BACHELOR OF SCIENCE (B.Sc.)

Term-End Examination'

Juo€, 2OO7

MATHEMATICS

MTE-3 : MATHEMATICAL METHODS

Time : 2 hours Maximum Marks : 50

Note r Qu estion no. 7 is compulsory. Do ony fourquestions from questions no. 1 to 6. lJse of

calculator is not allowed.

1 . ( a ) L e t f : R \ { - 1 } - + R , g : R - + R b e d e f i n e d b y

f(x) = h,

gk) :3x

Find the set of solutions of : (fog) (x) = (gof) (x) 3

(b) The probability that a regularly scheduled flight

departs on time is 0.83, the probability that it arrives

on time is 0,82 and the probability that it departs and

arrives on time is A.78. Find the probability that a

plane arrives on time given that it departed on time. 3

(c) Find two non-negative numbers x and y whose sum is

300 and for which P : xzy is a maximum. 4

MTE-3 P,T .O .

2. (a) A bag contains 7 red balls and 5 white balls. In how

many ways can 4 balls be drawn such that

(i) all of them are red,

(ii) two of them are red and two white ? 2

(b) If the sum of a certain nurnber of terms of the A.P.

25, 22, 19, .... is 116, find the number of terms. 3 t

(c) In a shop study, a set of data was collected to

determine whether or not the proportion of

defectives produced by workers was the same for the

duy, evening or night shifts worked. The following

data was collected :

shift Day Evening Night

Defectivcis 24 43 13

Non-defectives 3 1 57 32

Use the ^trz-test to determine if the proportion of

defectives is the same for all three shifts at 5o/o level: 5

I The following values of )f may be useful :

?Xfz,o.os : 5'99

x'r,o.rr: g'27

2XI, o.os = 7'82 |

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a

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i.i

3. (a)

4. (a)

(b)

(b)

(c)

A particle moves in the plane according to the lawx : t2 + 2t, ! = 2f - 6t. Find the slope of thetangent line when t : 0.

Find a unit vector perpendicular to the two vectors3i + 2i - k and i + j + k. Also find the area of thetriangle having the above two vectors as two of itssides.

The probability of getting a head in one tossing of adefective coin is p. This defective coin is tossed8 times. If the probability of getting a combination of4 heads 'and 4 tails is the same as the probability ofgetting a combination of 3 heads and 5 tails, use theBinomial distribution to find the value of p.

Find the point(s) on the cuwe y = xz at which thetangent line is parallel to the line y :i 6x - 1.

Compute the correlation coefficient for the followingdata :

Also find the line of regression of y on X.

(c) Evaluate the following integrals :

4

3n/2

j0

1

j0

(i)

(ii)

sin2 o. - - - d 01 + c o s 0

x 5 6 7 8 9

v 6 7 I 9 10

MTE-3 P . T . O .

(c) If X has the probability density

[t"-t* , for xf(x) : I

t 0 , elsewhere

5. (a) Find the point of intersection of the plane

3 x - 2 y + 3 z - 2 = 0 a n d t h e l i n e

x - l y + 1 z - I

3 =

z = - 2 '

(b) Verify Euler's theorem for the function

f(x, y) : a*2 + ZhxY + bYZ.

find

(il the value of the constant k

(ii) P(0.5 < X

(iii) mean of X

(iv) variance of X

6. (a) Find the domain and range of the function

f(x) : - zJi.

(b) Find

, .g#(c) If X is a Poisson variate such that

P ( X : 2 ) : 9 P ( X : 4 l + 9 0 P ( X : 6 )

find mean and variance of X.

3

2

2

2

3

4MTE-3

(d) The measurements of a sample of five weights were

determined as :

8'0 , L0'2, 9'4,8'5 and 9'7 kg, respectively.

(i) Determine an unbiased estimate of population

mgan.

(ii) Compare sample standard deviation with

estimated standard deviation.

7 . State whether the following statements are true or false.

Give reasons for your answer.

(i) x : 3 is an asymptote of the function

f(x) = lxz - 5x + 6') l(x - 3)

(ii) For a Rormal distribution, the mean is equal to the

mode.

(iii) The function f : R -+ R+ U tOl , f(x) : x2 is one-one

but not onto.

(iv) A E B ==r B *A forany two sets A and B.

(v) H#i ;i :ffi :"ffi:',: ;"?T3:,ru;" :'"

1 0

MTE-3

f{flq Frtrtr, (fr.qsfr.)v{iil qfrsrE'{, 2fiO7

TTfrTT

lp.*.t-s, rrffiq fEkd

Wlz{ : 2 q"f qfwndq €ji6 : So

r t e : w r d . 7 s r f f i d t w r H , 1 0 6 f 0 m t dar wr *?frrq t fuSda1 w Yqhr frFf 67+gaf rw ldr

l . ( s ) q F T f f i F q q t : R \ { ' 1 } + R , s : R + R ,

(x) : 4, g(x) = 3x HRI qm{rFo t Ix + I

(fog) (x) = (gof) gl t Ef, Trd fr.tf$rq I

(s) WH t f{rttR6 vsTIT Tt T{eIr{ e;rt q1 qTtrsnr0.83 t, sgn t Hrrq qt .r{qt +1 qrtrfrilr 0'82t sfr wl-{ + v'r{T vt x{rytq m.rt si.{ td€ qtHfqm,ilr 0.78 t I sSffi t {TrrT qt qgqi 4txrFr*dT {rd mfqq Elqfs sgn TFTzT qt xsIFTe F t A l\ efrfi {iqT( x eilE v {rfr frrqq rsrsT(r) ql 1\

ffi 3oo A stk ffi firq P : *'y 3Tflr*'ilqqFTS

E i l

MTE.3 P . T . O .

