lb cl

r.44' La+b+--+ -J --r -->l f--;c b+c cl=La

f-+ -) , --) -+La + b + c b + c

6166r ff,lmt6)l6 :

45. coscr + cosp + cos^y :0: sino + sinp + sin1, eraflei;

(, cos3a*cos3B+cos31=3"ot1o*U*r,

(ii) sin3o + sin3p + sin3"y:3tit,1., * U * r,

6166r 16lmr6)15.

If cosa * cosB * cos] : 0 = sinct + sinB + sin"y,

prove that

(, cos3cr+cos3p+cos3t:3"ot1**U*r,

(i0 sin3ct + sin3B + sin3"y:3t1rl1., * U *r,

4G. xy:c2 6t66tp Glsorr 6,rs et$)uporeoarup6]uit urcpg,b g19 q cirofl u9eu 6u6n[urut-]GlLb

Glpn@Gan@ r,y s{s6656rflei: GlauL-@Lb gairi$esn a, b erenqLb $uqeiraflu91en

GlarfuGenrrgein Gorr@Lb gJaitr@ocir p, q erenerl Lb $6ur-9loir aP+bq=0 erenri snL-@.

The tangent at any point of the rectangular hlperbola ry:C makes intercePts a, b and the

normai at the point makes intercepts p, q on the axes. Prove that aP+bq:0.

47. 5l(D pc19lb ornear$$lein parr F $leiir eoeinun@ , : i * l00reraflo psolugl6itr Slglo

Lo$uelru& snains.

Resistance to motion F, of a moving vehicle is given by t = i * 100r. Determine the

minimum value of Resistance.



77

48.

4e. (0

(ii)o$ u r9lareo earrid@o.

(, If u: y2 , show tf,rt ,! + yE =,r.dx 'dll

(ii) Find the differential dy for the given values of r and d.r.

y=x4-bp+x-1 ; x:); d:r:0.1

50. ! = sin2x ereirrp oteoono eay, x=0, r:rr [DEE]6 ,r_ .glse 96luollipneb .9{6DLUG1rb.s1pao$$ein u [u 616o6..r{i sn66drs.

Find the area bounded by the curve y: sin2x between the ordinates t=0,:r:r andr-axis-

51. 6ft{i6 : (D2 - 2D - 3)y: sirur cos:.

[ $19uqe / Turn over I

3621i;:.

L(D6D6fi ou qer9leunoor gO LG6l)rrsS 5J6mrq6b L6iT6n 4 rd.r-g. ergllL_gLb 12 dl.d. €gG]6, i6)snatrr- gO gl6o6nugl6Dar rfain@Lb er$lsuu@$p, etpeiir erglr_L_Lb 4.12 r-6i.rfL_rrne'sr$srf isuu@6pgr. 6p.in a'aooreune greoor$g er@6suuLL e-G6Dn*ssre'r Gpnqnu-rglaneoouri snairis.

A cylindrical hore 4 mm in diameter and 12 mm deep in a metai block is rebored to increasethe diameter to 4.12 mm. Estimate the amount of metal removed.

" = E + 1/' , eraaa ,$ - o# =u eren6 onr-@s.

6an@rieuurr_

-v:r{-3r3+:-1

r o$gLb d-r Lo$uqegag dy ein

; r:2 ; dr:O.l

Solve : (D2 - 2D - 3)y: sirur cos:.



3621 18

t1 ol i-l ol It ol l-1 ols2' L; l], 1; ;]'1, -,1'Lo -rl +dtu pineinlq etaoo-'sqeLb errridlu somrir

.gatflu Glrlg6se$ein ED gO erSeSlueir (96Dp6np elarLorilgib erenri snL-@s'

(pafl $pefl u na e1arfl u G u lgtioeuseir s nem el er 6l u-r Li Sl cn ar eu )

showthaitheserorrourmatricestl ?] t; ?1,[J j,l t; -i] rormanaberian

group, under multiplication of matrices.

(Detailed matrix multiplications are not necessary)

53. (p " q) -+ (p v q) aeirrg 919 ClLoureoLo erem6 snr@s.

Show thai (p " q) + (p v d is a tautology.

54, rgleinerg Lb 61"$p""-rl errn$$le +nriqri19 erJnerflaru.Jqb uTouf rrleouqLb sn6 ns

[3"-3'o.r.ol(v\: Ir^' Io ofGgrru6'9Lb

- 13"-3t o<xcaFind ihe mean and variance of the disiribution I (1) : 1 ^I u elsewnere.

