SAM PLE S,U5 STIO N PA PEi t-fl,

C£'>f.pJa.no.±ions_)

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( 0-) It
.
lo q, ve..n tho. ±
1. V
-, ()__•::. p q L i
' 1

. ~ a...- p)(q ')(q_

-and . b = p3 )( q_

~ b-::. .p~p "Xp -xq\

Thu s LC.M - p ')(· p ',( p ~ 9 'X cf =- pS 9_'l..

p (:x.) %: nurn be..v oF

~- CoJ. Tht. n.u.mbe.Y- c,f. ·'2. t;'YO es of- ,po 1'1n.orv.d oJ,
:x...- Q.)LJ ~. A \.so if th(_ ~Yo.ph
t.t:rnes g-Yo.ph of · p(.x.) i.n±e_ys e..c ts ,th.e.
-
l'f no m',a.1 does ·no't in te..-vs e.c. t !lt.-a
;)C.is, the n i:he. rw rn.b e..Y of
of .po
p(..x..J ' 2.e.:y--0 . A 'f
z.e.yoes t,f- tl-Le.. po \'!noMi cd IS,

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,
.J
- I
A- / .).
.,. ,
OU

'-- -- /
.
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I
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I
I 3. (c) Grve..n '--qu.o...tions a.Ye .x. + ~;1 - s =-o & - 3x- 6 '1 + t s -- -o
-
H~Y e. Q., =, b, = o2 I
_.

c, -::: - s -
--- .. I

_JI i ~ =-3 , b,, =- - I:, , ~-= - ,~ .

I
Al so, Q.., - b, ~ c. 0-$ l - .-<. t
- -s
Ql.. b'L c~ -3 6 ;
(S
-
9 These. eq.u crl{o ns m~ e.. i "' f ; ru'+eJ,..'I ~ l,o\u :lio ~.
....
·, .
......
'f I
 J·
t
- ;
' '
, 1

4· c~'> This. l
\S lhe.. Fo,rh a.x.1.+ b:(.+c. - o
of-
~he."re Q_::: ~ b=-.8, c:.- 8 I . I'
I ..
. . Dis c-ri mi na.Y\t , f) -= b2.....-Lt Q.C
I

(:-cJ)~- '1 ~ t.l ~ 8 -

I ~ 6 ½ - b ½· ==- o
' ' D =-o
. •. The ,yooi:s ~Ye. Ye.al,&, eqLU1.l. .. :

5. (._c.) la.± th.e C..00'\IY\ Ot'\ dJ.ffe-Y~ ~·two A·P be..°'

F-17& t te_TrY\ of· r A· P =: - 1 '

f-ou" i:::h. H ti 'tI.. A·P -;:, (-l-+3dJ '

f--i-Ts t ... ,II 1B • A·P ":.. .,..-9

f-ou_yth II 'U.- t\·P '::.. (-e+3d)
"

Then L'4).,... n - (.'Cl+)Tfrtt>

-.
-:. (-1+3dJ-(-S-t-3 dJ
- -1+3d+e -3d
- 1· -

6- Cf:) t.Lt th~ ~ o h'\ ' l>-'l~ rM L:rte. ~egmen± ➔ oi ""1nq

. V

• p.(..-3,lo) 0-Nj 8 ( €,, - S) ,· s d-.t.v: du;f b~ (l 1 ' ) be It ! I

-
T\t.Lirefo--re J -I ==- 6 k
~,
-t-1 (-3) ~~ ::. "7-k_
I
) ~ =- ~
?
=> c.2.:-.,...
- -
~

- .....
I~•.

I
- :l::h.e... v0.lt.Le..s
1-- (.d) l>!. i vtq be..cl.iof\ f-o'Ym.u.la- ~ b.ulob t.i tJ-t.:t.l N:1
- V 0

oP q)ven poin:ts i0 l i u!) e. ' h.OJJ'e.. I

- V
,
-:- I t 'j- 3 + ~ "1-- 1-- I
• l~lt -to< 'I- (-b J '\
/
I
\.. '-+ ~ t+.J.
-- I '':f . -e\
\.. 3 3 ,/

Th e -.--e. f-oYe it Lies •,n ISL" 9u.a.d"l'a.Rt.

