Relatt md Fumctons.

abt s am epuinleneerelato.
epuialen
b R aides
(a,: numbaw,
St IK q nQal
tt
show thak He ielton R

toamsitye
mnetnie nor
oR R deined ns
show thak the nelato R
mela've and hromnsi Ave but
R= 1 (a, b): a < b
not mmani

Ari det A2,2,3,4, s}

melan R mthe
5 Show tat the
lmenth ll,3,sj an
folahon shos thar all tu
elated to each ohe
Me a o to each ulbey, no eltmenls)
lmsnt e4f.
6
shiow ht RinRe is aso an pui lenen nehdn,
a delahn on tu etA ovdaned
R(uw)
euss posiHve hategs d ined by (a,9)

puvalene rela dan

8imilar toon tut det
8. how ther the relnhons
pleme is om
natNa nunbes amd
4. Lot N dentes the Jet a Al|
R be th halaton on NXN denod by (a,b) R(C,d)
ad btc)= bc (atd) ghow that R K an
 nelhion n AxA
10) et A = ,2,3....4} Ond R be te
bte for (a b), Cc4)
deined by (ab) RcA).s atd epivalence
b+c
is an
nalat
W Ax A. Prove thar R
the
Onat als0 obram the inlene elas l(2,s].
umbos g'vom lag R1a,b: a ii ivisr b?
natual -tromsiye. A oloternthe
eiaenee Telahoa,
hn the Be A=j XEZ:ogs2
12 how lhar the nelnthon
Rven b S-1 Ca,b):a,b
mequialanee Relahion.
deyined
IS. chece whethes the melaon R

hatleaive, Agmabie by (a,b)R Ce,d) ’

Heladnn R deg ined
Sho taarthh am quivalence
Bel N N
he
Ad b+c .on
omd
numbahs omd t, dtine aRy
1s. for mel
ey
Fnpbin Yo
Yom

: R - - ’ R be a tunetion daqned as
lb. Lat
s how taoat, :R-1’Range
3+4
x-Chacle
3 whelu
that ()=
17. Lot TR>Rach
or not.
2
cheek igecvtiy
18. Lat given b 4)
en by
14, check omd Swyechviy 4t:z

an
be tnhon deaed by t a) = *-x, MeA
Yonr wves,
 1nverSe Trisonomaic. Fnctions

I- Find he principal vale e s )+Sini

valie tan C)+e
2. write the pri cip

4, Fid tte valne

s. hyitk e principa alue a ccos )
6, wite the pineipal e to Can )

3. Fund the rincipal al

13. Evaluate ? Si (Sin 37 ) + c Cs) +tan (i)

l4. Evaluete cs(cs ))
1s. EVluate; S
(3in)+c(cs3)4tan ()
lb fmd tte oom ain q Sin'(4)
7. pino e olo mas md ange Cincp ale banch) I
tumchon a)= tan

Qo. Enprens tam -3<<n in the kiplect form

2
 al. Prove that s()+s ) 77
a7. Pid the vale sin(es()+ n (9))
R&. Proe bat si (2)= Sin ts)
. Proe tyant s(+e(2) C)
30,
31. show rat: toan ( sn
t-.
32. Poe the to lbdiy: 3

38. P that (ton'CSlet')) 4

St Prove Ahats
toin'()÷ -,xe ((T,)
4

3s. Prave bat: tav/t

tanA+}csa)+tan(-o)
s, Proe thar ct
- , x E (o,
Nitsinn-sina
 2

chate matrices omd determi nart.

