BACHELOR OF SCIENCE (B.Sc.) Term-End Examination ' Juo€, 2OO7 MATHEMATICS MTE-3 : MATHEMATICAL METHODS Time : 2 hours Note r
1.
Maximum Marks : 50
Qu estion no. 7 is compulsory. Do ony four questions from questions no. 1 to 6. lJse of calculator is not allowed.
(a) Letf :R\{-1}
f(x)=
(b)
(c)
MTE-3
h,
-+R, g:R-+R
bedefinedby
gk):3x
Find the set of solutionsof : (fog) (x) = (gof) (x)
3
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 probabilitythat it departsand arrives on time is A.78. Find the probability that a plane arrives on time given that it departed on time.
3
Find two non-negativenumbersx and y whose sum is 300 and for which P : xzy is a maximum.
4
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
(b)
(i)
all of them are red,
(ii)
two of them are red and two white ?
2
If the sum of a certain nurnber of terms of the A.P. 25, 22, 19, .... is 116, find the number of terms.
(c)
3
In a shop study, a set of data was collected to determine whether or
not
the
proportion
of
defectivesproduced 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
31
57
32
Use the ^trz-testto determine if the proportion of defectivesis the same for all three shifts at 5o/olevel: I The following valuesof )f may be useful : ? : 5'99 Xfz,o.os
x'r,o.rr: g'27 2
XI,o.os= 7'82|
MTE-3
t
5
t
3.
(a) A particle moves in the plane accordingto the law x : t2 + 2t, ! = 2f - 6t. Find the slope of the tangent line when t : 0.
(b) Find a unit vector perpendicular to the two vectors 3i + 2i - k and i + j + k. Also find the areaof the triangle having the above two vectors as two of its sides.
\ I
a
1 J
t'
(c)
\ i
. i
The probability of getting a head in one tossing of a defective coin is p. This defective coin is tossed 8 times. If the probabilityof getting a combinationof 4 heads'and 4 tails is the same as the probability of getting a combination of 3 headsand 5 tails, use the Binomial distributionto find the value of p.
4.
(a)
Find the point(s) on the cuwe y = xz at which the tangent line is parallelto the line y :i 6x - 1.
(b)
Compute the correlation coefficient for the following data :
(c)
x
5
6
7
8
9
v
6
7
I
9
10
Also find the line of regressionof y on X.
4
Evaluatethe following integrals :
3
n/2 (i)
j
sin2 o d0 .--1+cos0
0 1
(ii)
j
0
MTE-3
P.T.O.
5.
(a)
Find the point of intersectionof the plane 3x-2y + 3z-2 =0andtheline y+1 z-I x-l
=
3 (b)
(c)
=
z
-2 '
Verify Euler's theorem for the function f(x, y) : a*2 + ZhxY+ bYZ.
3
2
If X has the probability density [t"-t* f(x): I t
,
for x elsewhere
0 ,
find
6.
(a)
(b)
(c)
(il
the value of the constant k
(ii)
P(0.5 < X
(iii)
mean of X
(iv)
variance of X
Find the domain and range of the function
f(x): - zJi.
2
,.g#
2
Find
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.
MTE-3
4
3
(d)
The measurementsof a sample of five weights were determined as : 8'0 , L0'2, 9'4,8'5 and 9'7 kg, respectively.
7.
(i)
Determine an unbiased estimate of population mgan.
(ii)
Compare
sample standard deviation with estimated standard deviation.
State whether the following statements are true or false. 10 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 ==rB *A forany two setsA and B.
(v)H#i;i:ffi:"ffi:',:;"?T3:,ru;" :'"
MTE-3
f{flq Frtrtr,(fr.qsfr.) v{iil qfrsr E'{, 2fiO7 TTfrTT
lp.*.t-s,
rrffiq fEkd qfwndq €ji6 : So
Wlz{ : 2 q"f
r t e : w r d . 7 s r f f i d t w r H ,1 0 6 f 0 m t d ar wr *?frrqt fuSda1w YqhrfrFf 67 +gafrwldr 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 I x+I
(fog)(x)= (gof)gl t Ef,
Trd fr.tf$rqI
Tt T{eIr{e;rt q1 qTtrsnr (s) WH t f{rttR6vsTIT 0.83t, sgn t Hrrqqt .r{qt +1qrtrfrilr0'82
m.rtsi.{td€ qt t sfr wl-{ + v'r{Tvt x{rytq
Hfqm,ilr0.78 t I sSffi t {TrrTqt qgqi 4t qt xsIFT xrFr*dT{rd mfqq Elqfs sgn TFTzT eFtAl \ (r) ql 1efrfi {iqT( x eilEv {rfr frrqq rsrsT \ qFTS firq P : *'y 3Tflr*'ilq ffi 3ooA stk ffi Eil MTE.3
P.T.O.