I

8, (9.) 'In . 6AB C , DElf BC .• . AD -· At

D,S E.C

B'I l.LSI' ~ prop o-r tio noJ..1.~ -\:he.o~.

(I -
~ x..
:x:..-~
- x.+~
:t.-1
~ -:c..(-x- I) -::.. l:x...+~) (::c.--1_)

-?> x.1-- :,c. :: : ( L_ lf -:::) ;x_-:::. 4

q._ (c) Ac.c.oydi~ t-o p --rope.Q±i es of c.)"rc.\e. ~

- -
The OYlSJ le. be-.t-w~en the. ¼:~u-tt 0-r\.d a.. c.ho"rd
= . -
- C he...re. L <9-P'< 3 0°) IS. e..q_u.o...L to the. 0-Y\.;J le. ALLbten d e.d by
tha.i -c.ho'Y d Cch ocyd p s..') on. th.e. oppos,u ~tde of lh.e
C.l'YC Le ( h_e_ye.. lPB6\)
0

ThuS I LPr.3~ ~ L ~P'< -=.'30 I Now , L PB'( ::.. \ 80'° - l&P'1

0
H cnc.e , Lt° B S -:: :. I Sb

--
l!tl,,
 \ "--

,,
. L . ' j I l I
'
I
'

1A· Cd) t~t c.irrc le !-ou.c..h .the Aide p e.

-lhf2 a.l: A, $
~rt ok ~ ' (.2_~ o..l c_. & Sf a±- .DI ' o/ ~
/
' . AP::: PD, DS=SC., ,e._c = ~B . a.n.'d 1J.\ Q: Bf;)... ~/ ,~--
"'
~

l,
Since -kh.e l.2.ngth.s of -l:he- • ~, I

c..
t
t::CV\ q~S , d)\,o...vJn.
u
f-'¥0 hf\\ CL polv-± bl cJ-y-cJe. ~ '
)

a..1\,.CL e.q u.o.l.

..
' - ,. .., • 4, •
~ \'-- / Q ' '

•
•• f\P+PQ.. = P Q.. =- PD-t- Q.S • • • • (j)
"'·
e
ct. +' cs =- ~s = R..B+ oS • r "! {ii) • ' ~ I ! \

AG\di~ '
(I) ~oi) p 9- +-. r&' ~ ~ ~ +p s . He.nee. P 9- tt.S -::: SR-+ Ps
(/
. ;
. J '. • .''I r ·' I'' I I

. '

ll• (c) y j ve.n c.os X. ~ Cl. ),

& i Y' -:;t =- .J t- cos2=:;e :., ft- {'a._ \L :- ~ t - ct.2...

'- b / b'l..
I

Nou:> ton ::x., = \

~,n.t,:__
Co~:X... '
', 1

~ i an :X:.. - ✓ b2- Q..2 - ✓ b2.-~1.

,b
41 I ) Q

lj
a./b l l

,·
I
I
'
I
. -
-.
' I •. I

-
--.. ,,~ ,
 \~. Cb) t. Cl()~ e - +- I - ~ec.t..e_, ~+I
\ +bee. e l + S~c.e ,

\ ' <
' =- • > (bee S -\-.1 ),(bc..G {;)-I) 4-- '
(l~sec. e)

·. -- LC~ec e - 1) ;- \ -
' , -::... ~ e...c.e
'
co.se

12>· (Q.) H ei.ghi:. of tM wn.ll =- B c.. C

V
bi I"' 3 o•
-- Be
\5
L ..
"/bo
~ ~ -=, Sc '
' ' ''(..o/

I
~
..~·- 5 .' , /
~ e,c..-:= \S
cot.
~ l·Sm· I I
' Pl/ e,
I
: !

' Lt·
(Q.) The lt2..n.gth
.... oF Q.")'C ond o..,s--e..a. A of- 0.. c.i.Ycle 0~ -Ya.cl ;us
'Y O..Ye given b'J ~ I ..
1 I,
- -
Q. = Q. y ~ 7'.-Y o..nd A -=..! 6) 'I,. 'f\°Y2. I
3 bO.., 3 bo 0
...