55. (r) gg s$lrfluBsu Glungofle,Sl(D$gl geuJ.un Slaeirsoir eureflurns 20 $)dlr sneu

$tarrGleuafl u91ei 5 o-em lL6puu@61p5. u nuten:nein Ll [6)j6D6DU r-]ueinu@$$l

gflurgu 20 PrfL $enrGlouoflu9lei;

(0 2 e-rfl$seir

(i1) 6qeogr$gu r+ib 2 zr-sBsqyrisnen pa;lpseoari snaims (e - s = 0'0067)

gl606D5l



1"93621

(b) If (a, + ibr)(a, + ibz).........................(an + ib,.,) = 41ig,ererflei->

(, (^? "ailGl *btr) . . (u?^ +a!,) = A2 +82

(ii) ""- lll* ",-,[9)+.................+ *n-,lb]= r''* tu,-,/I) p.7\,11 \az) l"^J N,"<"' lo,,erem $lqgGt

(u) Alpha particles are emitted by a radio act:20 minutes intervar. Using poisson dirt ibrti'o",.,uf:;Jii:;iffi[,.,?:r'ff:,:'*:,it"'(, 2 emission

(i, at least 2 emission in a particular 20 minutes interval (e-5:0.0062)OR

(b) If (a, + ibr)(a, + ibz).........................(a,, + ib,,) = 41ig,prove that

(, (,f + uf )("! + ui ) .... ........ ..(,1 + ufr) : A2 +82

(ii) *'(+).*-'[3)+ +,,,,-,(*,):r,n,--,(f,.)r.z

ugpfl-cTnanr_clgfluq : (i) oaoeuGu.rglLb u3!g orglearnGsqgai1g aglaruurafliisoqd 10x10=100(i, ofleon 6r

.,sr *"6*"a ffi # "c#,i ;f,ffi n_"ffi ; :*::*,_".ffNote : (, Answer any ten questions.(i, euestion No. 70 is compulsory and choose any nine questions from theremaining.

56. GprtLong +ltorfl snerureb goarpugleb r-geinouqgLb GlpngLrenu$ $ri6s :x-3y-82+10:03x+y:42x+5y+62:13Solve the following system by matrix inversion method :x-3y-82+10:0 ..

3x+y:42x+5y+62=12

[ $guqe / Turn over



'\. 3621

57. e(D GrtsGsnarr5p efu O$gJriGsn@srir gG 7 q cirofl ugl cn e$$tri; g Lb ereinu peneur Cl ouri L-n

rLoenmu9lo rElmtous.

"Altitudes of a triangle ate concurrent"- Prove by vector method.

58 zi +al +zi, zi -rl -i oggLb z7 + 7 .56lu.reufarp $)are., Glerrir;sn'rnori

GlsnainL- qeirafleeir atglGu GleelgLb pdrp66itr Glorsrri o$gLb a;nrir-Slu.lein

er-oeinu n@saoon6 snains:

Find the vector and cartesian equations of the plane passing through the points with position

-+ -+ --) -J -, --t -, -)vectors 3 i +4j +2k,2i -2i - k andT i + k.

59. (-Ji - n% $ein ereuon Lo$)uqsara-qLb snains.

Find all the values of (-.fs - i)% .

60. 6!(E oIle,o6n6q .s1eo 7 $eiror L-r arqal$$ld; e-oiror5. slp6in e{56u6 48 e1q' zulpLo

20 etq, parTuglerSlgpgr 10 errq e-u r1$$la: euenano,9lein stsa:Lb ereinen?

An arch is in the form of a semi-ellipse whose span is 48 feet wide. The height of the arch is

20 feet. How wide is the arch at a height of 10 feet above the base ?

61. sx+\zy:g ereingr GpnGsn@ et$lurlorareiru:b f -gf :g g$ Clpn@dlpgl erar $lgr-916s

GLogrn 6gn@Lb r-leiraflaruqLb snarurs. .

Prove that the line5x+72y=9 toaches the hyperbola x2 -9f =9 a1d find its point of contact'



362121.

62' B(D gerrsareror, parlugersl(D$gr Glerarg$pna GLoebGpnri6l6 6+g$grLbGungr l,Gprp6loGleebgLb o-wytb 'x' ereina. gpein +oeinun@ :r = 100 t _

? ,, *uona

(i) elqsarar.u9ein GlpnL_6s $aoeGeusLb.

(i, 6r6l{s6n6rd.r zzi+ lu:7$eop ayarr_qLbGung .srpein GpTLb

(iii) q-6rl6a)6nr =s1atL_q Lb zd,+ zutTLb

(iv) q-6rls6D6nr p6i)[6rr)u] geo r_q LbGu ng .etpein $areGorsLbgdluouf alp6 enanro.

