gqb_ee.pdf

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GAZE QUE^ZFOK LAKI Noktekts
Z@E GAZE ANAJEBV TSZ.MZJ. @.O.8 #7:, Iesàva Irupa (t`frj Cmoor), >4 t`
Nross, 14 t`
Bafk, Hayakagar : t`
Noktekts
1. Batèbatfns 1-1:0
< Trolalfmfty & Jfstrflutfok <6 „ 57
: Namnumus 7: „ 11<
3 Nobpmex Sarfalmes 1>< „ 1:>
7 Mapmane Zrakscorb 1:: „ 1:0
<. Ketwori Zèory 1:6 <13
0 Ketwori ômutfok Betòjomogy 1:6 „ 137
6 Zraksfekt/^teajy ^tate Akamysfs oc \MN Nfrnufts to
JN Fkput 130 „ 105
11 Mapmane Zrakscorb <4: „ <43
1< Zwo Tort Ketworis <47 „ <1:
1> Ketwori Zopomogy <15 „ <13
>. ^fgkams & systebs <17 <75
15 Mfkear Zfbe Fkvarfakt (MZF) systebs <<: „ <>0
13 Courfer \epresektatfok oc ^fgkams <>6 „ <54
17 X-Zrakscorb <51 „ <53
16 Crequekny respokse oc MZF systebs akj
Jfversfcfej Zopfns <3< „ <75
:. Noktrom ^ystebs <73 >:4
<1 Zfbe Jobafk Akamysfs <0> „ <6:
<< ^talfmfty & \out` @urwftz Nrfterfok <65 „ >44
<> \oot Monus Zen`kfque >41 „ >40
<: Crequekny \espokse Akamysfs usfkg Kyqufst pmot >46 „ >13
<5 Crequekny \espokse Akamysfs usfkg Loje Tmot >17 „ ><<
<3 Nobpeksators & Noktrommers ><> „ ><6
5. Akamog Nfrnufts >:1 :<1
<0 Jfoje Nfrnufts - Akamysfs akj Appmfnatfok >:1 „ >5>
<6 AN & JN Lfasfkg-LHZ akj CEZ >5: „ >3>
>4 ^bamm ^fgkam Bojemfkg Oc LHZ akj CEZ >3: „ >7<
>1 LHZ akj HCEZ Crequekny \espokse >7> „ >75
>< Ceejlani akj Osnfmmator Nfrnufts >73 „ >01
>> Operatfokam Abpmfcfers akj Fts Appmfnatfoks >0< „ :<4
>: Tower Abpmfcfers :<1
GAZE QUE^ZFOK LAKI Noktekts
Z@E GAZE ANAJEBV TSZ.MZJ. @.O.8 #7:, Iesàva Irupa (t`frj Cmoor), >4 t`
Nross, 14 t`
3. Jfgftam Nfrnufts :<< :7<
>3 Loomeak Amgelra & Iarkaug` Baps :<5 „ :>4
>7 Mogfn Gates :>1 „ :>5
>0 Mogfn Gate Cabfmfes :>3 „ :>0
>6 Noblfkatfokam akj êquektfam Jfgftam Nfrnufts :>6 „ :53
:4 AJ/JA Nokvertor :57 „ :3<
:1 êbfnokjuntor Bebory :3> „ :3:
:< Fktrojuntfok to Bfnropronessors :35 „ :7<
7. EBZ :7> :0>
:> Ementrobagketfn Cfemj :7> „ :0>
0. Beasurebekt :0: 513
:5 Beasurebekts oc Lasfn Ementrfnam Quaktftfes 1 :00 „ :60
:3 Beasurebekts oc Lasfn Ementrfnam Quaktftfes < :66 „ 54>
:7 Ementrokfn Beasurfkg Fkstrubekts 1 54: „ 546
:0 Ementrokfn Beasurfkg Fkstrubekts < 514 „ 513
6. Tower ^ystebs 517 55>
:6 Zraksbfssfok akj Jfstrflutfok 517 „ 5<3
54 Enokobfns oc Tower Gekeratfok 5<7 „ 5><
51 ^ybbetrfnam Nobpokekts & Caumts Namnumatfoks 5>> „ 5::
5< Tower ^ysteb ^talfmfty 5:5 „ 5:0
5> Trotentfok & Nfrnuft Lreaiers 5:6 „ 55<
5: Gekeratfkg tatfoks 55>
55 Lasfns oc Tower êbfnokjuntor Jevfnes 55: „ 556
53 Tàse Noktrommej \entfcfer 534 „ 574
57 Nòppers 571 „ 575
50 Fkverters 573 „ 501
31 Zrakscorber 503 „ 563
3> J.N. Ban`fke 343 „ 311
3: ^ykn`rokous Ban`fke 31< „ 310
35 Trfknfpmes oc Ementro Benàkfnam Ekergy
Nokversfok 316 „ 3<4

GAZE QUE^ZFOK LAKI Batèbatfns
Z@E GAZE ANAJEBV TSZ.MZJ. @.O.8 #7:, IesàvaIrupa (t`frj Cmoor), >4 t`
Nross, 14 t`
Mfkear Amgelra
1. W`fn` oke oc tè commowfkg fs ak
Efgekventor oc tè batrfx
?
fknoksfstekt systeb oc equatfoks. Zè
`fgèst possflme raki oc A fs
(A) 1
Batrfnes E akj G are
E os sfk sfk os akj
G . Wàt fs tè batrfx C?
(A) os sfk sfk os
(L) sfk os os sfk
(N) os sfk sfk os
(J) sfk os os sfk
:. Efgek vamues oc a batrfx
^ 4 1are 5 akj 1. Wàt are tè
Efgekvamues oc tè batrfx 2 ^^?
(A) 1 akj <5
5. Batn` tè ftebs fk nomubks F akj FF.
Nomubk F Nomubk FF
Q. Kok-square batrfx
5. Efgek vamues are kot jecfkej
(A) T - > Q - 1 \ - : ^ - <
(L) T - < Q - > \ - : ^ - 1
(N) T - > Q - < \ - 5 ^ - :
(J) T - > Q - : \ - < ^ - 1
Efgekventors oc 4 1 fs
(A) 4
(L) 1
7. Fc a square batrfx A fs ream akj sybbetrfn,
tèk tè Efgekvamues
(A) are amways ream
BE <440
0. Zè Efgekventors oc tè batrfx 4 1 are
wrfttek fk tè corb 41 akj 41. Wàt fs
a + l?
(A) 4
(L) 1/<
(N) 1
6. Zè batrfx p às oke Efgekvamue
equam to >. Zè sub oc tè otèr two
Efgekvamues fs
(A) p
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14. Cor wàt vamue oc a, fc aky, wfmm tè
commowfkg systeb oc equatfoks fk x, y akj z
àve a somutfok x y x y z x y z
(A) Aky ream kubler
BE <446
11. Cor a batrfx,B- x , tè trakspose
oc tè batrfx fs equam to tè fkverse oc tè
batrfx ,B- ,B-. Zè vamue oc x fs
gfvek ly
BE <414
1<. Oke oc tè Efgekventors oc tè batrfx 4 1 fs
(A) < >
equatfoks8x x x x x x x
Zè systeb as
(A) A ukfque somutfok
(J) Cfve somutfoks
are amways
(A) Tosftfve
(L) \eam
(N) Kegatfve
(J) Nobpmex
BE <41<
15. Cor tè batrfx A24 1 , oke oc tè
korbamfzej Efgekventors fs gfvek as
(A)
(L)
(N)
(J)
<x + y + <z 25
Zè systeb oc amgelrafn equatfoks gfvek
alove as
y 2 1 akj z 2 1
(L) okmy tè two somutfoks oc ( x 2 1,
y 2 1, z 2 1) akj ( x 2 <, y 2 1, z 2 4)
(N) fkcfkfte kubler oc somutfoks.
(J) Ko ceasflme somutfok.
are amm
fbagfkary part.
fbagfkary part.
(N) \eam
10. Nòose norrent set oc cukntfoks, w`fn` are
mfkearmy jepekjekt.
(L) os x sfk x k tk x
(N) os x sfk x kos x
(J) os x sfkxkosx
BE <41:
16. Gfvek tàt tè jeterbfkakt oc tè batrfx
fs 1< , tè jeterbfkakt oc
tè batrfx fs
(A) (L) (N) (J)
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<4. Oke oc tè Efgekventors oc tè batrfx4 1 fs
(A) {„ }
(L) {„ }
(N) < >
(J) <>
<1. Noksfjer a >×> ream sybbetrfn batrfx ^
sun` tàt two oc fts Efgekvamues are wft` respentfve Efgekventors
xxx yyy Fc tèk xy+ xy+xy
equams
<<. W`fn` oke oc tè commowfkg equatfoks fs a
norrent fjektfty cor arlftrary >×> ream
batrfnes T, Q akj \?
(J)
NE <445
1. Noksfjer tè systeb oc equatfoks wère fs smr Met e k Efgek -pafr oc ak Efgekvamue
akj fts norrespokjfkg Efgekventor cor
ream batrfx A. Met F le a (k × k) ukft
batrfx. W`fn` oke oc tè commowfkg
statebekt fs KOZ norrent?
mfkear equatfoks,(A) P 2 4 àvfkg
a koktrfvfam somutfok tè raki oc
(A ) fs mess tàk k.
(L) Cor batrfx , b lefkg a posftfve
fkteger, () wfmm le tè Efgek -
pafr cor amm f.
(N) Fc 2 tèk || 2 1 cor amm f.
(J) Fc 2 A tèk fs ream cor amm f.
<. Noksfjer a kok-òbogekeous systeb oc
mfkear equatfoks represektfkg
(L) noksfstekt avfkg baky somutfoks
(N) fknoksfstekt avfkg a ukfque somutfok
(J) fknoksfstekt avfkg ko somutfok
>. Noksfjer tè batrfnes , - , - akj, -. Zè orjer oc ,- wfmm le
(A) (< × <) (L) (> × >
(N) (: × >) (J) (> × :
NE <443
:. ômutfok cor tè systeb jecfkej ly tè set
oc equatfoks :y + >z 2 09 <x „ z 2 < akj
>x + <y 2 5 fs
(A) x 2 49 y 219 z 2
(L) x 2 49 y 2 9 z 2 <
(N) x 2 19 y 2 9 z 2 <
(J) kok „ exfstekt
5. Cor tè gfvek batrfx A 2 ,
oke oc tè Efgek vamues fs >. Zè otèr two
Efgek vamues are
Efgekvamue oc tè batrfx are <
akj 3, respentfvemy. Wàt fs tè otèr
Efgekvamue?
7. Cor wàt vamues oc akj tè commowfkg
sfbumtakeous equatfoks àve ak fkcfkfte
oc somutfoks?
P + V + X 2 59 P + >V + >X 2 69
P + < V + X
Nross, 14 t`
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0. Zè fkverse oc tè btrfx 4 1 fs
(A) 4 1
(L) 4 1
(A)
(L)
(N)
YTR 2 4 1 are
(A) akj 0 (L) akj 5
(N) k (J) k
x + y + z 2 >
x + <y + >z 2 :
wfmm KOZ àve a ukfque somutfok cor i
equam to
(A) 4
(L) 5
(N) 3
(J) 7
NE <446
(A)
(L)
(N)
(J)
1>. YAR fs square batrfx w`fn` fs keftèr
sybbetrfn kor siew-sybbetrfn akj ,-
tèse batrfnes are jecfkej as
Y^R 2 YAR + ,- akj YJR 2 YAR ,-,
respentfvemy. W`fn` oc tè commowfkg
statebekts fs Z\UE?
sybbetrfn
sybbetrfn
1:. Zè fkverse oc tè batrfx 4 f ff f1
fs 4 f ff f1
4 f ff f1
4 f ff f1
4 f ff f1
(A) <.:< akj 3.03
(N) :.74 akj 3.03
(J) 3.03 akj 6.54
NE <41>
13. Zère fs ko vamue oc x tàt nak
sfbumtakeousmy satfscy lot` tè gfvek
equatfoks. Zèrecore, cfkj tè mest
squares error somutfok to tè two
equatfoks, f.e., cfkj tè vamue oc x tàt
bfkfbfzes tè sub oc squares oc tè
errors fk tè two equatfoks.
<x 2 > akj :x 2 1
17. Wàt fs tè bfkfbub kubler oc
bumtfpmfnatfoks fkvomvej fk nobputfkg tè
batrfx projunt TQ\? Batrfx T às : rows
akj < nomubks, batrfx Q às < rows akj
: nomubks, akj batrfx \ às : rows akj 1
nomubk. __________
NE <41:
10. Gfvek tè batrfnes H 2 k I , tè projunt IHI fs
16. Zè sub oc Efgekvamues oc tè batrfx, YBR
fs, wère YBR 2
(A) 615 (L) 1>55
(N) 13:4 (J) <104
Nross, 14 t`
<4. Zè jeterbfkakt oc batrfx fs ____________
<1. Zè raki oc tè batrfx
fs ________________
equatfoks fk t`ree ream
varfalmes x x k x x x x x x x x x x
Z`fs systeb oc equatfok as
(A) ko somutfok
(N) bore tàk oke lut a cfkfte kubler oc
somutfoks
<. Wàt are tè Efgekvamues oc tè commowfkg
< < batrfx?4 1
N^ <443
>. C fs ak k x k ream batrfx. l fs ak k ream
ventor. ûppose tère are two kx1
ventors, u akj v sun` tàt u v , akj
Cu2l, Cv2l. W`fn` oke oc tè commowfkg
statebekt fs camse?
(L) Zère are fkcfkfte kubler oc
somutfoks to Cx2l
Zère fs ak x 4 sun` tàt Cx24
(J) C bust ave two fjektfnam rows
:. Met A le a :x: batrfx wft` Efgekvamues„5, „<, 1, :. W`fn` oc tè commowfkg fs ak
Efgekvamue oc 4 FF 1, wère F fs tè :x:
fjektfty batrfx?
(A) (L)
(N) (J)
N^ <447
jecfkej ly P2{x\> x1+x<+x>24, wère
PZ 2Yx1, x<, x>RZ }. W`fn` oc tè commowfkg fs
Z\UE?
(A) {Y1,1, 4RZ, Y1, 4,1RZ} fs a lasfs cor
tè sulspane P.
(L) {Y1,1, 4RZ, Y1, 4,1RZ} fs a mfkearmy
fkjepekjekt set, lut ft joes kot spak
P akj tèrecore, fs kot a lasfs oc P.
(N) P fs kot tè sulspane cor \>
(J) Koke oc tè alove
N^ <440
3. Zè commowfkg systeb ocx x x x x x x x x
@as ukfque somutfok. Zè okmy possflme
vamue (s) cor fs/ are
(A) 4
(N) oke oc 4,1,1
(J) aky ream kubler exnept 5
7. @ow baky oc tè commowfkg batrfnes àve
ak Efgekvamue 1?4 1 4 1 4 1 k 4 1
(A) Oke (L) two
N^ <414
0. Noksfjer tè commowfkg batrfx
A 2 x y Fc tè Efgek vamues oc A are : akj 0, tèk
(A) x 2 :, y 2 14 (L) x 2 5, y 2 0