2 3 Bsve the
-5 - 4 - 2
+ - Q2=3

-4 2 t 2 3
2

Wse this t o r e Hi ystem'q eguation

+ Qz= M-y=3, 2 + 3 t+42=I7.
2 3I
3. '2 t Aand hemie Bolve tne
-3

4ullawig sytem guaonsi n+g-3z =13;

31 4+29+z=4: t+Qy-Z8.
A. DeteTmine hs pooinct
S, -3 2

md wse it to olve he systero equahon,

2
2
-|

Bve
hen
6. Evaluate the product Ae, 2 2 - 4
A= B= 2
Ssfre
2 -4
5

fhndons
M-3, 21+3y +42-17, J+az3 =7
-1 I
and henee
2
 matrix a ovdan 2,
idenyfinA!
A=4A- 3T.ece
thonso tatA=4A-39
2 31
show that A-23A - o 0
3 then,2I
2

value 9 A-3A +2 I.
l0.
2
thes ind the
Haey calculate
-6 o 8
7-8 o Ac+Be.
Veit tat(A+B)C =
Ac, B and cA+B) C" Aso psd a madri'a
B=(od c-(3 ) d,
|2 3),
Jet A
DBueh tharcD-ABZ O. 2

3 4 4
|3, A= 2 2 6/
7S 4
stewymmtic matara,
I4. matri b

th

a and b,

- 6 4
Jand B= hotty tha r
lb. A (3
(AB)'= A.
 2
+2 IS.
+3* 2n+ =4,ay-2z X+2=t,-
aguahons; ;: gem the sone Ao
to 't
-3-1 S
3 | Determmnei l4,
and
-=+ + Ssfve (3.
+3-z|, 32 slre 12.
Qn-9+a2
=9,
A
) A'
= toatshw
=C. AxB tat ).8- GiA
ven:
(22), (amd c=
natin te Fd 4.
which tor A
-tann = Aainen
n &i2
AA anhow
that
f
aj aj Fand 7.
2%
(Cs o) 2 A=(adia)iy (AB)-
' !
venith -3) B=
2
Raak
=(adjA)A= AladjA)
thar
tiat 8how
ence A-3A-1T,
=0,
Cosd -Sind
())= (Fthat Bhow Snd
2-2 I
A
2
etemani 2 amd Matice
 Hence, stre Ie followly
=|| and x-ay42so
| -2
a4g+2 sb, yt3z
yter 3 opuahow;
Sad adj Aadvemity thar
|7.

A(adjA) =dj)A = IH{E, 2 2

B= 3.
D: -2
ls-2

followw matics Aomd s,

19. For the
Ae)'.a'. A=

A= 5A+kT 2 4
AB.
2
aud B

Henci, ssve the ystem

-4= 3, 22 +31+ 4z =(1,
Q2, fo wtat valne
y+2)
an+l 3

tte natn A
nt,2 4
Q3.F d tre vales
AA=î2
sahstes the puahon
CsN S amol A+ A J~I, tuy
). o<neT
2

tind the yal

then how tAr.(A +D) CA- 41) 0

26. T (&n o,tind ty valu

27. T} A'= )Omd B

 Applieation q Iwteaion,
| Draw the onph seatch

the heion tounded by tte cu e trom

the ordinates T to and 4he a ans.
6 3
te e v e
2. Fml he aOa the agon boundded
4x+4= waig ntegakon,
3. Make a ough lceteh o tae egion ):o4y<x+,

metod a mdegrahon,
egion, wag the 1 +2| .
gaphg ye la42|. henee aluete -S
4: Stetoh the
S. pind lbe

6, pnd the ane

bamded
y2

tegahn the ane 7 the

7. Using line ty=2, Hhe
Buadnt enclased by the
y a n the dmai.

len meudd tbeheen the

+h

Hhe meon bomled the panabta

10 firol tee anee
and the e 2y=4
bomdod by the cire
|, p d tthe

12. Make a ough scateh

tegaon? o4243}
1 (a,9): oe X+3, ßey <n43,
 trst gundranh
amd A deDtes he n a n koumded by
y=4x, nz4,td A:Az i.
l4, Mak Seatch
the

Aes n the egion wny mtegaon.

melnded heWeen Hhe
Cwwe tya3 a m d t o - e24 3M+|2
l6. P d the asea tto Bmalle hgion boumded
and the shraht
a
2
maller ea enclosed by Hthe cicle y=+
17Fb the
and te line't =2 2

18. Fid the aea encleed by he wve y=-and

the shaguteme ty2 =o

the Aagton wng intagladon!

