MATHEMATICS

MTE-3 : MATHEMATICAL METHODS

:Time: 2 hours Martmum Morks ; 50

ific{e : Question no. 7 ls conpulcrcrrr1t' Do any tou1

. questions trom qu(Ijtlons no' 7 to 6' U* of

cslculoior is not ollouqd'

ure uaa"a to thes€ three numbels res"€ctiv€Iy' ltrer't

th€t/ form a G'P. Find x, Y and z 3

I (u) ri,eighs more tlnn 65 kgs ?' 5

P.T.O'

tilr) -.0.8413, 0{1.1) = 0.8643{(1.e) - 9'e713, al} - o.s7i, Qlz.tl = o,geztl

3

2. lal

(b)

(b)

(c) lf y- +, proverhat fr-*2l fr -ry+ l.J l - x '

tiFind the maximm ,ralue of

'n x where, 0 < x < @.

x

Consider the set X.= lx, y, z, v, wl and let A - (x, yland B - [y, z, wl be sub,sets of X. Verifo theDe Morgant la.qn for A and B.

(c)

;' (a)

X 6 , 8 9 . r - a : - 1 2Y

: - . . . . _ . . . . . . . . . E : ! g

3 4 5 6 7

MTE.3 2 : : , r i . l

(c) Frc bags of rbe wetgh .102 lqgs each and another

eight bags u,€tgh 98 kgs eactr. What i" Ur" u*rage

r.veight of the 13 bags ? 2

(d) Show that

[ 0 , i f x < lI

f(x) - { rl__j=, if x > II x - r

ls disconlirnrous at x = 1. 2

3 , , , 3 r a * ,{. (a) Irt tt*, * = ^

Y; i'*' , s}row that I,s, = fo. . 3

(b) Truo identical coins are to€sed simulfar-reordy

. 100 times. TWo heads appear .30 times, turo tailsappear 30 times and one head and one tail a. ppears40 btno. Does this reSult agree with the hypothesis . .that the tossing is random at 5% level of

Irio*= J.8a, yl.** = s.99, 7315* = 7.s21

(4, the. mgan and standard deviation of a ,Binomialdisbibutton with parameters n and p are l0 and.116 respecttvely. Find the values of p and n. z

significance ?

MTE-3 F.T.O.

5. (a)

5. (a)

(b)

Obtain the equation oI ltre gphefie havins cer.rhe on

x u z .the [n€ := +-+ and passirg through t]le

5 ; L - J

points (0, -2, - Al'at:d' (2, -1, -11.

Three bo:res have

81 :5Red 5 Black

82 :4Red I B lack

8 3 : 3 R e d 6 E a c k

balls respectiveb. One box is chocen at random and

a ball is drawn, whtch ls fo:nd to be trlaclc Fird the

probabiltty that ttre bo:( Bg uras cho€€n.

Consider a pofliaton oI firp unib r A, B, C, D, E.

Lst'all posdble saqles oI size 2 dram from the

above population withorn replacenrent.

Flrid tlrc aamptotes of the q.rr,e

Y h - 9 3 - v k - 9 - 2 .

The probabitty of gettllg "o

*i.pnimf fn a pqe of a

bool{ rs t 4. What b tre probabev t}rat a page

contalns more than 2 misptlnb ?

c

(cl

(b)

4MTE.3

(c) A reading test is given to an elernentary school classthai consists of 12 ghls and 10 bor4. The rasults ofthe test are :

Girls Bot/s

Mean 74 70s.d. 8 10

Ii the difference between the means of tuNo groupG$gnlficant at 5% level of signiffcance ? g

t bo, o.os - 2'086, trr, o.ou - 2'014,gr. o.* - 2.080 |

Slate whetlrer th€ folort |rrg slat€rnenb ate true or Jalse.' Giw reasons for gu.r anslers. ZxS-70

(i) The funaion f : R -+ R defined as fk) = x> is inhrtfve.

{il) Suppose X has the unfform dishibudon on Ia, bl, then

the mean of x is b-a.

2

The dornain of the real valued function

fH = frf b the set ot a[ real nunbers.

The functbn f(x) - sin x is monotonic in the Interval[0,.d.

M Thi' triro vorbbl€s X and Y arc podltuely correlat€dthen the co,nebbon coefffcient between X ard y lieswithin Ff , 11.

(il)

(iv)

itTE-3

frinq Frrc, (fr.qs *.)mk qfrmqF, 2OO8

ITfuR

qf.*.f.-g ' qffiq ftffi

grlzJ : 2 u1v/ etfuqdc aiE- : so

q?z. : '

I , f{) crr dfrq *, y, , \Ftid{ *"fr C t sfrr F{drfi'rsf, rs t r sR v frt dsrefr t mqflTr-, -3,.e +sr qrs il F{+ qd TfrR }oft xrqt t f r i t x , y d c l l z f f i d f r S r - J

(q) Ss 6if,q s Ern 6 crt irr qdtFrFT da d'rqsrqq6i qFa ss ftqr t frr cr+6 fu+eq s i+qr* t

{{ Frd sr nrcfidr Hr E Is qfi 6t:f 4r qR

(i) 60'5 f+.fl olh e+.s f+'*r + +s d ?( i i ) esfrn*off isd ? s

MTE-3 P.T.O.