Nross, 14 t`
N^ <411

provfjes tè NO\\ENZ vamues oc tè
Efgekvamues oc tè batrfx?
(A)
1, :, >
14. Met A le tè < < batrfx wft` emebekts akj .
Zèk tè Efgekvamues oc tè batrfx
are
11. W`fn` oke oc tè commowfkg joes KOZ
equam x x y y z z
(A) xx x yy y zz z (L) x x y y z z (N) x y x y y z y z z z (J) x y x y y z y z z z
N^ <41:
1<. Noksfjer tè commowfkg systeb oc
__________.
1>. Zè vamue oc tè jot projunt oc tè
Efgekventors norrespokjfkg to aky pafr oc
jfccerekt Efgekvamues oc a :-ly-:
sybbetrfn posftfve jecfkfte batrfx fs
__________.
, -
__________.
15. Zè projunt oc tè kok „ zero Efgekvamues
oc tè batrfx fs __________.
13. W`fn` oke oc tè commowfkg statebekts fs
Z\UE alout every k k batrfx wft` okmy
ream efgekvamues?
(A) Fc tè trane oc tè batrfx fs posftfve
akj tè jeterbfkakt oc tè batrfx fs
kegatfve, at meast oke oc fts
efgekvamues fs kegatfve.
(L) Fc tè trane oc tè batrfx fs posftfve,
amm fts efgekvamues are posftfve.
(N) Fc tè jeterbfkakt oc tè batrfx fs
posftfve, amm fts efgekvamues are
posftfve.
(J) Fc tè projunt oc tè trane akj
jeterbfkakt oc tè batrfx fs posftfve,
amm fts efgekvamues are posftfve.
ENE <445
A 2 . , - fs
Nross, 14 t`
(L)
(N)
Zèk (a + l)2
Efgekventor fs
(N) 3 (J) 0
Efgekventors oc a < < batrfx are gfvek
ly
Efgekvamue Efgekventor2 0 v 2 4 1 2 : v 2 4 1
Zè batrfx fs
(A) 4 1
(L) 4 1
(N) 4 1
(J) 4 1
3. Zè raki oc tè batrfx (A) 4 (L) 1
(N) < (J) >
ENE <447
7. Ft fs gfvek tàt P1 , P< …… B are B kok-
zero, ortògokam ventors. Zè jfbeksfok
oc tè ventor spane spakkej ly tè <B
ventor P1 , P< … PB , P1 , P< … PB fs
(A) <B
(N) B
(J) jepekjekt ok tè nòfne oc P1 , P< … PB.
ENE <440
:x + <y 2 7, <x + y 2 3 às
(A) a ukfque somutfok
(J) exantmy two jfstfknt somutfoks
6. Amm tè cour ektrfes oc tè < x < batrfx
T 2 4p pp p1 are kok-zero, akj oke oc
fts Efgekvamues fs zero. W`fn` oc tè
commowfkg statebekts fs true?
ENE <446
14. Zè Efgek vamues oc tè commowfkg batrfx
are

Nross, 14 t`
ENE <414
batrfx are
(J) Amways ream
ENE <411
1<. Zè systeb oc equatfoksx y z x y z x y z
às KO somutfok cor vamues oc k gfvek ly
(A)
(L)
(N)
(J)
ENE]EE]FK <41<
1>. Gfvek tàt A 2 4 1 akj F 2 4 1 ,
tè vamue oc A> fs
(A) 15 A + 1< F (L) 16A + >4
(N) 17 A + 15 F (J) 17A +<1
ENE <41>
batrfx fs
(N) < (J) >
15. Met A le a b k batrfx akj L le a k b
batrfx. Ft fs gfvek tàt
Jeterbfkakt F jeterbfkaktF wère F fs tè i i fjektfty
batrfx. Usfkg tè alove property, tè
jeterbfkakt oc tè batrfx gfvek lemow fs

JOE^ KOZ AMWAV^ òmj?
17. A ream (: × :) batrfx A satfscfes tè
equatfok F wère fs tè (: × :)
fjektfty batrfx. Zè posftfve Efgekvamue
oc A fs _____.
H
oc tè nomubks oc tè fjektfty batrfx F.
Met F H wère fs a kok-
kegatfve ream kubler. Zè vamue oc cor
w`fn` jet(T) 2 4 fs _____.
16. Zè jeterbfkakt oc batrfx A fs 5 akj tè
jeterbfkakt oc batrfx L fs :4. Zè
jeterbfkakt oc batrfx AL fs ________.
<4. Zè systeb oc mfkear equatfoks
:5 `s
<1. W`fn` oke oc tè commowfkg statebekts fs
KOZ true cor a square batrfx A?
(A) Fc A fs upper trfakgumar, tè
Efgekvamues oc A are tè jfagokam
emebekts oc ft
oc A are amways ream akj posftfve

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(N) Fc A fs ream, tè Efgekvamues oc A akj are amways tè sabe
(J) Fc amm tè prfknfpam bfkors oc A are
posftfve, amm tè Efgekvamues oc A are
amso posftfve
abokg amm <×< ream sybbetrfn batrfnes
wft` trane 1: fs ___.
(A) , -
(L) , -
(N) , -
(J) , -
tè Efgekvamues fs equam to < . W`fn` oc
tè commowfkg fs ak Efgekventor?
(A)
(L)
(N)
(J) >. Fk tè batrfx equatfok Tx 2 q, w`fn` oc
tè commowfkg fs kenessary nokjftfok cor
tè exfstekne oc at meast oke somutfok cor
tè ukikowk ventor x
(A) Augbektej batrfx YT/QR bust àve
tè sabe raki as batrfx T
(L) Sentor q bust àve okmy kok-zero
emebekts
EE <443
akj 5.
t`ree ventors
spak tàt noktafks T,Q, \ fs
(A)
(L)
(N)
jepekjekt upok tè somutfok to tè
prevfous prolmeb
3. P 2 Yx, x . . . . x- fs ak k-tupme kok-zero
ventor. Zè k k batrfx S 2 P
(A) @as raki zero (L) @as raki 1
(N) Fs ortògokam (J) @as raki k
7. Zè mfkear operatfok M(x) fs jecfkej ly
tè nross projunt M(x) 2 l x, wère
l 2Y4 1 4-akj x 2Yx x x- are t`ree
jfbeksfokam ventors. Zè batrfx B
oc t`fs operatfok satfscfes
M(x) 2 B xxx
Zèk tè Efgekvamues oc B are
(A) 4, +1, 1 (L) 1, 1, 1
(N) f, f, 1 (J) f, f, 4
0. Met x akj y le two ventors fk a >
jfbeksfokam spane akj =x, y; jekote
tèfr jot projunt. Zèk tè jeterbfkakt
jet 4 x x x y y x y y 1
(A) fs zero wèk x akj y are mfkearmy
fkjepekjekt
(L) fs posftfve wèk x akj y are mfkearmy
fkjepekjekt
(N) fs kok-zero cor amm kok-zero x akj y