21, pd Hhe ahea tt epn boumded by te pmasta
y=pan and its katno lechm.
Ce gesx,
-ais be ween a0 to a 2.
Omol the
beundea
Q3. Usiwf adegiaton t dthe aea
- 22S the nes 2 and 2 and tu
qn+a5Y
M-a.

oe boundod by thi e v e - a s and ke ordies

=y amd x ivided to wo
as. The aea beees
 Applicabn o integra ealelus
Boan bas R stims,

Cwhen . wmy eyahon,tdthe ave

to k= N an d Mio -onis,
1om the onalnates x
bij y4x,
Me reglon houncded by
tu
2. en
Aj denotes he area
aant and A
lao the -onis mtho 1ngt
denotes the anea n hagãion bumdud by y=4, Mat,
d A,:A
3. Maee a):0y<,

Aafion, wng te method idahon,

4. Make a oph Seoeh

He smalle Rafeon tonnded by

y
e ellip% a2
y

7. prd tue amaln a e encossd byy

&. pnd te a
atitas
9, pid Hhe ane te min Begt tu

y2
 ntega,
bn bonded by tu paratota
andis lalas rechin.
4ax
anea ienclde by tecuwe yaCsn.
13. p d tii
ais bereen
amd
bounded by Hee cuce x+yel6
14. prd tte anea rst gudanb,by
ana te ine 3 n n te
wi stegsadsn
 Vector

=3+ ane Such Hhat b+ s pependlclen

to a,Yhen find the vale g
) A
Afon samding at olo,o, o) s watehingA(4,0,3).
aetoplame which is caordinate pomt
at te
At he same t e i e Bw abird at the coornate
BA mates
8(o,o,). Fvind the angles whidh BÀ
point
the wechr e, = - + .
3) An omt s
oving long tthe nechr
= 30-aj tk hich is melined o an mglaowik.
tthe nechr hihen nd
Baler l.
f: The posiibn Vechrs vetees q AABc
AABc ane
A (atj4)
al he ABc

value q Bo that ttee wecrs band a b

olto gona.
6, pidl tie dinechhon atio and dinechon cosines ) a
paHallel to the
eine baale ne whase eguaons
eguaoms ane
b x =34+4
and bane mit wechs amd & s the
la+.
ngle behwas
then har prove thak Cs B
8. pind the anen 9ts panallelogom h adjncent
Bides ane tu' echYs ä= i-j+3k and
IU-j + k
7. fnd the ve chov whose keno is 3 io amal wthich s
lo. o vechvs j+Ê amd
ABc

find the lingk he madiam through A,

JE]=NE
|3.
î4 a nd'Bae two vectns Buch hak lal=
and .E-, d the amgle bebreanand.
1+b|=6o, |ab|=40. and la|=22, then
l4. 29

Ts. The co ord inte pamt

) ond 6(S,,-2) s 4, Pnd hz-Coordin
pomB P2,2,
He dheohn csinesn the line Whch s pepandiela
16, Pu d popovhnal to
to Ue lmes wit diretbn Cosines
<l,-2,- 2> amd<o,21)
|7.
bott n
a unit wecor berpend'cular t
Ihe vecoe ( a - ) a n d ( )
a=ij+3k amd
Side ae airen by the wechs

19. = 3i-j-ak amdB= 2l+2j +k, hem tn d

(7+2b) x (aa-L).
Jo. ha two adiacent sids 4 patallalam
ie nereserted by Ita vechws 4)- md
at+af+3. pind nit wechovs paralLed to its agps,
te aponal vechows,ind the anea the paslldojm
tu ownb A(2a,3),
atb
B(3,-5, b) amd c-,,4) ae col (nga?;
99 4 a lnemakeg amgled, Bamd Yw ote Co ovdinate
Qasfhes tmd the alua , css2B+es2.
eyu cmgles wiwi to the Co ovlinde a s
 Q4) Tn A (a, 3, 1), B( 2, 3, 3), c (-, -2, s), 4i
lhe pont D Such thak Ae amd co
Dn te Co ovdinates n he bomt A, B,c
A, B,Cano D
as) (2,9,2)
e ll,2,3), (4,S, 7),(4,3, -b) amd
esprctively, Hhen find the omgla bateen o
3

