DU SOL Question Paper 2018 B.Com (Hons.) Business Maths

IThis question paper contains 24 printed pages.]

s98 Yottt' Roll \'l o

B.Com. (Hons.) / II G

Paper Code : B-i04

BUSINESS MATFIS - Paper XI

(Admissions of 2004 and onwards)

Time .- -l Hours l'[aximum Murks : 75

(Write your Roll l{o. an the top immediately

on receipt o.f this qttestion paper.)

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Note ;- Answers may be written either in English or in Hindi; but
the same medium should be usecl throughout the paper.

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# flfrg; a/a,T vrfr rrd' a-r qtLqtr yq # *{r q@ r

Attempt all questions.
Logarithmic Tables and graph
papers v,ill be supplied on demand.
Use of simple calculator is allov'ed.

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)
598

afnfuq- ?qa ,l'D-r rtr +q{ #
q-c Fd sP'i r

fiEIrw *aEdei t Yql'1r d a1++fr * t

given by R = 8x (where R
(a) A firm has revenue function
sold) and the cost
is gross revenue and x is quantity
function is given bY

c = t,5o,ooo. uo[th)'

the number of units to
Find the total Profit function and
be sold to get the maximum
profit. Also calculate the
profit.

OR

the average
If x be the number of workers employed,
cost of Production is given bY
:

AC=24x+

Determine the value of x that
will make average
emPloY 4 or 5
cost minimum' Will You advice to
(6)
workers ?

3
s98

(b) For the demand functions
of two commodities given
rvith
elasticities of demand
below, find the four partial
rvhether the commodities
respect to price and indicate
are comPetitive or comPlementarY
:

X' = 6 p,-utptt'',t

X, = 5 p,o'p,'o-

the two
quantities demanded of
where xr and xr are the
commoditiesatpficesp'andp.respectively.
OR

function :
Show that the Production

x = f (/'k)= 2J k

where x, /, k are units
of output' labour and capital
returns to scale and diminishing
respectively, gives constant (6)
returns to inPuts'
when output is x units
is
(c) The marginal cost function
given bY
MC:*z-27+5
and average cost function
if
Find the total cost function
show that the slope of
the
the fixed cost is {30' Also'

average cost functrol
ls MC-AC '

- P.T.O.

598 4

OR

Thc demand and supply functions, respectively. under
perf-ect competition are

Po: 16-x:

P':2x2+ 4
Find the mirrket price, consumer's surplus and producer's
su rp lu s. (6)

(d.) The denrand cur\:e for a comrnodity is given
by

x:20-lp- p,

u'here p'r and x are the price and the quaniitv in rlemand
respectively. Find the elasticitl,' of elemand for p .= 2.5.
For u'hat demand the elasticity raill be unity,l

OR

A department store sells 2500 refrigerators per year,
with sales occuriing at a relatively constant rate. The
annuai holding cost per reliigerator is {10. To reorder,
there is a basic T20 service lee per order. Hor.v many
times per year and in what size should the store reorder
to minimize the total annual inventory cost ? (6)

598 \

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rrn-d rtErFT ofu x ffi d qrer *) 3rlT aFRT siffi gst ver
qom *,

c = 1.50.000*60r-I-)
\e00 i

Ed i{riT q-o-+ ah oTEro-f,q Frrq rrrkT fi+ + frq +* vr*
ffi $-a fr rr{qr are dfrg r

$t TqT

qR x fr€rq\T q-.ffim * <-cqr i A iiq-d JsIrE=r ar.n grt
an *fr t,

AC=24x+-
-
2\x-a)

x tnT TeT ftiifrn *Fsq ffi
sftsm dr{ra Tttrc qrq * r

tFII i{Iq 4 qT s 6.ffid 6} fuffi 6.d +t ?rara ? tt
(q) fti frq {q A c"d + fr{r q,ffi + frq, qR BTrfrm fq
diif +} frfld + {qtr + 66 dfr{ *t e-dr-{q fr } co-q
vffirns-o i qt qco,

P.T.O.

598 6

Kr =6p, '"p:t"

x, = 5 p,utp,-"-

trei x, itt x, * ct*- fr +'fr rri ffir{ *-rHT: p, +tq-d sltt
Prfi{dtndr
etelqt

Tdtr-d dfrq fr Ycqre{ s-tr{

r=l(/.k)=2tf1

urei r', / 3iR k F-rrsT: Jflt-e;t atq im {fi efr {F e i i
qrcrfr Jirun qr fr+r cRw.-d .p 6N + frq {trgqrq yRs-f,

tlir
(.r) fua vrrra q?Fr, iG[ iteflt-r x 1ffi 6l +, g-{+ vra *tr
*.