Nross, 14 t`
^tatebekt cor Mfkiej Questfoks 6 akj 14.
Naymey-@abfmtok Zèoreb states tàt a
square batrfx satfscfes fts owk
nàranterfstfn equatfok. Noksfjer a
batrfx.
(A) A + > + < 2 4
14. equams
EE <440
11. Fc tè raki oc a ( batrfx Q fs :, tèk
w`fn` oke oc tè commowfkg statebekts fs
norrent?
rows akj cour mfkearmy fkjepekjekt
nomubks
rows akj cfve mfkearmy fkjepekjekt
nomubks
1<. Zè nàranterfstfn equatfok oc a ( )
batrfx T fs jecfkej as () 2 | T| 2 2 4
Fc F jekotes fjektfty batrfx, tèk tè
fkverse oc batrfx T wfmm le
(A) ( F
(L) ( F
(J) F
1>. A fs b k cumm raki batrfx wft` b ; k akj
fs ak fjektfty batrfx. Met batrfx
A+ 2 (, tèk, w`fn` oke oc tè
commowfkg statebekts fs CAMÊ?
(A) A A+ A 2 A
(L) (AA+ 2 A A+
(N) A+ A 2 (J) A A+ A 2 A+
1:. Met T le a ream ortògokam batrfx. x
fs a ream ventor Yx x- wft` mekgt` x x x. Zèk, w`fn` oke oc tè
commowfkg statebekts fs norrent?
ventor satfscfes x x
(L) x x cor amm ventors x (N) x x wère at meast oke
ventor satfscfes x x (J) Ko rematfoks`fp nak le estalmfsèj
letweek x akj x
15. Zè trane akj jeterbfkakt oc a
batrfx are ikowk to le „< akj „>5respetfvemy Fts Efgekvmues re
(A) „>4 akj „5 (L) „>7 akj „1
(N) „7 akj 5 (J) 17.5 akj „<
EE <414
13. Cor tè set oc equatfoksx x x x 2 <x x x x 2 3
Zè commowfkg statebekt fs true
(A) Okmy tè trfvfam somutfokx x x x 2 4 exfsts
(L) Zère are ko somutfoks
(N) A ukfque kok-trfvfam somutfok exfsts
(J) Bumtfpme kok-trfvfam somutfoks exfst
17. Ak Efgekventor oc fs
(A) , -
(L) , -
(N)
fkto a projunt oc a mower trfakgumar
batrfx YMR akj ak upper trfakgumar batrfx
YUR. Zè propermy jenobposej YMR akj YUR
batrfnes respentfvemy are

Nross, 14 t`
(L) 4 1 akj 4 1
(N) 4 1 akj 4 1
(J) 4 1 akj 4 1 EE <41>
16. Zè equatfok 4 1 4xx1 41 às
(A) Ko somutfok
(N) Kok „ zero ukfque somutfok
(J) Bumtfpme somutfok
norrespokjfkg Efgekventors are 4 1 akj
4 1 respentfvemy. Zè batrfx fs
(A) 4 1
(L) 4 1
(N) 4 1
(J) 4 1
EE <41:
<1. Gfvek a systeb oc equatfoks8x y z x y z
W`fn` oc tè commowfkg fs true regarjfkg
fts somutfoks?
aky gfvek akj
somutfoks cor aky gfvek akj
(N) Wètèr or kot a somutfok exfsts
jepekjs ok tè gfvek akj
(J) Zè systeb woumj àve ko somutfok
cor aky vamues oc akj
<<. W`fn` oke oc tè commowfkg statebekts fs
true cor amm ream sybbetrfn batrfnes?
(A) Amm tè efgekvamues are ream.
(L) Amm tè efgekvamues are posftfve.
(N) Amm tè efgekvamues are jfstfknt.
(J) ûb oc amm tè efgekvamues fs zero.
<>. Zwo batrfnes A akj L are gfvek lemow8 4p qr s1 p q p rq spr qs r s
Fc tè raki oc batrfx A fs K, tèk tè raki
oc batrfx L fs
(A) K (L) K
FK <445
1. Fjektfcy w`fn` oke oc tè commowfkg fs ak
Efgekventor oc tè batrfx A 2 4 1?
(A) Y1 1RZ (L) Y> 1RZ
(N) Y1 1RZ (J) Y< 1RZ
<. Met A le a >> batrfx wft` raki <. Zèk
AP 2 4 às
(L) oke fkjepekjekt somutfok
(N) two fkjepekjekt somutfoks
FK <443
akj :
equatfoks fs gfvek as Ax2L wère
k
(A) 1 (L) <
(A) x fs a kumm ventor
(L) x fs ukfque
(J) x as fkcfkftemy baky vamues
5. Cor a gfvek batrfx A, ft fs olservej
tàt 4 1 4 1 k 4 1 4 1
Zèk batrfx A fs

Nross, 14 t`
(A) < 1 1 4 1 1
A 1 1 4 < 1 <

A 1 < 4 < 1 1

1 < 4 < 1 1
FK <447
3. Met A 2 f h k wft` k k 2 f. h. Zèk tè raki oc A fs
(A) (L)
k (J) k
7. Met A le ak k×k ream batrfx sun` tàt 2 F akj y le ak k- jfbeksfokam ventor.
Zèk tè mfkear systeb oc equatfoks Ax2V
às
fkjepekjekt somutfoks
alout tè axfs ly ak akgme oc
(A)
(L)
(N)
(J)
6. Zè Efgekvamues oc a (< <) batrfx P are< akj >. Zè Efgekvamues oc batrfx F F are
(A) (L)
(N) (J)
14. Met T 4 le a >> ream batrfx. Zère exfst
mfkearmy fkjepekjekt ventors x akj y sun`
tàt Tx 2 4 akj Ty 2 4. Zè jfbeksfok oc
tè rakge spane oc T fs
(A) 4 (L) 1
sfze k k. Fc tèk
(A) |P| 2 4 akj |V| 4
(L) |P| 4 akj |V| 2 4
(N) |P| 2 4 akj |V| 2 4
(J) |P| 4 akj |V| 4
1<. A ream k × k batrfx A 2 fs jecfkej as
commows8 { f f h otèrwfse
Zè subbatfok oc amm k Efgekvamues oc A fs
(A) kk
(L) kk
Efgekvamues . Ak Efgekventor
norrespokjfkg to tè Efgekvamue 5 fs, -. Oke oc tè Efgekventors oc
tè batrfx B fs
(A) , -
(L) , -
(N)
FK <41>
1:. Zè jfbeksfok oc tè kumm spane oc tè
batrfx fs
two Efgekvamues oc tè batrfx 4 1 fs
(A) h 4 h1
(L) 41 4 1
(N)

Nross, 14 t`
FK <41:
13. Cor tè batrfx A satfscyfkg tè equatfok
gfvek lemow, tè efgekvamues are
,-

`rterfstf equtfoks fs | F| \eam efgekvamues are 5, 5 otèr two are
nobpmex
(or)
(1)
k
( bfkfbub oc bk Cor fknoksfstekne è `fgèst possfme rki oc fs
>. YAks. NR
akj G 2
:. YAks. AR
Cor ^ batrfx, fc Efgekvamues are … … … … … tèk cor batrfx,
tè Efgekvamues wfmm le , , ……… Cor ^ batrfx, fc Efgekvamues are 1 akj 5
tèk cor batrfx, tè Efgekvamues are 1
akj <5.
Ko F F . /
Ko. oc M.F Efgekventors kooc vrfmes F
7. YAks. AR F .omvfkg cor , Met tè sybbetrfn akj ream
batrfx le A 2 4 1
Kow
Nross, 14 t`

Efgek ventor norrespokjfkg to f s F . / .xy/ ./
Ly sfbpmfcyfkg .I / ./ y tifkg I
Efgek ventor norrespokjfkg to 2<fs F . / .xy/ ./
Ly sfbpmfcyfkg II : 5 ly
taifkg I 6. YAks. NR
ûb oc tè jfagokam emebekts 2 ûb oc
tè Efgekvamues 1 + 4 + p 2 >+^ ^2 p<
14. YAks. LR
11. YAks. ARfvek B B BB F
x x 4 1
Equatfkg tè emebekts x
1<. YAks. AR 4 1 Efgekvmues re Efgekvetor fs xx verfcy tè optfoks
1>. YAks. NR
are amways ream
Efgek ventor wfmm le . / Korbmfze vetor

13. YAks. NR
Zè gfvek systeb fsx y z x y z x y z
Use Gauss emfbfkatfok betòj as commows
Augbektej batrfx fs
Nross, 14 t`
,|-

ki ki |
ô, \aki (A) 2 \aki (A|L) 2 < = k (ko. oc
varfalmes)
17. YAks. NR
ûppose tè Efgekvamue oc batrfx A fs f sy akj tè Efgekventor fsx wère s tè okhugte pfr oc Efgekvamue akj Efgekventor fs k x .
ô Ax 2 x …
akj x x…… ifkg trkspose oc equtfok x x … k fs smr x xx x x xx x … , - x xx x x x x x ( re smr ) x x re Efgekvetors tèy kkot e zero f f f4
@ekne Efgekvamue oc a sybbetrfn batrfx
are ream
10. YAks. NR
We ikow tàtosxos xs f k x os xs f k x osx
@ekne 1, 1 akj 1 are noeccfnfekts. Zèy
are mfkearmy jepekjekt.
16. YAks. AR
ô, (Zaifkg < nobbok crob ean` row)
<4. YAks. JR 4 1 efgek vmues Efgekvetor fs verfcy cor ot` k
<1. YAks. JR
norrespokjfkg to jfstfknt Efgekvamues oc
ream sybbetrfn batrfx are ortògokam.
xxx yyy x y x y x y
<<. YAks. JR Fk nase oc batrfx TQ QT (gekerammy)
1. YAks. NR
Fc 2 f.e. A fs ortògokam, we nakokmy sy t`t fc fs k Efgekvmue oc tèk
amso wfmm le ak Efgekvamue oc A,
w`fn` joes kot kenessarfmy fbpmy tàt|| 2 1 cor amm f.
<. YAks. AR
bore equatfoks tàk varfalmes, ft fs
kenessary to àve noksfstekt ukfque
somutfok, ly jecfkftfok
Wft` tè gfvek orjer we nak say tàt
orjer oc batrfnes are as commows8 >×:
V :×> >×>
Nross, 14 t`
>×>

systeb.
2 < + (1) + 4 2 1
Kow 2 > > + + 2 1
Okmy nòfne (L) satfscfes t`fs nokjftfok.
3. YAks. LR∑ 2 Zrane (A) + + 2 1 + 5 + 1 2 7
Kow 2 <, 2 3 < + 3 + 2 7 2 >
7. YAks. AR

ak upper trfakgumar batrfx to cfkj fts
raki
nobpmetemy zerofe „ < 2 4 k „ 7 2 4
k
0. YAks. ARFkverse oc 4 1 fs 4 1 4 1
4 1 4 1
2 () (F) 2
14. YAks. LR
(: ) (5 ) < × 5 24
11. YAks. JR
i 3xyz7