\I ..
• l-=- I s o _'/.. ~ "'I- ~~ )(~ \ =- ss C.l'Y'\
3 bo 1
- II
II
L_ - - -
·-
\ I I
 ,s. .

"'-· Cb') q, \/'fn. e~ 9 o• and,

t,.:) e.. ILno~ tho±, Lltf\ SY th of Q.f\ a:zrc. -::- (9 '}{ tJ..""'
- 3bo 0

~ l,7'. == 9o 0
~ ~ '1\~
3 b0°

⇒ b - o2..I ~

~ 'Y =: (~CM: '

.
I b• (d) Tottl O'\o...~b)~ =- · \ t + 4-+ 1 .~ et-~ .

q-y-ee.n nlo..Ybles ~ \ I I blOt..Ck. t')')o.,,bl es '= 4

V

pu-rple rno..Yb/e..s =- 1
p(not pU--rple) - 11 +4
e)__
- \5
-t.~
- O· 69

lr• Cd) To-loJ. C Ol,d Si == s ..2.. I

-

T oL,.l no. oF a.c e. c::a.rds ::. <-t

-
P ( d'tOLwi nSJ ~ce.. ca~ds) :: <-t
5~
- I
13
i
~

I
..
~

-
!
IQ. (d) The. c u.rnm.u...Lo..± t.ve. f'Yeq ve.nc..'j to.hie ,· s :
...
,
'
.. ' I

CI a.s$ I f',yeq u e.nc~ -cumv l·o.±i ve... f-'re't.

o-5 fa lo
s-ro IS « 5'
lo-IS"
'~ 3:,-
Sr
15-~(:) ~o
~o-o25 9 l,l,
;

.. He....,e.. , N -.; 6 C:, ~·-33 . , l

-~-- ~ 1
' '

. Th.us 51 is ~ust • ~....,...eo.fe.Y th.a.rl 3 3 ,. \

' -
. ' I

• rne.-d.io.n c. La..s.s ~ lo--\ 5""

• • '
- mod oJ.
C-\a.ss - 1s-oto
.
The. uppe.,r L:Wti't ,s l'S ~d f-o, m.odol c...lo.S.S ~o.
. ,. .
SuM of upp~--r U.m.i tS OA~ of mecJ...io.n cl.ass o...nd modal.
c..1a.ss - lS+~o ==- 3S
I
 t "J. (.b) Bo th , A~A~,rlion CJ-) .o:re.. -bYu e b.u±
(Pt), o..nd 'Yi"e.a5o f\ ,
t"("e-0..SOn (¥-) is not c. o "f"'rec.t e.>< p I o.no.±to r'l ~F As.se.¥-cion l_A-) '
I . f,\~ea.. bµ:> e..pt : .b'I mitu.l.t-e ha.n.d ,n t 5 mi n.u.le~.

- e
3€,0
")( 7\ '"Y 2.

-. - ..

. --- 3o ~ ~et )(. ~~"=f-

3bt:> "":f
' '
c;- c:...t'Y'~
. ·. :::. 1,7/- • \.., ::. I l ..t '-
6

~o- (1>) So±h o..1>~ e.irrio nond ,(Le.a.so(\ CR-) Cl.Ye. t'Y.u.e.. o.n.q
(.A)

')'"e...oc.s o n (IL) is not -the o:>~ec.t e..~ P Lo.na..±i.o n of A·

A'YeA. bv-)e.pt by. minµ_!:,e.. h.o...t'LC1 ,n ' $" mi ru.Ltes.
~ e ~ ]\"•r''-=:... 3 0 x ~~ x:,.. x=,-
3'9 3bo 7-
I
I'::' 1 ':J, I ~ \ol ~ c.. n-,. ~ .
6 - I

I . ' ' ' I . I

~ ,. Le..t Cl. = ½ 0 ½ o.nd b:::9b

I
404 :- ~ ')( ol "}(. \ 0 t '

I 9b - ol ")( ot ){ ~ "'!- ~ )(Gt,.')(3

•
• • LC.M of Yo4. OU'ld ~b is c:t. ~ °'- )( ~ 'X ~ )(. ~ ~ \ o \ X 3 - 9 b 9.k

HCf oF 4-0 ~ o..nd 9 b ,· s ot. )(~ =- 4. , : . HC F ">(.. L.Ct-,t\ -= i a !r€J ½.