A missire fired from ground level rises ir metres vertica,y upwards in,t, seconcls and-r_-lOCr_i,..r*0,

(i) the irjtial leloi:ih of the missile

(ir) the time wher the height of the missile is maximum

(iii) the maximum height reached and

(iv) the velocity with which the missile strikes the ground

63. y ='t2x2 - 2x3 - xa. ere-^ --:..---Ip snnlr r6ngl er$p $tarlGleuaflsofleb qg g! or1 .9a ar rdleinp eoereiirupenelrq Lb Lopg Lb atenonq Lon$gu q eirafl e aoorq Lb eneuns.Find the intervals of concavity and the points of inflexion of the function y :1Lt2 _ b3 * f .

64. y = f +1 ereinp ouenen 6l6D [6Drr] 6u6D[s.

Trace the curve ,/: /3 + 1_

65. q.Va ,a' L6nl-ru er.l rr$pein esponeoeuri Glpnelre6snermeb lgeopugeb snarro.Find the perimeter of the circle with radius ,a, by using integration.

[ $]1gr-rqe / Turn over



3621 22

b6. x:a(t+sint), y=a(1 +cost) 6r6inE) 6 LL L(D 6ir erl 6D 6n uElein gg e6lcb e$leo eor e1p ein

el qu u&e$enp u (r- etri's) Gl u ng $5r +p pp orp neb 6rD u 0 ib $1 L- u Gl u n6ofl eiir

oreooruryLtaruti sraine.

Find the surface area of the solid generated by revolving one arc of the cycloid r : a(t + sint),

y=a(1+cost) about its base (x-axis).

6j. dy: x3dy +3x\d:r + secx(secx * tan"r)dx ereingl a.reoeriGlsqg eLoeiiru nL-6drl$ $nris;.

Solve the differential equation

dy: x3dy + 3x2ydr + secx(secx + tanr)dx.

68. C : {2" f " e ,l 6,raq 56o1l-Dn6drgt Gluqy&serSleiir E$ gg 6rserSlu.lein g6Dp6Dp etarLorilSrir

eroni snu@e.

Show that the set 6 : is an abelian group under multiplication.{2" 1",,}

69. gg Glsnoirseu$$lo + Goueiraron L-o$g Lb 3 Eeuuqu u$5elg6 e-eirenor.3 u$Soeoor

gou G o-J neinp ns er@ts1g6Gu n5 Eer u q $) p u u$greofl ein erirdli:eosuSlel $ls Bpseq u

u pareb ($1enpr} enriq)

(, $l1gur9 arar&grj: gargru9lei;

(ii) $lgur9l arou6sn grarpu9leu snains.

An urn contains 4 white and 3 red balls. Find the probability distribution of number of red

balls in three draws one by one from the um

(0 with replacement

(ii) withoutrePlacement.



70. (a) g(D par 1o r-r-$6 *s)Opgt 7.5 16^ ru p$$ eb p6a r & (5 $) enarmu ns G r n1g$p Lr u rL_919 lgpru9ler5lOggJ Gl6!6rflGugrrn $n perlrcorg

3621

Gp$nei'r

ar9lfitdlrnsr.

(b)

(u)

q\,, \1u,,*,"",\9lrgr \orburorr\oulUlLD pn per, Teou_t3i Cipn@Lb Ltn6Dp g(Euloreooru.,$arg s-pu0pg,dpgj GLogLb @$p u To.raroru:u r narpugein goeoengpru9leiir er.rnuSleb elarLodlpg. gp,nrl or_L_p$[g z.s r-6 gGp

$rfloi,r urureJnarglgpnu9ein (p6o66r eug!unsti GleeigLb 6,-* 15pgl?i; Ge nL_rq$1g3 rjrr-ri gtTrpdd reirorg,

1aflc;ir, lgpgl{tJ Ganr*Lgersi1g$5l crerororq grupdp(g

91 u u neb 6 f neorgl penqugl eb orgl 19 6 creinueopB e,rarls.-

gla)a)gl

6euuu$lenaro 15.C z eiron gO s{6opug6b elrotrie uur_@eironGleuuupeoeu 100.c 966 ergr 5 pr6lrrue6rfl6b 60.C +6 156np,EglGogtd 5 $rJlL-6 sg!$gr Gp$n6ir Gleuuu$leoareourri snains.Assume that water issuing from the end of a horizontal pipe, T.S ryr above the grounddescribes a parabolic nath Th; vertex. of th" pr.ubo;;;th is at the end of the pipe.At a position 2-5 m belorv the line of the pipe, th" fr..;'"; water has curved outward3 m beyond the vertical line through the .r,i of tf," p;. ;* far beyond this verticalIine will the water strike the grouad ?

OR

A cup of coffee at temperature 100'C is placed in a room whose temperature is 15"Cand it coors to 60"C in 5 minutes. Find its temperature after a further interval 0f5 minutes.

(b)

-o0o-



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Transcript of --- · 9/12/2013 · 3627 i I I I 4. a2 u1-; = I 6rdnp 6i6n6rT6.r6i)[6oru lgSp+Cleneoru Glung$gr...