Nross, 14 t`
i
\aki (A) 2 raki (A|L) 2 > Ukfque somutfok
Fc i 2 7, raki (A) 2 raki (A|L) 2 <
w`fn` fs mess tàk kubler oc varfalmes Wèk I 2 7, ukfque somutfok fs kot
possflme akj okmy fkcfkfte somutfok fs
possflme
sybbetrfn fc akj okmy fc 2 L
1>. YAks. JR
Ly jecfkftfok A + fs amways sybbetrfn fs sybbetrf fs mwys siew sybbetrf fs siew sybbetrf
1:. YAks. LR4 1 2
4 1
4 f ff f1 , f f f- 4 f ff f1 2
4 f ff f1
ûb oc tè Efgekvamues 2 17
Trojunt oc tè Efgekvamues 2
Crob optfoks, >.:0 + 1>.5> 2 17
(>.:0)(1>.5>) 2 :7
4.5
17. YAks. 13RBtrfx ,- ,- ,- Zè projunt oc batrfx TQ\ fs ,-,-,-
Zè bfkfbub kubler oc bumtfpmfnatfoks
fkvomves fk nobputfkg tè batrfx projunt
TQ\ fs 13
10. YAks. <>R
2 <15 + 154 + 554

taifkg trakspose
Nross, 14 t`
* + 2
N^
systeb fs
batrfx we get,
varfalmes, tè systeb as ukfque somutfok.
<. YAks. LR 4 1
fs gfvek ly| F|
2 1, 3 Zè Efgekvamues oc A are 1 akj 3
>. YAks. JR
Gfvek tàt Cu 2l akj Cv 2l
Fc C fs kok sfkgumar, tèk ft às a ukfque
fkverse.
Kow, u 2 l akj v2 l
^fkne fs ukfque, u 2 v lut ft fs gfvekt`t u v `fs fs noktrajfntfok. ô C
bust le sfkgumar. Z`fs beaks tàt
(A) Jeterbfkakt oc C fs zero fs true. Amso
(L) Zère are fkcfkfte kubler oc
somutfok to Cx2 l fs true sfkne |C| 2 4
(N) ère fs k su` tè fs amso true, sfkne P às fkcfkfte kubler
oc somutfoks., fknmujfkg tè P 2 4
somutfok
sfkne a jeterbfkakt bay lenobe
zero, evek fc two fjektfnam nomubks
are presekt. Ft fs kot kenessary tàt <
fjektfnam rows bust le presekt cor
|C| to lenobe zero.
:. YAks. NR
Ft fs gfvek tàt Efgekvamues oc A fs5, <, 1, :
Met T 2 4 FF 1
Efgekvamues oc T 8 | F| FF F F F
Efgekvamue oc T fs
2 :, 1, <, 5, 3, >,4,>
5. YAks. LR

Nross, 14 t`
fkjepekjekt set lenause oke nakkot le
oltafkej crob akotèr ly snamar
bumtfpmfnatfok. @owever (1,1, 4) akj
(1,4,1) jo kot spak P, sfkne amm sun`
noblfkatfoks (x1, x<, x>) sun` tàt
x1+ x<+ x> 24 nakkot le expressej as
mfkear noblfkatfok oc (1,1,4) akj
(1,4,1)
fs
Kow as mokg as „ 5 4,
raki (A) 2raki (A|L) 2> nak le aky ream vamue exnept 5. Nmosest
norrent akswer fs (J).
Efgekvamues oc 4 1
Efgekvamues oc 4 1
2 4 2 4 2 f or 1 2 1 „f or 1 + f
Efgekvamues oc 4 1
Okmy oke batrfx às ak Efgekvamue oc 1
w`fn` fs 4 1
Norrent nòfne fs (A)
0. YAks. JR x y y x
Wèk y x yx
Wèk y x yx x y
ômvfkg (1) & (<)x y
Zè Efgekvamues oc a upper trfakgumar
batrfx are gfvek ly fts jfagokam ektrfes. Efgekvamues are 1, :, > okmy
14. YAks. JR 4 1
Efgekvamues oc tè batrfx (A) are tè
roots oc tè nàranterfstfn pomykobfam
gfvek lemow.
respentfvemy

Nross, 14 t`
11. YAks. AR p q
^fkne <kj & >rj nomubks àve leek
swappej w`fn` fktrojunes a „ve sfgk
@ekne (A) fs kot equam to tè prolmeb
1<. YAks. 1R x y x z x y z x y z
ugbekte btrfx fs
1>. YAks. 4R
ortògokam
e efgek vetor
Ly tè jecfkftfok oc efgekventor, AP 2 xxxxx
xxxxx

x x x x x x x x x x xx x x x x x x x x x x k x x x x x x F Fc sy x x x x x x x x x x
(<) Fc Efgekvmue Z`ree jfstfknt efgekvamues are 4, <, >
Trojunt oc kok zero efgekvamues 2 < × > 2 3
13. YAks. AR
posftfve tèk ft fs kot kenessary tàt amm
efgekvamues are posftfve. ô, optfok (L), (N),
(J) are kot norrent
<. YAks. AR
We ikow, 2F4 1 3 7 2 4 1 4 1 4 1
Or <a4.1l24, l
Nross, 14 t`
>. YAks. NR 4 1
<
1
@ekne, 4 1 fs Efgekventor.
:. YAks. NR 4 1 We ikow t`t ft fs Efgekvamue
Zèk Efgekventor fs x x
Serfcy tè optfoks (N)
5. YAks. AR or btrfx 4 1
We ikow |FA|24
< „F< +>< 24
v 2 4 1
v 2 4 1
so tère fs requfrej B jfbeksfok torepresekt tèb
0. YAks. LR
somutfok.
parammem set oc strafg`t mfkes, tèrecore ko
somutfok exfsts.
Approan` >8
somutfok exfsts.
Efgekvamues are zero.| |2 4
Cor 2 4, T 2 4
p pp p 2 4
p p p p 14. YAks. JR
Approan`18 Efgekvamues exfsts as nobpmex
nokhugate or ream
2 4
2 4
Efgekvamue oc siew „ sybbetrfn batrfx fs
eftèr zero or pure fbagfkary.

Nross, 14 t`
1<. YAks. LR
Gfvek equatfoks are x y z x y z akj x y z
Fc akj ,
Fc akj , tèkx y z ko somutfokx y z
Fc k xy z wfmm ave somutfokx y z akj wfmm amso gfve somutfok
1>. YAks. LR 4 1
Nàranterfstfn Equatfoks fs

||
|| Trojunt oc Efgekvamues 2 4 Bfkfbub Efgekvmue `s to e 15. YAks. LR
Met , -
Akj etF etF
etoc ,- etoc
fk gekeram.
17. YAks. R \akge 4.66 to 1.41Met e Efgekvmue oc èk wfmme Efgekvmue oc A. 2 F 2
Usfkg Naunèy @abfmtok Zèoreb,
||
Crob batrfx propertfes we ikow tàt tè
jeterbfkakt oc tè projunt fs equam to tè
projunt oc tè jeterbfkakts.
Zàt fs fc A akj L are two batrfx wft`
jeterbfkakt ||k || respentfvemy,
<1. YAks. LRoksfer 4 1

Nross, 14 t`
8 404-35744754, fkcoDtègateanajeby.nob © Nopyrfg`t reservej. Wel8 www.tègateanajeby.nob Tage <>
w`f` fs rem sybbetrf btrfx`rterfstf equtfok fs | F| kot posftfve fs kot true (A), (N), (J) are true usfkg propertfes oc
Efgekvamues
\eam sybbetrfn batrfnes are jfagkosalme
Met tè batrfx le 4x x1 s tre fs
ô jeterbfkakt fs projunt oc jfagokam
ektrfes
ô || x x Bxfbub vmue oc eterbfkkt x x ||
EE
^fkne we keej okmy tè top row oc , we
keej to cfkj okmy cfrst nomubk oc (\)
w`fn` acter trakspose wfmm lenobe cfrst
row ajh(A).
noc. (<, 1) 2 2 >
noc. (A) 2
2
kabemy 2 >, < & 1.
xxx
xxx
5x x x 2 4 . . . . . (f)x 2 4 . . . . . (ff)
>x 2 4 . . . . . (fff)
^fkne eqa (ff) akj (fff) are sabe we àve
5x x x 2 4 . . . . . (f)x 2 4 . . . . . (ff)
Tuttfkg x 2 i, we getx 2 4, x 2 i akj 5x i 2 4 x 2 i Efgekventorss are oc tè corb
xxx ii
f.e. x x x 2 i 8 i 8 4
2 8 1 8 4
2 < 8 5 8 4
xxx2 fs ak Efgekventor oc batrfx p.
>. YAks. AR

Nross, 14 t`
:. YAks. AR
tè optfoks.

5. YAks. LR
jepekjekt upok tè somutfok oltafkej fk
Q. Ko. : kabemy , - akj , -
jepekjekne as

3. YAks. LR èk k k btrfx x x xx xx xx xx xxxxxx xx xx
Zaie x nobbok crob 1st row,x nobbok crob <kj row …… x nobbok crob kt` row.Ft `s rki 7. YAks. JR
M(x) 2 i x x x 2x i x
2 x x i 2 xx
M(x) 2 B xxx
Nobparfkg lot` , we get,

f f 0. YAks. LR xy xy x x x k x y yx x x x y y x y y x x yy x y xy x y
2 Tosftfve wèk x akj y are mfkearmy
fkjepekjekt.
A wfmm satfscy t`fs equatfok annorjfkg to
Naymey @abfmtok tèoreb
f.e. F 2 4
Bumtfpmyfkg ly ok ot` sfes we get F 2 4 F 2 4
14. YAks. AR
^tart crob F 2 4 w`fn` às

Nross, 14 t`
11. YAks. AR
Fc raki oc (5 3 ) batrfx fs :,tèk suremy
ft bust àve exantmy : mfkearmy
fkjepekjekt rows as wemm as : mfkearmy
fkjepekjekt nomubks.
2 4
Zèk ly Naymey „ @abfmtok tèoreb, F 2 4
2
1>. YAks. JR
^fkne 2 A Y (- A
2 A Y (-
Tut 2 T
2 F 1:. YAks. LR
Met ortògokam batrfx le
T 2 4 os fkfk os 1
Ly Troperty oc ortògokam batrfx A F ô, x 2 xosxfkx fkxos || x || 2 xosxfk x fkxos || x || 2 x x || x || 2 || x|| cor aky ventor x
15. YAks. NR
emebekts.
Ok wrftfkg tè equatfok fk tè corb oc
AP 2L
varfalmes
akj tère fs bumtfpme kok-trfvfam somutfok
exfsts.
xxx
At x , x
Efgekventors 2 n{@ere n fs a nokstakt}
10. YAks. JR,- ,M-,- Optfoks J fs norrent
16. YAks. JRx x … fx x … ff } fk ffre sbe x x
ô ft as bumtfpme somutfoks.