sfu s q+q tt t r ffi 2. (s) lrs +A t 7 HreT trEd t +t'q +At t Fffirmqr ffi t ftilsS
fr
(i) q Hri, eflTrEr, (ii)
(rsl qR sqiilr *ofr zs, 22,ts, ....*, Ss qqt 6r +'rq-drro d, d qiit +1{Gqr;p6 ffiq r 3 ('r) qo cn+i + fdq fr qs rfq fr g{6, rrrq *{ rn * mf it orq q,,trt q,rffi era cerRa lilrFrrilc,r+1rrnrqqH sEgrdt t qr r& q6 s{tq{i fuqr qqr I silzrfi fr xq ffi
Frqtubrt : RTE
Edd
I${lct
24
43
13
H-&
31
57
32
{ITTT
q.G t fdq fr so/o ztE.RTT t wefq.ar Kr rR qS tr+ Mt fr wqrkacrFr qrcTs'r q-{qrd !trwrq t rz-qfrqilrr il x,fr'rdfqq I s qraqTqtfirq *+fr A F*.A lr't ffi
t
: 5'gg' x\,,o.ou x'r,o.or:g'27 x 3 , o 'o=u 7 '8 2 1
MTE-3
.
3. (q) lf$ s'UTdef t Ffqq x = & + 2t,y = 2!3- 6t *
qgsn wlcT t I t = o Ai rR H{t tqr s1
stwrdTTnfrqifqq I (rs) rrd qs-s HRSITF ffifqq S A qRqii 3 i + z i - k e h i + i + k v t a i qA l s s m 5 w
s't STF€S {rd dfqq ms-*1E} gqlq 3;,n Rqrq*srq$dl
(rr) lfs {firq ffi t q-s qR s61f,f rR fd qIqT 41 8qR H.{f fi srtrsrr p t I {s lstlEtffi tre qR rrqr sT €q}qq qrq ssteTl r + Fa s+{ 4 q-ri q1 qTtrfiilr, 3 f{fr $il{ b tre sT {tqtff :il-q md ql vrktrdr t q{r,in A, H} f{T(-Eieq qI xdT qnt p sT qrq Srn ft1ffiq I 4. (6) qfr y = *2 qt r{ R€ flf, fifqq Fr rR HYt tcT,tury:6x-ltgqrffiRtl
Bffi ffifqq :
(€)ffi qffin
t
fitq \TEHq*r Tlis
X
5
6
7
8
9
v
6
7
8
9
10
tg1 ,fi {rd q1ffiq I x t1.rv *1 {rr{Try{rq guil?Ffrt rn+ {rd fifqq : ('{) ffi
4 3
n/2 (i)
j
sinZ o
ffido
0 1
(ii)
j
0
MTE-3
dx
F P.T.O.
5. (s')dm 3x-2y+32-2=O sfu t{gr
*-1 = u*t = 3 2 _ 4
2
qr sftt+Er*g vm
dFq r (q) qffi
3
(x, y) = ax2+ 2hxy + byz+ fdq s:frqgf
s+q HHIFRqftfqq
2,
(T) qR x sr yrfd,dr FI-GT saF{ f ( x=) { u r * * , x ) o + f m q [
sfeTqT
0 ,
d, * FrqRfudtr;6{frq. ;
s
(i) wR k 6,r rnT (iil P(0.5<x<1) (iii) x fi
qrq
(iv) X fi'l Y{Rut
6. (t$) qv+ f(d = - 2Jx +ifqq l
(rl)
,.ri,n^ #==
q,r nid aft qfrrt ild
vradRe r
2
2
(r) qR x rEniifuR d srh P(X = 2l = 9P(X = 4) + 90 P(X = 6)
* x qr qte ek rtnur ild +iFTq MTE-3
10
s
:
(q) fu t Fq fr frq rrq qfq Tt EFrqrq{ ffirrw g.7ffiqrq t I 8'.0,L0.2,9.4,8.53ft (i) {rqE qTEqiFTsffiFffid silErf, Trd *1Frq I ( i i )f f i q F F F E q i H t f u r { F l t F fqffi ffi gsTr dfqq I r
7.
6er;iit t etat $q;T(m t s+{+tt
ffi
3:t(m ? 3{q} fiR *',qTtor qfl5q I (i)
1A
r = 3 rFeFTf(x): (*2 - 5x + 6l /(x - 3) St 3Fiil{T{fr
t|
',r l Ei-cn* frq, rTIszI{gifs t q{rqt Af,r
(ii)
t l ( 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 R
sil@ffi Tfr | (iv) (v)
MTE-3
d t A r r g - u rAi fs l k B * f u A E B + B * A . Rrcirq-dfcd qt {wr Efqrrfi Tfr{q fnq qT G {fnnt-erEqT*qil {sr t cnrn Afi t I
11
10,000
t
t
a
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