26) ghow toat the line wito d e c nCines

12
I3

A ane mtuny ferpondiculan.

besiton ecuys the three ventioes
a7) Ta Hhe
A, B and a a
taiangeaj +3ka

respectielg,

ectr the linechbn

28) d a mit
=-a2
whene a =-ìj+k omdb

Jusrify o ansrey,
30) î1 AC,3), G (2,-3, 4) and c(-i,2, 7), ti
poit D }uch that AB and o ae
pa where Q,A
anol B ae the points -2,,3), (o, 2,s), 4,-3,o)
Omd (7, -S,-)fapecvely.
B2) A
mit ecNY a makes amgles
X-ais and d-ais Sespecely amol 4an acute argk
wik Z-anis, en find &omd the(scalkr and ec)
Comp1nerts
.

34. Pind 2|4 (-a)(a)=12,whene

.
a is a uni echor.
3s. fnd the ce chr n magnitde 6, wshich
perpomicular bo bo the we chovs i+aA
M and 4oj+2k
WechovS, Shw Hhak he wecosa+ amol
( a )ae perpondialar toeach ohar.
2 Ri he dinechn Casines s 'a
ttroy the fomt (l,3,s)and (2-, +;6)

hen thol the value

= t-a2k,b

Ma

4. For onn veehr d, nve hat

(axi)+x(axi)+
+aj+3,E a-j+t,c= 3i+)+
42.
Gmd ax(xc)= at mbtn C , ' t
Vanes l, mamd n,
43. Sho thar the e ponallloa
honing digonals s+j-ei amd
 hechon Sheo q teline
I) Fmal the
2
+2 2-1 = so, Hrl tt nechy gpuan
2

lAee eies.
tue ine which sterechs te
9Fisd the egunhon
4
2
4
omd passesr thnogh (y,). Suehh tat
two vecos aand'baie

|al= 2, and
t r any two vecosaam
4) Pnyetthar
9-3

S) T1A deula

to . ueehy witto wecbs

qid the wecor.

S, te
no B d o e h
-, b6 and taa
7) Wecs a,

)
 AN type 15 mank esios n Thnee dimansiogeomay
font
(S,4,2) to he 3

2 Mis ponfndielat Find e imag

Rne
(-l,2, ) amd paallel to tie toaight line

8hortet i t ance beweer the lines

3.Show thar ta Rimes 2X3

tersechhen
&hotest dstamee between tie 8raih
Find u

7-(2t+)-at) +(a+45 st)

S. p d the wechor ond CotteiAm equs hon qaragt
Aine pasng thogh (S,2,
(S 2, 3) amd is porpeodiculan to
the saighr lines = tj- + ( - a )and

Fnd the Hre eme which îs ae mirror

given
(,6,3) amd paale to te line ,
thoph the point 8hortet distme
7 fino he lgtb omd epune
bereen he
2+1 amd yis
-6 -2

a'= Hit+3j+ak partale Ao tu hach b at+3j+bt.

Ite pes hendicula draw on Iais
e Ma pomnt Pith posin Vectbv Y=C+j+3k,
 9. Fod. heeahon hie line wieh tects lae
y-3
lins 3

amd tases thmoyh he bont C,,, ).