~

.'
-
 ct~- :tn A ABC , LM" ec ~

B'I p-Y-opo-r honctl.i ty_ ~eo ,rem, r-1\~

""/-
~

~
V

PIIVI ::. AL _(\) f\~l;.. I'

-
,AB AC
~
I.n 6ADC., LN 11 c.~
~~ V
D
B'I p-yop e-y-iionoJi. ty the.o Ye.JV' , -
-
AN - AL ~ -·
AD Ac. I
I

F--Yorn [) & ~) AM ::::- AN p

AB AD
I\
I
I
I \
al 3. To p'Y0O'ie... : PS:::\ ( P G + 9-fl +Pls)· ' Al A \ Q
.;i_ y ~1-L.
P.Mof-: -
P..8 - P (_
c·To.r::Je.>ttS d,ycu,.:>n f'l'OYY\ o.n ~tt-rr,.OJ.
oo, ~ P i:o Q ci"'Tdr_ a.~e ~vo.1.
J Bl \
'
G--8 -=- G-A , r<. c. ==- Fr~ /\
I
I \
. ~

NO v) 1 ct p e, -= PB-t-f>8 .. ' . '· ; . ~_/ \

J

- PS+Pc.
- PQ.+&..B+Pc ~ C.P&+Q.S)-+(_Pie...+ ~c)
. -
~ c.=- A-R_J
I - f> Q...+ (9.8 -t-f> f<. --t-A ft [ • :
-==-- p S--+ p I<-+ [. Q.S +A-r.?.-)
I --
:::. pg. -t~R.. + (.~A +A-~) (. ~ SB= Q.A.)
l -
(.: l9-A rf\-fl. ~ <9-R) Henu.,
_rs -:::-- '/ 2-C_P G-+PP-+e f<-) p,y-ovd,
-- -
- -
-
,
~ li. .b) We n._O,l.Je1

~ ( COS2. Li S O -t to..1'2 6 Oe) - 6,( bi ,,2.4 5°-.i:o.n L3 o•)

, \'1-
\
...

-- ~
~(
' \ 1.-+- ( ../3)~- -6 r ' J\.,_ - -(L
., \.. .[i J - ._ .... /_r;.. ~ J _. I

- -,
.'
- ~
' +3 - 6 ( r
\. ..l.
- \ \. \

... o.2. - .._ 37.....)

--- o<. / \+
I
~ \ - b ( 3- ~ \
.z_ .J . \_· 6 ~
==- 1---' ~ ~ .
'- I

, •1i

~S· b) r-=- lf CrY\ I e-=:-3o

A-"re.0- 0 (:.. ,g.e..c..Th'Y d""c. le.:: e- 'X 7'"Y 2 --' =- 3 o• "><! 2S:)( ½ )<y
36o~ 3b<S-

- -:::-- ' 7'" ~lb :;;. y ~ :::Lf· \86 C..n"lL

\ol '
3

A--re....Q.. oF ci.'Ytle. -A~ o F tno~ se.db-Y- :

. t'Ylel.4 O'Y
.be..c.±o..Y-- -z:- f'Y\
A-Ye.CA 0 J: . . •
,J

- /
'
4· \ 8-'
.
~'Y~-
I ,
I
.
I •==-f~., y')~x(Lf)'--~.' Q-t I .
(n<>jO'T

--
I
- 3·Jlf ){lb-½· t 8-' \ M )

-ll
- ½ b· oS~ CJ'\"\ 2. '-.Cmt "o 0/
-
,

-- ...
 ~I:,. ~(:X) - a..:x ~-+ b :::c+ C
.. .
.•. "<+'3 =:: ·-b .,, ol e- C. I

°'- , ,o... . I , •

~e.. h().)Je,
- -< + ,a. 2..
-
o{ ,, + ~ 2. I '

ol__
I 13 ol f3 ' '

-- (ol-tt3) - o'l~f3
~

c~+fr)
~I~ ~
'

I ---b \ '2... - r:fl._,'I, C

'=-
\... '
c,._ ) I
;
CL '
'

c_l°'-
-- b~ - °<-. C
Q..."2.- )
0....