MC: x2- 2x + 5

tiT dr{rd q-e-{ Bfr{ at{fr i{n-f, srFr HliT dfu of terrfr
FnqiT 30 t i r erq fr v-{ffif, dfrq fu 3i-fld dFliT siFt

q' ara qa * MC-AC I

X

598 7

Stelztt

nftT3lkqqr€wf,{TrfiYF|*FrdT+3l"Fi{F'qsr:}e

P.l -- 16 - xr

P,. = 2xl+ 4

qt-frR q1{fl,3q*ffiT 3{Driq efu ssre* 3{BrAq an
frfrq t

(q) t'n cu-q + Rq f{r s-fi gst aro *ar * '
x:20- 2P-P'
rn-d p ofu x q-qsT' fis-{ 3i{ d'{r fr qr*r d r p:2'5 +
ftq m{r m A-s ard dfrq r frs frlr + frq
+q $"4
*fi?
Stgral

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r'+ Gcrftc €tr eftHf zsoo tfil*a
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qrffi*. unsr

drrn lo t. I r ft-t Q 3ilin t+ + Rq cR 3TAn
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.c& +1
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g{: B{r?rr }+ sTF:q ilfu E-d qd$ erffifi f,r{rd
qqnq A

flgl

P.T.O.

598 8

(a) Explain (i) unbounded solution, and (ii) infeasibility
in
2.
Show these
the context of linear programming problems'
with the helP of rough sketches'

OR

problem :
obtain dual of the following linear programming

Maximize Z = l6x, + 22xr- 18x,
Subject to
3xr+6xr+2xr<36
4xr + 3x, + 5xr= 25
Kt, X2, *, > 0 (5)

(b) A company manufactures and sells three models of
large

sized pressure cookers for canteen use' While
market

demands pose no constraints, supply of aluminium
is

limited to 750 kg per week and availability of machine
time is limited to 600 hours per week which restrict
the

product-mix' The resource usage of the three models
and their profitability are given below :
Model

M1 M2 M3

6 3 5
Aluminiunr/unit

Machine-time/unit
60 20 80
Contribution Rs/unit

598 9

(i) Formulate the problem as an LPP and solve for
optimal solution'

(ii) Does the problem have multiple optimal solutions ?
If yes, identify another optimal solution'

(iii) Using the information in optimal solution tableau'
determine shadow prices of the resources'

(iv) Which of the models is not being produced and
why ?

OR

Given below is the simplex table for a maximization
type of linear programming problem in which
x' x2 and
x3 represent the number of units to
produce of the three

products A, B and C respectively while S'
S' and S'
stand ior the respective slack in three resources used:

ci+ t0
A
00 0

Basis Kl xt x.J sl s2 S" QuantitY

x2 0l 516 513 -116 0 20013

100/3
xl 10 U6 -213 116 0

i00
00 -20
1
A I
S.

P.T.O.

s98 10

Answer r,r'ith reasons the lbllorn'ing questions in relation
to the solution in this table :

(i) Is the above solution feasible ?

(ii) Is the above solution oPtimal ?

(iii) Is the above solutiou unbounded ?

(iv) Is the above solution degenerate ?

(v) Does the problem have multiple optimal
solutions ? If yes, give an alternate optimal
solution.

(vi) What are shadow prices of the three resources ?

(r'ii) Which of the products is not being produced and
why'i

(r'iii) What is the objective function of this problem ?

(ix) What are the optimal values of the dual
variables ? (9)

(o.) erc<r dftq: (i) w<roa E-{ 3i{ (ii) tFq-6 ffi'r t-rsrS
A v.d$ i :iqrtqer I FB +- att qt 3{Rr+ 6 qee t y-dtrd
etftK r

s98 11

iltzrdt

trqRR+o tRq-6 ffirr €rl-€ll 6l iu cr.d frRq:

Maximize Z: l6x, + 22x, - 18x,

Subject to

3x,+6x.*2x,S36
4xt+3x"+5x,=25
Xt, X,' x-, > 0

(e) vo ffi st 3{rfiR + irR q_+.{t.. + fi-'T ffi 6T ffvr
ofu ffi q-{fr * fr +&r * e*{r
n- 3Trt * | q{R qrqn

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fr qwr€ zso
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+ qsftr+ 3qd-{ 3ik 3-i-S dTrttcilT fti A ur rfi t:
fr'E-a

MI M2 M3

5
Rqqfiqq/1F-c 6 J

qfi-+ vqqzqF-c 3 4 5

qtrr€r+ n zqRc 60 20 80

P.T.O.

598 t2

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+frqwadfrqr
(ii) wr rffisr * e-go Eqir{T e-a * i iTR d tr erq Eqffr
ao +) ddr{g I

(iii) Eqf,rT Ed H i ocffir {d{r 6T y*q s{+ tr€Erfr
q1 6fud *{a Rtrifrn frfuq r

(iv) gt ffi- q + q.t{fir eqrqr rfi dT {6T + *{ d ?