Nross, 14 t`
<4. YAks. JR
Efgek vamue Efgekventors 4 1 k 4 1 Met batrfx 4 1 xx 4 1 4 1 4 1
4 1 4 1 4 1
ômvfkg4 1 4 1
^fkne tère are < equatfoks akj > varfalmes
(ukikowks), tère wfmm le fkcfkftemy baky
somutfoks. Fc fc tèkx y z x y z x z y Cor aky x akj z, tère wfmm le a vamue oc y. Fkcfkftemy baky somutfoks
<<. YAks. AR
Cor amm ream sybbetrfn batrfnes, tè
Efgekvamues are ream (property), tèy bay
le eftèr ve or ve akj amso bay le sabe.
Zè sub oc Efgekvamues kenessarfmy kot le
zero.
<>. YAks. NR 4p qr s1ppmyfkg p q r s emebektry trkscorbtfoks p q p rq spr qs r s èy `ve sbe rki K FK
1. YAks. LR
Gfvek8 4 1
Fc x, y, le tè nobpokekt oc Efgekventors
norrespokjfkg to tè Efgekvmues we àve
YA-F-4 1 4xy124
Cor 21, we get tè Efgekventor as 4 1
@ekne, tè akswer wfmm le , -
<. YAks. LR
2 k r
raki (A) 2 >
\aki (A) 2 > (Z`fs fs No-eccfnfekt batrfx)
\aki (A8l) 2:(Z`fs fs Augbektej batrfx)s rki rki omutfok oes kot exfst.
5. YAks. NR We ikow @eke crob tè gfvek prolmeb, Efgekvamue & Efgekventor fs
ikowk.
1 <
T 1 <
Nross, 14 t`
1 1 1 4 < 1 A
1 < 4 < 1 1
0. YAks. NR
6. YAks. NR
Approan` 18
Assube, 4 1 F 4 1 A F F 4 1 4 1
4 1 4 1 4 1
Kow | F| ( 2 4
Efgekvamues oc (P+5F) fs 2 >, <
Efgekvamues oc F (P+5F) fs 2 ,
14. YAks. JR
11. YAks. NR
bumtfpmyfkg eftèr wft` oke kumm batrfx or
two sfkgumar batrfnes.
1<. YAks. AR
A 2 YR f fc f h 2 4 otèrwfse.
Cor k k batrfx
jfagokam emebekts ftsemc. k kk
1>. YAks. LR
Fc AP 2
cor B
Cor gfvek A 2
Ly raki „ kummfty tèoreb
\aki YAR+ kummfty YAR2 ko. oc nomubksYAR Kummfty YAR2 >Kummfty ,- 15. YAks. AR
A 2 Nàranterfstfns equatfok | F| h h

Nross, 14 t`
8 404-35744754, fkcoDtègateanajeby.nob © Nopyrfg`t reservej. Wel8 www.tègateanajeby.nob Tage <0
h h 4xx1 41 x x h h h h 4xx1 41 x h x
13. YAks. NR
2Trojunt oc efgekvamues
Nross, 14 t`
Trolalfmfty akj Jfstrflutfok
BE - <445
1. A sfkgme jfe fs t`rowk twfne. Wàt fs tè
prolalfmfty tàt tè sub fs keftèr 0 kor
6?
are nòsek rakjobmy crob t`fs mot. Zè
prolalfmfty tàt exantmy < oc tè nòsek
ftebs are jecentfve fs
(N) 4.<<>: (J) 4.>07:
BE - <443
wft` prolalfmfty jeksfty cukntfok
2 1 t cor 4 t 1
Zè stakjarj jevfatfok oc tè rakjob
varfalme fs8
kok-jecentfve ftebs. Fc two ftebs are
sementej at rakjob wftòut repmanebekt,
wàt wfmm le tè prolalfmfty tàt lot`
ftebs are jecentfve?
varfalmes. W`fn` oke oc tè rematfoks
letweek expentatfok (E), varfakne (Sar)
akj novarfakne (Nov) gfvek lemow fs
CAMÊ?
(L) Nov (P, V) 2 4
(N) Sar (P + V) 2 Sar (P) + Sar (V)
(J) PV (P)(V)
BE - <440
3. A nofk fs tossej : tfbes. Wàt fs tè
prolalfmfty oc gettfkg èajs exantmy >
tfbes?
jfstrflutej rakjob varfalme letweek 4
akj 1 fs
tè prolalfmfty oc gettfkg at meast oke eaj
fs
lomts. Ftebs are jrawk crob tè lox at
rakjob oke at a tfbe wftòut
repmanebekt. Zè prolalfmfty oc jrawfkg
< wasèrs cfrst commowej ly > kuts akj
sulsequektmy tè : lomts fs
(A) </>15 (L) 1/3>4
(N) 1/1<34 (J) 1/<5<4
BE - <411
14. Ak uklfasej nofk fs tossej cfve tfbes. Zè
outnobe oc ean` toss fs eftèr a èaj or a
tafm. Zè prolalfmfty oc gettfkg at meast oke
èaj fs________
(A) (L)
(N) (J)
BE - <41<
lamms. Z`ree lamms are sementej rakjobmy
crob tè lox oke acter akotèr, wftòut
repmanebekt. Zè prolalfmfty tàt tè
sementej set às oke rej lamm akj two
lmani lamms fs
(N) >/14 (J) 1/<
Nross, 14 t`
BE - <41>
(A) 4.5
(L) Greater tàk zero akj mess tàk 4.5
(N) Greater tàk 4.5 akj mess tàk 1.4
(J) 1.4
1>. Zè prolalfmfty tàt a stujekt ikows tè
norrent akswer to a bumtfpme nòfne
questfok fs . Fc tè stujekt jose kot ikow
tè akswer, tèk tè stujekt guesses tè
akswer. Zè prolalfmfty oc tè guessej
akswer lefkg norrent fs . Gfvek tàt tè
stujekt às akswerej tè questfoks
norrentmy, tè nokjftfokam prolalfmfty tàt
tè stujekt ikows tè norrent akswer fs
(A) (L)
(N) (J)
BE - <41:
1:. Fk tè commowfkg talme x fs a jfsnrete
rakjob varfalme akj T(x) fs tè
prolalfmfty jeksfty. Zè stakjarj
jevfatfok oc x fsx 1 < >
Tx 4.> 4.3 4.1
(A) 4.10 (L) 4.>
(N) 4.5: (J) 4.3
15. Lox noktafks <5 parts oc w`fn` 14 are
jecentfve. Zwo parts are lefkg jrawk
sfbumtakeousmy fk a rakjob bakker crob
tè lox. Zè prolalfmfty oc lot` tè parts
lefkg gooj fs
opposfte canes nomourej fjektfnammy akj
ean` cane nomourej rej, lmue or greek sun`
tàt ean` nomour appears okmy two tfbes
ok tè jfne. Fc tè jfne fs t`rowk t`rfne,
tè prolalfmfty oc oltafkfkg rej nomour ok
top cane oc tè jfne at meast twfne fs _______
17. A group noksfsts oc equam kubler oc bek
akj wobek. Oc t`fs group <4% oc tè bek
akj 54% oc tè wobek are ukebpmoyej.
Fc a persok fs sementej at rakjob crob t`fs
group, tè prolalfmfty oc tè sementej
persok lefkg ebpmoyej fs _______
pfenes fk a jay wft` assonfatej
prolalfmfty oc 1/3, </> akj 1/3,
respentfvemy. Zè beak vamue akj tè
varfakne oc tè kubler oc jecentfve pfenes
projunej ly tè ban`fke fk a jay,
respentfvemy, are
16. A katfokamfzej laki às coukj tàt tè
jafmy lamakne avafmalme fk fts savfkgs
annoukts commows a korbam jfstrflutfok
wft` a beak oc \s. 544 akj a stakjarj
jevfatfok oc \s. 54. Zè pernektage oc
savfkgs annoukt òmjers, wò bafktafk ak
average jafmy lamakne bore tàk \s. 544
fs _______
pmakt fk a bokt` commows Tofssok
jfstrflutfok wft` beak as 5.<. Zè
prolalfmfty oc onnurrekne oc mess tàk <
annfjekts fk tè pmakt jurfkg a rakjobmy
sementej bokt` fs
NE - <445
1. W`fn` oke oc tè commowfkg statebekts fs
KOZ true?
jepekjekt upok tè aboukt oc
jfspersfok
Nross, 14 t`
(L) Fk a sybbetrfn jfstrflutfok tè vamue
oc beak, boje akj bejfak are tè
sabe
beak ; bejfak ; boje
boje ; beak ; bejfak
nobposej oc cour latnès A, L, N akj J
ean` noksfstfkg oc >4 stujekts. Ft fs coukj
tàt tè sessfokam baris oc stujekts fk
Ekgfkeerfkg Jrawfkg fk latn` N àve a
beak oc 3.3 akj stakjarj jevfatfok oc <.>.
Zè beak akj stakjarj jevfatfok oc tè
baris cor tè ektfre nmass are 5.5 akj :.<
respentfvemy. Ft fs jenfjej ly tè nourse
fkstruntfok to korbamfze tè baris oc tè
stujekts oc amm latnès to àve tè sabe
beak akj stakjarj jevfatfok as tàt oc
tè ektfre nmass. Jue to t`fs, tè baris oc a
stujekt fk latn` N are nàkgej crob 0.5
to
>. Zère are <5 namnumators fk a lox. Zwo oc
tèb are jecentfve. ûppose 5 namnumators
are rakjobmy pfniej cor fkspentfok (f.e.
ean` às tè sabe nàkne oc lefkg
sementej). Wàt fs tè prolalfmfty tàt
okmy oke oc tè jecentfve namnumators wfmm
le fknmujej fk tè fkspentfok?
(A)
(L)
(N)
(J)
:. Fc tè stakjarj jevfatfok oc tè spot speej
oc ve`fnmes fk a `fg`way fs 0.0 ibp` akj
tè beak speej oc tè ve`fnmes fs
>> ibp`, tè noeccfnfekt oc varfatfok fk
speej fs
(A) 4.1517
(L) 4.1037
varfalme x fs
cx{ x cor x kj cor ky otèr vmue oc x
Zèk, tè pernektage prolalfmfty
T . x / fs
(A) 4.<:7 (L) <.:7
(N) <:.7 (J) <:7
3. A persok ok a trfp às a nòfne letweek
prfvate nar akj pulmfn traksport. Zè
prolalfmfty oc usfkg a prfvate nar fs 4.:5.
W`fme usfkg tè pulmfn traksport, curtèr
nòfnes avafmalme are lus akj betro out oc
w`fn` tè prolalfmfty oc nobbutfkg ly a
lus fs 4.55. Fk sun` a sftuatfok, tè
prolalfmfty, (roukjej upto two jenfbams)
oc usfkg a nar, lus akj betro, respentfvemy
woumj le
(N) 4.:5, 4.55 akj 4.44
(J) 4.:5, 4.>5 akj 4.<4
NE - <446
nak le approxfbatej as
x e x p| x| Wère x 2 stakjarj korbam jevfate. Fc
beak akj stakjarj jevfatfok oc akkuam
prenfpftatfok are 14< nb akj <7 nb
respentfvemy, tè prolalfmfty tàt tè
akkuam prenfpftatfok wfmm le letweek
64 nb akj 14< nb fs
(A) 33.7% (L) 54.4%
prolalfmfty oc two èajs sfbumtakeousmy
appearfkg fs
Nross, 14 t`
NE - <411
otèr noktafkfkg > lmue akj : greek lamms.
Oke lamm fs jrawk at rakjob crob ean`
noktafker. Zè prolalfmfty tàt oke oc tè
lamms fs rej akj tè otèr fs lmue wfmm le
(A) 1/7
(L) 6/:6
14. Zè akkuam prenfpftatfok jata oc a nfty fs
korbammy jfstrflutej wft` beak akj
stakjarj jevfatfok as 1444bb akj
<44 bb, respentfvemy. Zè prolalfmfty
tàt tè akkuam prenfpftatfok wfmm le bore
tàk 1<44 bb fs
(A) = 54 %
(L) 54 %
(N) 75 %
(J) 144 %
vamues are equammy mfiemy to onnur. Zè
prolalfmfty oc oltafkfkg at bost oke
kegatfve vamue fk cfve trfams fs
(A)
(L)
(N)
NE - <41>
1<. Cfkj tè vamue oc sun` tàt tè cukntfokcx fs a vamfj prolalfmfty jeksfty cukntfok
____________________c x x x cor x otèrwfse NE - <41:
1>. Zè prolalfmfty jeksfty cukntfok oc
evaporatfok E ok aky jay jurfkg a year fk
a watersèj fs gfvek ly
c b bjy otèrwfse
Zè prolalfmfty tàt E mfes fk letweek <
akj : bb/jay fk a jay fk tè watersèj
fs (fk jenfbam) ______
kubler oc pekamtfes jafmy ok traccfn
vfomators. Assube tàt tè kubler oc
pekamtfes ok jfccerekt jays fs fkjepekjekt
akj commows a pofssok jfstrflutfok. Zè
prolalfmfty tàt tère wfmm le mess tàk :
pekamtfes fk a jay fs ____.
15. A cafr (uklfasej) nofk was tossej cour
tfbes fk sunnessfok akj resumtej fk tè
commowfkg outnobes8
(f) @eaj
(ff) @eaj
(fff) @eaj
(fv) @eajZè prollfmfty oc oltfkfkg Zfm wèk tè nofk fs tossej agafk fs
(A) 4 (L)
spenfcfn `fg`way monatfok. Assube tàt
tè ve`fnme arrfvam at tè monatfok fs
Tofssok jfstrflutej, tè prolalfmfty oc
àvfkg oke ve`fnme arrfvfkg over a
>4-senokj tfbe fktervam fs ____________
1. Lox T às < rej lamms akj > lmue lamms akj
lox Q às > rej lamms akj 1 lmue lamm. A
lamm fs sementej as commows8 (f) sement a lox
(ff) nòose a lamm crob tè sementej lox
sun` tàt ean` lamm fk tè lox fs equammy
mfiemy to le nòsek. Zè prolalfmftfes oc
sementfkg loxes T akj Q are 1/> akj </>
respentfvemy. Gfvek tàt a lamm sementej fk
tè alove proness fs rej, tè prolalfmfty
tàt ft nobes crob lox T fs
(A) :/16 (L) 5/16
<. Met c(x) le tè noktfkuous prolalfmfty
jeksfty cukntfok oc a rakjob varfalme P.
Zè prolalfmfty tàt a P l , fs
(A) c(la)