all the sideo y the patalelog2ar
|o Fimd he epuahonsey C(-,-2, )
an A 4,7,8), B2,34),
whose verhces
ABc Ydines 2 ite
lso,
A tin the co
amol D (, 2, 5), cD.
rog Ato
3 the pþenoloulen
Pnd the echor amd caueoiom euahons a
i|
cl,2,-4) and parlle o
paswy traougn the pot A(3,3,-5),B;0, )
he line joining the ponts
distomee beheen tw ho eines.
d the
Hemce
12. f i l te
euahon the line psm thvvg
B.(3,S,9). Wenee, is the
poinb A G2,3) and td e which ale
l4
(4 mits roo tu porst B,
ar a dstame g te line segment
a portRl49,2) lias
PC2-3,4) a d (3,0, (o), fsd (he
roinind te point
dLstomee
hak the l e troupn A(o,-1, -) and B(4,S)
l4, Proe D(-4,tt)
terects thy ine throts c(a,q,4) and
I3. Fid tte Vector ama Canteoiam epuahs %
a line
point i-, ) ond peipendiela to the
throus the (i,-,0) amd C,2,-), (2,).
Kine join j the poite (4,3,2),
dieonas Hthe pata lzlagiam
l6. Find the eguaos
pRS whos rences ane PC4,2, -6),a(5,-3,),
Vse these
R (2, 4,5) am s(,9, - 2).
ko Bisd the pont ) sehan diagera.
digonals
17, he VerHces A (,2 1),e (6o, 3) md
C
c (4,2, 4). LABc aeo
ind Hhe lABe alb tid the epuad
medioms.
 Bubect
o Sn+7y 2=Minimie )
31+57<s,
°.y
Z= mise M'n 0)
to
sutct
to 3a+Sy miainide
= z 4)
Bbect
o Sn+4t Manimise
= 2 7.
set dont304 Marimie
z 6.
ams
24. -x+ Manimise
z= S.
nstrain tie to ect sub
to
X+ay z=
mibimise
Constaints te bubject ly. graghi
ca
probl gommig inean qollowig the ssa a,
pom two
minimum
Z q that
the goaphicaly Show
Bubject x+2=
minimige
poblem mming prga Lnear 4lowing the de
Consi 1.
Problem gramwig PrLinean
o
1) Maximist z = +24 subject to

4) Moniim z

Mai mie t = ISo 42soy subject to tto conshans

5
 2

a. ssfre the fteenh) oguaon (9+3

3. write the integoiy tnoy
(l+g)+(aay-cog)
A e: = y-tan ; lwhen o

(2= a(-Ias); y=l whem x=0

I+92
b.

dT4erenhouadon Csx - Cs3x-CS2%

7. sre the
skutin q the di4erenhalpunhen:
8. Fimd the gemeral
lozl = an+by

lo. F d Hhe gener stution

, J=T
Cas (2)=:aiveo thar when
2

|2, pnd tthe

dy = t-tan ), giveo tat g at
cotteew tal epnaon (+a')+2ay-40,
13. Sfve the Condition yco)=o
&ubjet to the miial

1b.
17. püe he patheulton
d Sec?xy = awee, 9ien thar yo)0
 (a+1) d +22g = Vot4

+aylana 3

al, sofe: (i+eJdyt(l-9) a =O, (3x 40),

(I-)Ax
+ytons
aa, sfre: ( t ) +

d
84. Stye the ditteemhalsmion (by) dy=(49

Where xE (-0,-) U U, o)
ob, pisd the fahenlar sbntn 9
dt
da

omogeneoweiFhdthe pon heer tuton

lkes d the ole

32 cs(4y)+ Sin (a49)

ve; d
Pnher ssution 1 the deyental phkn
33 in te 2ven ttas yo whana.
(i-y)+log)a + aay dy=o
34 p d e gRmoralRstuhon
(1+8+fot-e
 iveo hat

.sfei 3etany da +(9-)Sedy=O giteo Htah

36.
wien = O,
37. &e: xdy-d= Jaydx
show that thu ctenhial eghadonlae' +y)=
find the paihcular otnhon
oogneos
teenl apundon, gher What y=lwhen al,

eulassnn
40. P d te pih
(a log) ves har g
dx

(,2)
42.
Bthon the t e u l epua hon

43. S e x
+gt2gcot x= o(+o)

44Fnd tie paHh

Meo,
(1+)dy + (l49da= o,9iwen teok yalwhen
49. e, da (ay')y o

4, Slve

4S. Sstre

44, Bstre
9+*J)dx =o, iven lba whey
So Ssre; (2-yy+