3ler.n

fri tFq-+. ffirT qqsr * ereerirfr*-{ur frR + frq
ftr++R HrR-6r A rr$ t ffi x,, *, fr x, frr ysrd q-qr
A, B *t C q1*-qqr: q{r+ + fu qFa fr Tcqr qr frwur
q{t d *t S,, S, fr 53 egrtT fi-rr qgftr+ + e++t q'fi
+1 e-dr+ *,

ci- l0 6 4 0 0 0

SITET|-t xl x1 X.J s-l s2 S" ql-rT

x) 0 I 5t6 5t3 *t t6 0 200t3

xl I 0 U6 -zt ) U6 0 100/3

S. 0 0, 4 -2 0 I 100

598 13

6Rq qf,l'F-t FIqRRdd vr+ + sft ARq
S g{ ilk*.r
iaattrsEn*'
(i) wr sr{m aa qrrq * ?

(ii) wr rv{-< aa sstrq I ?

(iii) wt sv{-m aa srqrarB-n i ?

(iu) wt o{m aa g<rwn I i
qR d fr ffi
(r) tst (rr€I[ + qga Estr{ ae d i
ssilq ra q} qf,Eg I

(t) st fr+ {sftrn fr o,R{d ffi wr t i
(*i) gT *qrd l- t +t{€r enqr ;rfi qt {6I i dt d
I

(viii) vt wr€n 6T q<q'o. q-e-a eur * ?

(it) au qeqfffi * Escq qrc trr i ?

(a) There are two families A and
B' There are 5 men' 3
3.
womenand3childreninFamilyAand2men,2women
recommended daily
and 3 children in Family B' The
P.T.O.

598 14

allowance for calories is Men - 3400, Women - 2900,
Children - 2000 and for proteins is Men - 65 gms,
Women - 50 gms and children - 35 gms. Represent the
information by matrices. Calculate the total requirements
of calories and proteins for each of the two families.

OR

A manufacturer produces two types of products X and
Y. Each product is first processed in Machine M, and
then Sent to another machine M, for finishing. Each
unit of X requires 20 minutes time in M, and i0 minutes
time in M" whereas each unit of Y requires 10 minutes
time in M, and 20 minutes time in M". The total time
available on each machine is 600 minutes. Calculate the
number of units of two products X and Y by using
matrix algebra. (5)

(b) A firm has three service departments S,, S, and S., and
two production departments P, and Pr. The direct cost
of each department and the percentage of the total cost
of each service department allocated to various
departments are as given below :

15
598
Percentage Allocation
of Total
Cost of the DePartment
q S, 53
Department Direct Cost
30 10
0
sr Rs. 25,000

0 10
Rs. 124,000 20
S,

10 0
Rs. 77,000 l0
S.t

30 30
Rs. 120,000 40
Pr

30 5U
Rs. 290,000 30
P2

100 100
Total 100

plus allocated) of each
Find the total cost (direct
algebra.
department bY using matrix

OR

relationship is given below :
A two-industry input-output
Final Demand Gross OtttPut
IndusttY

determtne
Using matrix notation'
P,T.O.

598 l6
(i) Technology matrix and test whether the system
is viable.

(ii) Gross output required to satisfy the new final
demand of 90 units and 135 units for industry
I
and II respectively.

(iii) Total labour days required.

(iv) Total value added if wage is Rs. 80 per labour-
day.
(6)

(m) a efu s A cFqR * r cfuR a i s $raft, 3 srti stt
s
qE+ +, B cRqR i z eirEft, 2 *i. .:fu s {d * r ddft*
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+
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l1
598
qFra + frq Mt qfiT I 20
fr{d 6I {qq
I rX * q-46
qft{ i fr+e 6I qq€T wrdr + trqfr Y
a.rsr I ${ Mr to
ft'-{e qT qcq 3t{ M'
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m q+*' qFc d M, t lo
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1e) t'+' m.{ + fr v-sq{ arrro ofu
q++. Asr
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ffqFt, R fiF{d ft{fr
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lq qttl 5rr -s.'

fiFrwq qRprf,dr ft{r{r
s2 S.J
JretH i4r{rfr "I
Gqr{T
10
0 30
sl 25,000(.
0 l0
20
s2 124,000n'
10 0
10
s3 ?7,0008
30 30
40
P1 120,000d
30 30
P2 290,000d
100 t00
Ag 100

P.T.O.