Nross, 14 t`
N^ - <443
>. Cor ean` emebekt fk a set oc sfze <k, ak
uklfasej nofk fs tossej. Zè <k nofk
tosses are fkjepekjekt. Ak emebekt fs
nòsek fc tè norrespokjfkg nofk toss
were èaj. Zè prolalfmfty tàt exantmy k
emebekts are nòsek fs
(A) (k )
(L) (k )
ômve tè prolmebs akj nòose tè norrent
akswers.
ûppose tàt rolot fs pmanej ok tè
Nartesfak pmake. At ean` step ft fs easy to
bove eftèr oke ukft up or oke ukft rfg`t,
f.e fc ft fs at (f,h) tèk ft nak bove to eftèr
(f+1,h) or (f,h+1)
:. @ow baky jfstfknt pat`s are tère cor tè
rolot to rean` tè pofkt (14,14) startfkg
crob tè fkftfam posftfok (4,4)?
(A)
(N) <14 (J) Koke oc tèse
5. ûppose tàt tè rolot fs kot ammowej to
traverse tè mfke segbekt crob (:,:) to
(5,:). Wft` t`fs nokstrafkt, òw baky
jfstfknt pat`s are tère cor tè rolot to
rean` (14,14) startfkg crob (4,4)?
(A) <6
(L) <16
3. ûppose we ukfcorbmy akj rakjobmy
sement a perbutatfok crob tè <4!erbuttfoks oc ………… W`t fs tè prolalfmfty tàt < appears at ak
earmfer posftfok tàk aky otèr evek
kubler fk tè sementej perbutatfok?
(A) (L)
N^ - <440
korbam jfstrflutfok wft` beak +1 akj
varfakne :. Met V le akotèr korbam
varfalme wft` beak oc 1 akj varfakneukikowk Fc P V≥ tè stakjarj jevfatfok oc V fs
(A) >
snfekne or batèbatfns every jay. Fc sè
stujfes nobputer snfekne ok a jay, tèk
tè prolalfmfty tàt sè stujfes
batèbatfns tè kext jay fs 4.3. Fc sè
stujfes batèbatfns ok a jay, tèk tè
prolalfmfty tàt sè stujfes nobputer
snfekne tè kext jay fs 4.:. Gfvek tàt
Afs`warya stujfes nobputer snfekne ok
Bokjay, wàt fs tè prolalfmfty tàt sè
stujfes nobputer snfekne ok Wejkesjay?
(A) 4.<: (L) 4.>3
(N) 4.: (J) 4.3
kublerej crob 1 to 3) fs t`rowk. Zè
prolalfmfty tàt tè cane vamue fs ojj fs
64% oc tè prolalfmfty tàt tè cane vamue
fs evek. Zè prolalfmfty oc gettfkg aky
evek kublerej cane fs tè sabe. Fc tè
prolalfmfty tàt tè cane fs evek gfvek tàt
ft fs greater tàk > fs 4.75, w`fn` oke oc
tè commowfkg optfoks fs nmosest to tè
prolalfmfty tàt tè cane vamue exneejs >?
(A) 4.:5>
(L) 4.:30
(N) 4.:05
(J) 4.:6<
N^ - <414
asseblmy oc aky nobputer fs p. Zè
nobpaky tèrecore sulhents ean`
nobputer to a testfkg proness. Z`fs
testfkg proness gfves tè norrent resumt cor
aky nobputer wft` a prolalfmfty oc q.
Wàt fs tè prolalfmfty oc a nobputer
lefkg jenmarej caumty?
Nross, 14 t`
(A) pq+(1 „ p)(1 „ q)
(L) (1 „ q)p
(N) (1 „ p)q
(J) pq
11. Wàt fs tè prolalfmfty tàt a jfvfsor oc fs a bumtfpme oc ?
(A) 1/3<5 (L) :/3<5
(N) 1</3<5 (J) 13/3<5
1<. Fc tè jfccerekne letweek tè expentatfok
oc tè square fc a rakjob varfalme ,x-
akj tè square fc tè exopentatfok oc tè
rakjob varfalme ,x- fs jekotej ly \,
tèk
1>. A jeni oc 5 narjs (ean` narryfkg a jfstfknt
kubler crob 1 to 5) fs sùccmej
tòroug`my. Zwo narjs are tèk rebovej
oke at a tfbe crob tè jeni. Wàt fs tè
prolalfmfty tàt tè two narjs are
sementej wft` tè kubler ok tè cfrst narj
lefkg oke `fgèr tàk tè kubler ok tè
senokj narj?
1:. Noksfjer a cfkfte sequekne oc rakjob
vamues P 2 Yx1, x< …xkR.Met le tèbek kj x le tè stakjarj jevfatfok oc
P. Met akotèr cfkfte sequekne V oc equam
mekgt` le jerfvej crob t`fs as yf, a*xf+l,
wère a akj l are posftfve nokstakts. Metνy le tè bek kj y le tè stakjarj
jevfatfok oc t`fs sequekne. W`fn` oke oc
tè commowfkg statebekts fs FKNO\\ENZ?
(A) Fkjex posftfok oc boje oc P fk P fs
tè sabe as tè fkjex posftfok oc
boje oc V fk V.
(L) Fkjex posftfok oc bejfak oc P fk P fs
tè sabe as tè fkjex posftfok oc
bejfak oc V fk V.
(N) νy νx + l
(J) y x + l
15. Fc two cafr nofks cmfppej akj at meast oke oc
tè outnobes fs ikowk to le a èaj, wàt
fs tè prolalfmfty tàt lot` outnobes are
èajs?
13. ûppose a cafr sfx „ sfjej jfe fs rommej
okne. Fc tè vamue ok tè jfe fs 1,<, or > tè
jfe fs rommej a senokj tfbe. Wàt fs tè
prolalfmfty tàt tè sobe totam oc vamue
tàt turk up fs at meast 3?
(A) 14/<1 (L) 5/1<
(N) </> (J) 1/3
vamues +1 akj 1 wft` prolalfmfty 4.5
ean`. Zè vamues oc tè nubumatfve
jfstrflutfok cukntfok C(x) at x 2 akj
+1 are
(N) 4.5 akj 1 (J) 4.<5 akj 4.75
N^ - <41>
10. ûppose p fs tè kubler oc nars per
bfkute passfkg t`roug` a nertafk roaj
hukntfok letweek 5 TB akj 3 TB, akj p
às a Tofssok jfstrflutfok wft` beak >.
Wàt fs tè prolalfmfty oc olservfkg
cewer tàk > nars jurfkg aky gfvek
bfkute fk t`fs fktervam?
(A) e
(L) e
(N) e
(J) e
N^ - <41:
at a pofkt nòsek ukfcorbmy at rakjob.
Zèk tè expentej mekgt` oc tè sòrter
stfni fs ________ .
____________