- 67 (A,_
. - I
C./ q_
,
C/Q... I

.,
:
- b'l.- a..c: -
a..c I
• I

H~c_e.. o< -+- f3 -- r-> ~-C\...C. .

(3 ol.. CLC.

....J,,
~

--
\ ~ ~ ----.......z__"-- '~
'1-- ~ L
 ~-=,.. _) - (
'
A t'\'}i
point on the i:c-~i.s h<l.S. '-:I 1-- c.. o o')'c:Un ct±e ts O·
-
So, ihe poi\"\t
0...
-v
on the. ot: a.::c.1's be A-(:x,o) t,,0h1ch is o.:t- a..
d.ib-bo.n. c.e... eF s units f-'YOM s-(-1,4 )·, I

lJ6\n.9
'V
oli6to.nc.e. F-o.,rn u..l a.. .J •C:x.. -=c.tl·.+ IC<:l 2..--'-J ' ) ~
==--
V v·

No\.l!l, d.it:) to.nee_ be.±wee.n. e-(_~\'; 4) and ·an.y p·o, nt- on. th..e..
u
ot- o.x..is C:x, o) is I~
: I

-
I
Di t:,-¼:an ce. ::: j (..::c-+1)2-+ 4 2.

I s ==-- ..J ~.2+,2.;-- ~+ t b

s I = J xi + ct.::t +' -=,-
' ,

Squ..o..Yt ng both.• ~19-es· •

~ 5...,- ~2,.. + ot .::x:.+ r~- I

::::- ~~:+ .J.:x...-:-; .B

0 - I l

Po_..c:l:o~i '2-t ~ H-t~ o,.bev~ e.q u..o.± L·c n ~

~
V
cx,+4)(,x, -~) :=-0 ' ..

OC, '2::---4- Cl.1\.d ~===-.t.

H e..nce.. the... -1:wo points <l..'Ye .
(_-4, o) o..n..d {~ o) •
' -
- • I

j
I

I
 ~9- L.Q. t i:he.. ten's d..l~tt no be. oc. Qnq one's di.git no-be. \j •
..,
- V

So, the.. n.o. lA.,tll be.. =:-\ox.+~

Give..n : x..+'a = 9 .. - -. CJ)

9 (f ox.+y) -::: ~Ct o\j -t-~J ~ a ex.-, ''rt =-- o - ... - - di)
- .v -
On ~olvit'\..9
....
tf) o-nd tti) ~ i n"\.u.lt:a..n co u..s LJ.;f, 'fOU. t>Ot\l •
-
Sl c..t ::x.-==-1 d "'I, ==-- B . '
().n.
'tJ

Thexe-f-d¥"e dui.ire.d no. i's I 9, I I

at~· b) l~i:: AB be. QoF Q.. C-ho-v-d C. .t.'rrc...Le..----

~A
~i:1:h cen..i--e -r 0, o.n.d u.± AP OJ\..d / -~
I
BP be ch.Q. to.n~u"Lts cu Aa..nd 8. o_ {..\
~ -
~·
j ' I

Su..ppose... t.he. ~..,en-ts m.e..e.t- o.t P. '

\ )
~ .,,?,--
Join o.P. I• -------D...

To p-yoove : L PAC..=- LPSC

:t:n AP c.f\ 0-r'\.d L!:l p C B I ~ .e... h..cut-e..
f'Pt==-P8 .

LAPC. ~ LBPc.

r ·.. the. L a.ng etds o...,e.. e..qu..o..L.Ly_ tn c. ,u4 to ltne_ 1otnf~

V
u. V

e..,~ 're.Tnal p O j flt_ and c.e.n.rre oF c.i'YcL. 1 ' p C. -=- Pc. CC.OYV"f'O'V
- ....,
5o,6y 5 A.S, we... ho..we.. APAC::: llPSC. ~ LP AC :::. LPf>C .
L - -- -
I
 30 -(g..) bin A--~ binS A I

~ cos'A - cos p.