2. (s) lrs +A t 7 HreT sfu s q+q tt t r ffitrEd t +t'q +A t t Fffirm qr ffi t ftilsSfr(i) q Hri, eflTr Er,(ii)

(rsl qR sqiilr *ofr zs, 22, ts, .... *, Ss qqt 6r+'rq-d rro d, d qiit +1 {Gqr ;p6 ffiq r 3

('r) qo cn+i + fdq fr qs rfq fr g{6, rrrq *{rn * mf it orq q,,t rt q,rffi era cerRalilrFr rilc,r +1 rrnr qqH sEgrd t t qr r& q6s{tq{i fuqr qqr I silzrfi fr xq ffiFrqtubr t :

RTE Edd {ITTT

I${lct 24 43 13

H-& 31 57 32

ztE .RTT q.G t fdq fr so/o t wefq.ar Kr rRqS tr+ Mt fr wqrka crFr qrcT s'r q-{qrd!trwrq t rz-qfrqilrr il x,fr'r dfqq I s

lr't ffi qra qTqt firq *+fr A F*.A .t

x\,,o.ou : 5'gg '

x'r,o.or: g'27

x3,o'ou = 7 '821

MTE-3

3. (q) lf$ s'UT def t Ffqq x = & + 2t, y = 2!3 - 6t *qgsn wlcT t I t = o Ai rR H{t tqr s1stwrdT Tnfr qifqq I

(rs) rrd qs-s HRSI TF ffifqq S A qRqii

3 i + z i - k e h i + i + k v t a i q A l s s m 5 ws't STF€ S {rd dfqq ms-*1 E} gqlq 3;,nR q r q * s r q $ d l

(rr) lfs {firq ffi t q-s qR s61f,f rR fd qIqT

H.{f fi srtrsrr p t I {s lstlEt ffi 41 8qRssteTl rrqr r qR + Fa s+{ 4 tre sT €q}qq qrqq-ri q1 qTtrfiilr, 3 f{fr $il{ b tre sT {tqtff :il-qmd ql vrktrdr t q{r,in A, H} f{T(-Eieq qI

xdT qnt p sT qrq Srn ft1ffiq I

4. (6) qfr y = *2 qt r{ R€ flf, fifqq Fr rR HYt

t c T , t u r y : 6 x - l t g q r f f i R t l

(€)ffi Bffiqffin ffifqq :

t fitq \TEHq*r Tlis

x t1.r v *1 {rr{Try{rq tg1 ,fi {rd q1ffiq I

('{) ffi guil?Ffr t rn+ {rd fifqq :

sinZ of f ido

dx

F

4

3

(i)

n/2

j0

1

j0

(ii)

X 5 6 7 8 9

v 6 7 8 9 10

MTE-3 P . T . O .

5 . ( s ' ) dm 3x -2y+32 -2=O s fu t { g r*-1 = u*t = 4 qr sftt+E r*g vm3 2 _ 2

3dFq r

(q) qffi (x, y) = ax2 + 2hxy + byz + fdq s:frqgf

2 ,s+q HHIFR qftfqq

(T) qR x sr yrfd,dr FI-GT saF{

f ( x ) = {u r * * , x )o+ fmq[ 0 , sfeTqT

d, * FrqRfud tr;6 {frq. ; s(i) wR k 6,r rnT

( i i l P (0 .5<x<1)

(iii) x fi qrq

(iv) X fi'l Y{Rut

6. (t$) qv+ f(d = - 2Jx q,r nid aft qfrrt ild+ifqq l 2

(rl) ,.ri,n^ #== vra dRe r2

(r) qR x rEnii fuR d srhP(X = 2l = 9P(X = 4) + 90 P(X = 6)

* x qr qte ek rtnur ild +iFTq s

MTE-3 1 0

r 7 .

:

(q) fu t Fq fr frq rrq qfq Tt EFr qrq{ ffirrw

8'.0, L0.2,9.4,8.53ft g.7 ffiqrq t I

(i) {rqE qTEq iFT sffiFffid silErf, Trd *1Frq I

( i i ) f f i q F F F E q i H t f u r { F l t Ffqffi ffi gsTr dfqq I

ffi 6er;ii t t etat $q;T (m t s+{ +tt3:t(m ? 3{q} fiR *',qTtor qfl5q I

(i) r = 3 rFeFT f(x) : (*2 - 5x + 6l /(x - 3) St 3Fiil{T{fr

t | ' , r l

(ii) Ei-cn * frq, rTIszI {gifs t q{rqt Af,rt l

1 A

( i i r ) t f t F T f : R + R + u { 0 } E f d f ( x ) = x 2 , q f f i t r Rsil@ffi Tfr |

d t A r rg-ur i f As lk B* fu AEB+B*A.

Rrcirq-dfcd qt {wr Efqrr fi Tfr{q fnq qTG {fnnt-erEqT* qil {sr t cnrn Afi t I

(iv)

(v)

1 1MTE-3 10 ,000

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