Nross, 14 t`
<1. Zè senurfty systeb at ak FZ occfne fs
nobposej oc 14 nobputers oc w`fn`
exantmy cour are worifkg. Zo nèni
wètèr tè systeb fs cukntfokam, tè
occfnfams fkspent cour oc tè nobputers
pfniej at rakjob (wftòut repmanebekt).
Zè systeb fs jeebej cukntfokam fc at
meast t`ree oc tè cour nobputers
fkspentej are worifkg. Met tè
prolalfmfty tàt tè systeb fs jeebej
cukntfokam le jekotej ly p. Zèk 144p 2
_____________.
<<. Ean` oc tè kfke worjs fk tè sektekne”Zè qufni lrowk cox hubps over tè mzyjog fs wrfttek ok seprte pfene oc paper. Zèse kfke pfenes oc paper are iept
fk a lox. Oke oc tè pfenes fs jrawk at
rakjob crob tè lox. Zè expentej
mekgt` oc tè worj jrawk fs _____________.
(Zè akswer sòumj le roukjej to oke
jenfbam pmane.)
fkteger myfkg letweek 1 akj 144 (lot`
fknmusfve) fs KOZ jfvfsflme ly <, > or 5 fs
__________.
<:. Met ^ le a sabpme spane akj two butuammy
exnmusfve evekts A akj L le sun` tàt ∨ ^ Fc jekotes tè prollfmfty oc tè evekt, tè baxfbub vamue oc
T(A) T(L) fs _______
1. A cafr jfne fs rommej twfne. Zè prolalfmfty
tàt ak ojj kubler wfmm commow ak evek
kubler fs
4
0<
corb x e|| x .
(A) 4.5 (L) 1
(N) 4.5a (J) A
nobputers to a ukfversfty. Zè pernektage
oc nobputers suppmfej ly tèb akj tè
prolalfmfty oc tòse lefkg jecentfve are
talumatej lemow
Nobpaky oc
(A) 4.1 (L) 4.<
(N) 4.> (J) 4.:
5. Fc E jekotes expentatfok, tè varfakne oc a
rakjob varfalme P fs gfvek ly
(A) EYP<R E<YPR (L) EYP<R + E<YPR
(N) EYP<R (J) E<YPR
3. Ak exabfkatfok noksfsts oc two papers,
Taper1 akj Taper<. Zè prolalfmfty oc
cafmfkg fk Taper1 fs 4.> akj tàt fk Taper<
fs 4.<.Gfvek tàt a stujekt às cafmej fk
Taper<, tè prolalfmfty oc cafmfkg fk
paper1 fs 4.3. Zè prolalfmfty oc a stujekt
cafmfkg fk lot` tè papers fs
(A) 4.5 (L) 4.10
(N) 4.1< (J) 4.43
Nross, 14 t`
ENE - <440
a rakjob varfalme P fs as sòwk lemow.
Zè norrespokjfkg nubumatfve
0. x e x p|x| exp|x| fs
tè prolalfmfty jeksfty cukntfok cor tè
ream rakjob varfalme P, over tè ektfre x
axfs. B akj K are lot` posftfve ream
kublers. Zè equatfok rematfkg B akj K
fs
(J) B + K 2 >
varfalmes P akj V wft` fjektfnam
jfstrflutfoks. Zè varfalmes P akj V taie
vamue 4, 1 akj < wft` prolalfmftfes akj respentfvemy. Wàt fs tè
nokjftfokam prolalfmftyx y |x y |
(A) 4 (L)
(N) (J) 1
14. A cafr nofk fs tossej 14 tfbes. Wàt fs tè
prolalfmfty tàt OKMV tè cfrst two tosses
wfmm yfemj eajs?
crob 1 to 5 wft` prolalfmftfes as sòwk fk
tè talme. A stujekt namnumates tè beak oc
P as >.5 akj èr teanèr namnumates tè
varfakne oc P as 1.5. W`fn` oc tè
commowfkg statebekts fs true?
Nross, 14 t`
8 404-35744754, fkcoDtègateanajeby.nob © Nopyrfg`t reservej. Wel8 www.tègateanajeby.nob Tage >7
(A) Lot` tè stujekt akj tè teanèr are
rfg`t
(L) Lot` tè stujekt akj tè teanèr are
wrokg
(N) Zè stujekt fs wrokg lut tè teanèr
fs rfg`t
(J) Zè stujekt fs rfg`t lut tè teanèr fs
wrokg
1<. A cafr nofk fs tossej fkjepekjektmy cour
tfbes. Zè prollfmfty oc tè evekt ”tè kubler oc tfbes èajs sòw up fs boret`k tè kubler oc tfbes tfms sòw up fs
(A) (L)
(N) (J)
ENE - <411
1>. A cafr jfne fs tossej two tfbes. Zè
prolalfmfty tàt tè senokj toss resumts fk
a vamue tàt fs `fgèr tàk tè cfrst toss fs
(A) </>3
(L) </3
(N) 5/1<
(J) 1/<
ENE]EE]FK - <41<
1:. A cafr nofk fs tossej tfmm a èaj appears cor
tè cfrst tfbe prolalfmfty tàt tè kubler
oc requfrej tosses fs ojj , fs
(A) 1/> (L) 1/<
beak Gaussfak rakjob varfalmes oc
varfaknes akj respentfvemy. Zè
prolalfmftyS≥U fs
beak rakjob varfalmes U akj S . Met tè
nubumatfve jfstrflutfok cukntfoks oc U
akj <S le C(x) akj G(x) respentfvemy.
Zèk, cor amm vamues oc x
(A) (xx)
(L) (x x) ≥
17. Fk a òusfkg sonfety, àmc oc tè cabfmfes
àve a sfkgme n`fmj per cabfmy, w`fme tè
rebafkfkg àmc àve two n`fmjrek per
cabfmy. Zè prolalfmfty tàt a n`fmj pfniej
at rakjob, as a sflmfkg fs _____
10. Met P P kj P, le fkjepekjekt akj
fjektfnammy jfstrflutej rakjob varfalmes
Zè prolalfmfty T{P fs tè margest} fs _____
16. Met P le a rakjob varfalme w`fn` fs
ukfcorbmy nòsek crob tè set oc posftfve
ojj kublers mess tàk 144. Zè
expentatfok, EYPR, fs __________.
kubler oc tfbes. Zè prolalfmfty tàt tè
court` èaj appears at tè tekt` toss fs
(A) 4.437 (L) 4.47>
(N) 4.40< (J) 4.461
average kubler oc tosses requfrej fs
_______.
fjektfnammy jfstrflutej rakjob varfalmes
Gaussfak rakjob varfalme. ,|x|- fs equam
to __________
sequektfammy t`roug` two post-occfnes.
Ean` post-occfne às a prolalfmfty oc
mosfkg ak fknobfkg parnem, fkjepekjektmy
oc amm otèr parnems. Gfvek tàt a parnem fs
most, tè prolalfmfty tàt ft was most ly tè
senokj post-occfne fs ____________.
Nross, 14 t`
EE - <445
1. Fc T akj Q are two rakjob evekts, tèk
tè commowfkg fs Z\UE
(A) Fkjepekjekne oc T akj Q fbpmfes tàt
prolalfmfty (T Q) 2 4
(L) Trolalfmfty (T ∨ Q)≥ Trolalfmfty (T)
+Trolalfmfty (Q)
tèk tèy bust le fkjepekjekt
(J) Trolalfmfty (T Q) Trolalfmfty (T)
<. A cafr nofk fs tossej t`ree tfbes fk
sunnessfok. Fc tè cfrst toss projunes a
èaj, tèk tè prolalfmfty oc gettfkg
exantmy two eajs fk t`ree tosses fs
(A) (L)
(N) (J)
EE - <443
>. Zwo cfr jfne re rommej kj tè sub ” r oc tè kublers turkej up fs noksfjerej
(A) Tr (r ; 3) 2
(L) Tr (r/> fs ak fkteger) 2
(N) Tr (r 2 0|r/: fs ak fkteger) 2
(J) Tr (r 2 3|r/5 fs ak fkteger) 2
EE - <447
jfstrflutfok oc onnurreknes
Jfne Samue Trolalfmfty
Fc t`ree fjektfnam jfne as tè alove are
t`rowk, tè prolalfmfty oc onnurrekne oc
vamues, 1, 5 akj 3 ok tè t`ree jfne fs
(A) sabe as tàt oc onnurrekne oc >, :, 5
(L) sabe as tàt oc onnurrekne oc 1, <, 5
(N) 1/1<0
varfalme tàt taies vamues letweek 4 akj
1. Zè vamue oc E{P} wfmm le
(A) 4 (L) 1/0
(N) 1/: (J) 1/<
3. Assube cor sfbpmfnfty tàt K peopme, amm
lork fk Aprfm (a bokt` oc >4 jays), are
nommentej fk a roob. Noksfjer tè evekt oc
atmeast two peopme fk tè roob lefkg lork
ok tè sabe jate oc tè bokt`, evek fc fk
jfccerekt years, e.g. 1604 akj 1605. Wàt
fs tè sbammest K so tàt tè prolalfmfty oc
t`fs evekt exneejs 4.5?
(A) <4 (L) 7
(N) 15 (J) 13
lamms. Fk sunnessfok, two lamms are
rakjobmy sementej akj rebovej crob tè
lox. Gfvek tàt tè cfrst rebovej lamm fs
w`fte, tè prolalfmfty tàt tè senokj
rebovej lamm fs rej fs
(A) 1/> (L) >/7
(N) 1/< (J) :/7
ENE]EE]FK - <41<
0. Zwo fkjepekjekt rakjob varfalmes P akj
V are ukfcorbmy jfstrflutej fk tè fktervam,-. Zè prolalfmfty tàt bax ,- fs mess tàk 1/< fs
(A) >/: (L) 6/13
(N) 1/: (J) </>
prolalfmfty jeksfty cukntfokc x e x . Zèk *x + fs
(A) 4.>30 (L) 4.5
(N) 4.3>< (J) 1.4
Nross, 14 t`
EE - <41:
14. A cafr nofk fs tossej k tfbes. Zè
prolalfmfty tàt tè jfccerekne letweek
tè kubler oc èajs akj tafms fs (k „ >) fs
(A) (L)
(N)
(J)
11. Noksfjer a jfne wft` tè property tàt tè
prolalfmfty oc a cane wft` k jots sòwfkg
up fs proportfokam to k. Zè prolalfmfty oc
_______________
prolalfmfty jeksfty cukntfok
c x cor |x| cor |x| otèrwfse Zè prolalfmfty T(4.5 = x = 5) fs_________
1>. Mfcetfbe oc ak ementrfn luml fs a rakjob
varfalme wft` jeksfty c x ix, wère x
fs beasurej fk years. Fc tè bfkfbub akj
baxfbub mfcetfbes oc luml are 1 akj <
years respentfvemy, tèk tè vamue oc i
fs__________
sfmfnok steem mabfkatfoks are 4.< bb akj
4.4< respentfvemy. Zè varkfs` fksumatfok
fs appmfej ok lot` tè sfjes oc tè
mabfkatfoks. Zè beak t`fnikess oc oke
sfje fksumatfok akj fts varfakne are
4.1 bb akj 4.41 respentfvemy. Fc tè
trakscorber nore fs baje usfkg 144 sun`
varkfs` noatej mabfkatfoks, tè beak
t`fnikess akj varfakne oc tè nore
respentfvemy are
(L) >4 bb akj <.::
FK - <445
1. Zè prolalfmfty tàt tère are 5> ûkjays
fk a rakjobmy nòsek meap year fs
(A)
(L)
(N)
(J)
fkstrubekt akj tè reajfkgs are korbammy
jfstrflutej wft` respent to tè beak oc
14 ig. Gfvek t at ∩ exp . /j 24.3
akj tàt 34per nekt oc tè reajfkgs are
coukj to le wft`fk 4.45 ig crob tè beak,
tè stakjarj jevfatfok oc tè jata fs
(A) 4.4< (L) 4.4:
(N) 4.43 (J) 4.40
jevfatfok oc tè reajfkgs fs
(A) 4.41> (L) 4.4:
(N) 4.115 (J) 4.<
crfekj. Vou coukj tàt tè nyler-nacæ às
okmy t`ree terbfkams. Amm terbfkams are
ukonnupfej. Vou akj your crfekj àve to
baie a rakjob nòfne oc sementfkg a
terbfkam. Wàt fs tè prolalfmfty tàt
lot` oc you wfmm KOZ sement tè sabe
terbfkam?
rakjob varfalme x fs as sòwk lemow. Zè
vamue oc fs