=- oln A (1-ol$ln~~ )
c.os f\ C~ c_os2 A-- 1)

- .bin A (1-"'~inLA ) [ ..' c.os2...A:.:: \-.s.in'L'+]

cos A (Cl{ C..1-Si n~A)-1)
..
- ~in A (I-~ s.in2.A)
cos A ~ - ~~int.A-I)
.t,ln A (1-otslt'\t.A)
- co~A ( I - al s1' n '- A J
I

- Sl'f\A I

cos A

- tQJ"\A.
.-

.•- .blnA-eJ..sin3 A - i:OJ'\ A

~ c.as:3 A - c.os A

LHS ~ £HS
He¥1c.e. p-...-ovcd.

-
'
--
-
...--,,, -
 31. I
I --
C.lo.SS- 1n-\e'W'iaJ M,'olpo l n:t Lt•:::.:x.t-~s
~vine..~ I ,-,._ • Ft J.tt
,_, 0
s 3 -cot -b
11--lo IS t,, -, -'-,
s:J l-3o ots- ~ 0 0
31-Lfo 3S s I s-
.
Lfl-So Lj 5" ~ I
d._ \ .t.
Toto...l o(O -I 0
I
-=I-
I

He..,e. o.s.su f'Y\ed m.e..o.n ~~-s (A)

Appl~it"l9 bte.p ol evi o.±ion. f 0tm.u.LQ:
-
M t.M =- (A+h'\ ')( 2. ~ t 1-'C
£ f( I

\,,l h..e..Ye. A":- ~5

h. ~ lo
~ ~t~.· ~ - I:,+ L-½) -t'o +s + te<-: 1
s ub6 U.t.u± ln_q V
,::h e.l)f. vo..Lu.e...s ,•n...to t-oYrnu.l.Q :
.
I
i
Me..o.n =- oL 5 -\-\ 0 ')_( 1
oio

j - ol5+3· S
I

]I ~
ol8·5
I He.nc.e me.a.n i's ~ 8 ,5 ·
I
I

"
I
 •

"31-. fb) - A c~0 Ydt n9 -h, 9..u es4 ion , -

I
Alq e..b ~c. mod ·e_\ ?l.+'-t ,-;;_ 9 0 c...f) - .. '.,
- ,;r 'r ~
-
,
u

O•'L ?'\-+ O•S° '-1 =- 3 b - ~-- -·0 ')

6-0
-
So\ v-f n 9 e..9 \..l. ~ro n cf) foT ~ o.nc\ OU J-lt fpl~ •e.C>Lh -2>ide oF
e_q u.a.±1.on (11 ) •b '1 lo, ~o th.o :t _ft-: c.on ..to lra on\ " rn.t~e--~-~-
• I ••
-
Le.tJA ~e . ~u.b &t!i ::LL tl'O n -to bO\V e th.e ,:,'Is te.N '\ .
~·=- 9 o- '-I •••
- - ~ - - {J\1) .
2.7'\ + ~'I =:: 3 bO - ... - • c.r") '
I 2.:( °t 0-' I ) + s '-/ '=- 3 6 0

Su. .b.s tftu. tl~ 9 0-' j F-o~ c-n ln. e.q_u.o :tton (4.).,) , \A!) e. <a e..t
V

IS o - 2'-J + ~\.J -:;_ 36 o

3'1 =-) So

\./ -=- 60
'
Sol vfn. C\ f-o...- '-I •
V
'
n =- 9 o - •b o -:::. 31 o .. ' J

1-t e.nc.e. oo mL o ~ ~o\ uuo n A o.nd 6 o mL of- &o\ u-\: .lon B

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we ire u..se.d.
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. th.e.o'Ye..m b i::o.±e.s tho1, i.F a. lln e.. ,.s
prropoy i:io f\.oJiP-1
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p O.."f"CLLle.. L to Cl. ~id e coF
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othe""t" bid e.s into two ~tinc. l points, the.n llie. L:ne.. divides
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bide.s of the. h,'o.nflle. in A
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Since., t I='" .l._ A-B JI

El is the h.ught of t-y i o...n 51 \ e.s A-DE and DB~.

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~e.. know I me.o..n Coe.) - 2: f;c.:xe

zF
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~ Lt o ==- ~ F ~ ~ ==- :to .

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o.nd A be.. the post. t.io"n of- St i-r \ • e,
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q ''"c..n condition LBAC.=45~
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⇒ BC-==- 6l o/.E_
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ihe ~ IVef\ p'b'"oblern ,s so\ve.d.