{s dds il lqqfttude 6 + t :l0G) = o's413, tO'U - 08643+(1.9) - 0.9713, ll2l - o's772,0(2'1) = 0'9821 I

. - l

( r ) qRy-H. ,dRrq-SRCfr I' r l l -x"

O - -1 #

-ny+1 ' z

(+) jj qt qf6dq q-a 516 frFvq,x

q d o < * < - . 3

(€) qgqc x = lx, y, z, q wl dRE ft qn difuCA = lx; yl dr s = ly, z, wl, {ff x * sqsgaf;t r n efrt s * Rs s +t{ f{qq {if,Ifud t q r 3fin f<q {q ffit * tee xct vd'{Tqltnlutt s 'f,rd

x 6 8 9 1 0 t 2

3 4 5 6 7

3

3

ei6s Bq+,fr tu6 A

(r)

MTE.3

(q) ql-{n d qk dRdt + fr dt rnl qR 102 ksi sth srq srta dfid t r-*,+ *fr qr qne8 ks t I Ff t3 dfcit cr qtsd qR Frddfrq t

2

( v ) ReE{ f "Fx - l rR

[ 0 , { R x < r

f(x) - { rl _ i ? , { R x > 1Ix - . r

en{ird A t

(6) cr4 dRq4.

R€Rq fr. ' f = f g

lsJ yx

'(€) i qqn trfiT d rooqn q{ srq ggrd}.c( t

{g qffiav+r t to cnr t f+ ss6* qtf6dl-R{\r R-*$t.d rsrctr qEbs t. ? 5

tr|'* - 3'8a, r!,0* - s's.r., a!.o* = 7'sz1

(rr) crq€ n dt o srA cs ncq dfi * qre qt{tIFFs fuqeq FqqF ro dr Jo t r p S< n *

2

P.r.o.IHTE.3

qFr f,R frfdq I

(s) rq +4. Er qfurq W dfqs fqffd AEt€r + =+=+ qr ftqa t silr + fgsil3 2 - 5

(0, -2,4t Sr tz, -r, -D + +fi{ srdr i r 5

(q) frq *it i xqmBr :5E l t t f 5 f f i82:4CI |ET 8 iFId8 3 , 3 E i l q I 6 S f d

F t r Sq {fiil qr{€r+ fuqr.rrqr t *i ssn.,fr C6 'rq Fffirfr rrql t S fq 6rA 'i,r +1 Fffi e | {s erfr +1 Hfus-dr 6 +1frq frY-{r TqT iFRII 83 c|I I s

(r) lfq E€Eqi . A, B, c, o, e qd F HqEdfre r xificrw f6s fur scr d qqE +frq qs {r{s ? + qfr' {Tq cftq{i frfus r e

6. (s) {fi

y ( x - y ) 3 - y ( x - y l - 2

fr sriilsfrFf vrc sfrfqS r 4

(q) q+ gR-s * qd yB cr *$ fr rrdd c i+qfi srtr*-dr ;4 t r cd y€ c{ z t srfusrdffi A{ d,qrtr+.dr flit ffdq 1, .., s

1 0MTE-3

I

(q) q6 qm +1 cqqtr Rn * wit +r, Fqdrz vgfu'd sfrr ro Erg+ t, w tBi )€ l6crr r q r r l e * q l r q r q i t ,

iFn ss6 * Hfs-dr-Rr qr qt$ efr + qrgfr ++q i le i f lH l {s t ? 3I tzo, o.os = 2'086, t2z, o.os = 2.074,

t2r. o.* = 2.080 |

?. f{qRfud Hqn d t +lc t 6eH rdctgfrr +{ tirsil | s[cl sf,r * +rqq ccrEq | 2xi=t0(i) fH =f * qq t qRqrRf, rirFr r: R+nGd

i r(D qtq dfuS f+ h, ulw x6t g6 qq,qqc dr+ t,

ildt xi6r ve f t r

(ifi) qrgk{ qH r6?tq fl*) = fl-Z* ry ria wfi: ' l + xqrwt{fi {snsit 6r vga+ t r

trrl sid.{E{ t0, :rl fr qffc (x} : stn * q*Rs t r' (v) d qt xeft vqqrs+ qc t w-{aikd d d x

"ft f * frq +r vaftfq- {qri+ rr, rt t ftqaf r n t r

MTE-3 t t B,ooo

€gfrEi €-9tq|q 74 70

qr+*fuqca 8 10

t.