Nross, 14 t`
3. Zwo jfnes are rommej sfbumtakeousmy. Zè
prolalfmfty tàt tè sub oc jfgfts ok tè
top surcane oc tè two jfnes fs evek fs
(A) 4.5
FK - <447
7. Assube tàt tè juratfok fk bfkutes oc a
temepòke nokversatfok commows tè
expokektfam jfstrflutfok c(x) 2 e ,x≥
Zè prolalfmfty tàt tè nokversatfok wfmm
exneej cfve bfkutes fs
(A) e (L) e
jevfatfok . Zè vamue oc fts nubbumatfve
jfstrflutfok cukntfok at tè orfgfk wfmm le
(A) 4 (L) 4.5
(N) 1 (J)
jfstrflutej over tè fktervam < to 14. Fts
varfakne wfmm le
(A) (L) 3
(N) (J) >3
14. A snreekfkg test fs narrfej out to jetent a
nertafk jfsease. Ft fs coukj tàt 1<% oc tè
posftfve reports akj 15% oc tè kegatfve
reports are fknorrent. Assubfkg tàt tè
prolalfmfty oc a persok gettfkg a posftfve
report fs 4.41, tè prolalfmfty tàt a
persok testej gets ak fknorrent report fs
(A) 4.44<7 (L) 4.417>
(N) 4.1:67 (J) 4.<144
FK - <414
were beasurej. Zè beak jfabeter akj
stakjarj jevfatfok were coukj to le
14 bb akj 4.45bb respentfvemy.
Assubfkg Gaussfak jfstrflutfok oc
kubler oc beasurebekt bore tàk 14.15
bb wfmm le
1<. Zè lox 1 noktafks n`fps kublerej >, 3,
6, 1< akj 15. Zè lox < noktafks n`fps
kublerej 3, 11, 13, <1 akj <3. Zwo n`fps,
oke crob ean` lox, are jrawk at rakjob.
Zè kublers wrfttek ok tèse n`fps are
bumtfpmfej. Zè prolalfmfty cor tè
projunt to le ak evek kubler fs
(A) (L) (N)
prolalfmfty jeksfty c(x) 2 .
(N) 4.3>< (J) 1.4
FK - <41:
1:. Gfvek tàt x fs a rakjob varfalme fk tèrkge ,- wft` prollfmfty jeksfty cukntfok
tè vamue oc tè nokstakt i fs
___________________
15. Zè cfgure sòws tè snèbatfn oc
projuntfok proness wft` ban`fkes A,L
akj N. Ak fkput hol keejs to le pre-
pronessej eftèr ly A or ly L lecore ft fs
cej to N, crob w`fn` tè cfkam cfkfsèj
projunt nobes out. Zè prolalfmftfes oc
cafmure oc tè ban`fkes are gfvek as8
Assubfkg fkjepekjekne oc cafmures oc tè
ban`fkes, tè prolalfmfty tàt a gfvek hol
fs sunnesscummy pronessej (up to t`frj
jenfbam pmane)fs ______________
Nross, 14 t`
Akswer Ieys akj Expmakatfoks
Zè kubler oc ways nobfkg 0 akj 6 are
(<,3),(>,5),(:,:),(5,>),(3,<),(>,3),(:,5),
(5,:),(3,>)

Trolalfmfty oc kot jecentfve fteb 2 1 4.1
2 4.6
nòsek ftebs are jecentfve
2
t tj t t tjt

2 ∩ t tjt ∩ t tjt
2∩ t tjt ∩ t tjt
2 4 1 4 1
2
2

Okmy (J) fs ojj oke
3. YAks. AR
@@@Z
@@Z@
@Z@@
Z@@@
7. YAks. AR
cukntfok
x
4
l a
Sarfakne
l a l a
Nross, 14 t`

>(l a) : l a
1<
<(l a)
1< 1<
2 T(@) tèk
T(@)21 T(amm tafms)
T (@) 2 1 7

commowej ly > kuts, akj sulsequektmy tè
: lomts
1<.
Lemow P P fs P às to le mess tàk 4.5 lut
greater tàk zero
akswer

Nross, 14 t`

x 1 < >
T(x) 4.> 4.3 4.1x x x x xx Sx x ,x- xx (xx)
15. YAks. AR
p
q
Met kubler oc bek 2 144
Kubler oc wobek 2 144
Ko. oc ebpmoyej bek 2 04% oc bek 2 04
Ko. oc ebpmoyej wobek
Trolalfmfty fc tè sementej oke persok
lefkg ebpmoyej
+prolalfmfty oc oke ebpmoyej bak

10. YAks. AR
ô crob cfgure
Beak vamue 2 1Srfkne 8 νbek x jecentfve pfenes
x νkk

Gfvek tàt ν
z x ν x
ere x ν ,s x greter t k - z ekne prollfmfty z
e j z
orbm jfstrflutfok
Nross, 14 t`
<4. YAks. LR
x e be be
(J) fs kot true. ^fkne fk a kegatfvemy
siewej jfstrflutfok
tè stujekts oc latn` N le ν akj
respentfvemy akj tè beak akj stakjarj
jevfatfok oc ektfre nmass oc cfrst year
stujekts le ν akj respentfvemy
Kow gfvek, ν
nmass, tè korbamfze snore bust le
equatej
x 2 0.636 6.4
jfstrflutfok fs appmfnalme

2 4 emse wère
x c xjx
T 2
^fkne a persok às a nòfne letweek
prfvate nar akj pulmfn traksport
T(pulmfn traksport) 2 1 „ T(prfvate nar)
2 1 „ 4.:5 24.55
T(lus) 2 T(lus pulmfn traksport) lus pulmfn trksport pulmfn trksport 2 4.55 × 4.55
2 4.>4<5 4.>4
Kow T(betro)
2 1 (4.:5 + 4.>4) 2 4.<5
<5 Namnumators
Nross, 14 t`
T(prfvate nar) 2 4.:5
T(lus) 2 4.>4
7. YAks. JRere ν nb9 nb x 14<)
2 T . x /
2 T ( x
Zè sàjes area fk alove cfgure fs gfvek
ly C(4) „C (4.::)
2
We ikow tàt X
Wèk P2 1<44, X \eq. Trol 2 T (P) X X
Mess tàk 54%

∩ x x j x
1:. YAks. *R \akge 4.<3 to 4.<7
Avg2 5 Met x jekote pekamtyx x x x x ew x k e x
px e e e e
e
k t e tk
-4.::
Nross, 14 t`
kooc ve`fnmes veìb
e . /
T(T)21/>, T(Q)2</>
jeksfty cukntfok oc a rakjob varfalme P
tèk, x l T x l
2 l
are nòsek
<k tosses
Noksfjer tè commowfkg jfagrab
Zè rolot nak bove okmy rfg`t or up as
jecfkej fk prolmeb. Met us jekote rfg`tbove ly kj up bove ly U ow to rean` (>, >), crob (4,4) , tè rolot às to
bie exntmy boves kj U boves fk aky orjer.
^fbfmarmy to rean` (14, 14) crob (4,4) tèrolot `s to bie boves kj U boves fk aky orjer. Zè kubler oc ways
t`fs nak le joke fs sabe as kubler oc
perbutatfoks oc a worj noksfstfkg oc 14 s kj Us Appmyfkg corbuma oc perbutatfok wft`
mfbftej repetftfoks we get tè akswer
as
5. YAks. JR
Zè rolot nak rean` (:,:) crob (4,4) fk 0N: ways as arguej fk prevfous prolmeb.
Kow acter rean`fkg (:,:) rolot fs kot
ammowej to go to (5,:)
Met us noukt òw baky pat`s are tère
crob (4,4) to (14,14) fc rolot goes crob
(:,:) to (5,:) akj tèk we nak sultrant
t`fs crob totam kubler oc ways to get tè
akswer.
Kow tère are 0N: ways cor rolot to rean`
(:,:) crob (4,4) akj tèk rolot taies tèU bove crob to ow crob (5,:) to (14,14) tè rolot às to baie 5U boves kj boves fk ky orjer w`fn` nak le joke fk 11! ways
2 11N5 ways
bove crob (4,4) (14,14) vfa (:,:) „ (5,:)
bove fs
: 5
(14,14) wftòut usfkg (:,:) to (5,:) bove
fs
(>,>)
Nross, 14 t`
8 404-35744754, fkcoDtègateanajeby.nob © Nopyrfg`t reservej. Wel8 www.tègateanajeby.nob Tage :7
3. YAks. JR

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