598 18

+Fm ffim qr y-fr.r q.{+ e-+q. fun{r 6} qf, RrIrf,
(y-sel ur frfuin) an dfrq r

a-=-flH- + fi+{T-ieTqq {su fr} frq .rq *
'

fFw +asFr or vfrq m* numo dfrs:
(i) eHfr+l+fn ilk qts dfrq fr zra xvr& fi-+ltrq
ir
(ii) rrfi-cT se{reor fr T+rr I e}r II + frq irrrel; e0 zrfrd
efu rss $-d A r$ BTfr.c {lr d gR' + frq
enqr{s'* r

(iii; td ryq F+t qit i{rtr-{zr-F-dr I

(tu) E-m t€ +fun qtr r.w(fr oo n cR r{ fr-+e e I

s98 t9
A
a. Solve the following transportation problem and determine
the minimum total cost of transportation.

Markel

PIant vIt NI, M3 M4 Sapply

Pr 20 l0 t+ t6 600

P2 t2 8 t6 10 s00

.A
P3 l8 z+ 20 t4 900

Demand 440 320 760 480 2,000

OR

A production manager wants to assign one of five new
methods to each of four operators. The following table gives
the weekly output in units from different operator-method
combinations. Find the maximum output per week. Which
rnethod remains unassigned ?

ll/eekly Output

Operator M1 M" M3 M4 VI,
a-
A 25 27 42 )l 30

B 26 29 +l 40 30

A1
C 30 34 42 -- -A
)+

D 27 27 30 JZ 28
(11)
P.T.O.

598 20

frqRft{d cR{6T {qrqr q1 Eil dfrg iir qtqET fr
5o ETffiq
RrIRT rnr ftrfrq qifrq r

d7iFlT

azn Mr M2 M3 M4 SR?TF

Pr 20 10 l4 l6 600

P2 t2 8 I6 t0 500

P3 l8 .A 20 l4 900
ril-fT 40 320 760 480 2,0w

BIEIET

FF" ytqrer y-sus. qn y-*r+} + + k+n 61 cis ;r{ frEd + t
gs"fr ffi tar * r frcrRR+c Rrfunr Fr;iv-qrrm frRr *s+
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tTqtra iilRT dftN r frs Afu d fud Rqr rrqr * I

arratt?a giqrrf,
wflarca Mr M2 M3 M4 M5

A 25 27 A'
37 30

B 26 29 47 40 30

C 30 34 42 42 34

D 27 27 30 32 28

598 2l

5. (a) A computer whose cost is {4,40,000 will depreciate to
a scrap value of {24,000 in 5 Years.

(i) If the reducing balance method of depreciation
IS used, find the rate of depreciation.

(ii) What is the book value of the computer at the
end of third year ?

OR

Explain the relationship between nominal. effective rate
of interest and force of interest. Also, calculate the
nominal rate of interest convertible hslf--ysarly when
the effective rate of interest is 6o/, per annum' (5)

(b) Frnd the present value of a sequence of payments of
{3,000 made at the end of every three rnonths and
continuing forever, if money is worth 80,6 per annum
converted quarterly. If the payrnent of {3,000 is made
at the end of each year and continuing forever' find the
present value if money is worth 8% per annllm converted
quarterly.

OR

A nran borrowed <5,000 at compound interest of 1oh
per annum and has to repay the money in 20 equal
P,T.O.

598 ))

yearly installments. What should be the installment if
repayment has to start three years hence. (5)

(c) Ram borrowed {25,000 from a money lender but he
could not repay any amount in a period of 5 years.
Accordingly, the moneylender now demands {35,880
from him. At what rate percent per annum compound
interest did the latter lend his money ?

OR

A has taken a loan of t20.000 at a rate of interest of
4oh per annum payable half-yearly. He repaid {'1,000
after two years, <6.000 after a turther period of two
years and cleared all outstanding dues at the end of
seven years from the commencement of the transaction.
What is the final payment made by him ? (s)

(?F.) qfi i5-Erd{, frrs-fr drrkT 4,40,000 n *, qt au vc+r dq
sd+Sc {cq + rq q zaooo {. r-6 vrfr * r

(i) uR 1u qrru fr q{€flq iq fr& q"r r*rr frqr vrar
**1uqrrufrqranddfrqt
(ii) gfiq sf * {crfu q-{ 6qrr qT €ldl rrFI wr e ?

irTelst

598 23

irrlqTT, vrirft qrn-qi 3fu
qrET qT eqrq t Tqu fr qrqt
*frq r qrq -& 3{ti.'qrffi-fi w t qffiftq qr{rqn dIfGT-E{ 6I
qRq-tr{ mhq ffq ETrtr dil eqffi qt 6% YR ed * r

(re) 3000 d. ef rqrqft qT qdqra {s Erd dfrq d rt fi-+
qra

* T{rk qr * 6rfr * d{ T}q + ftlq vrt *fr qR m
q< fra qra i q{e[ fi s% Yfr q{ qrq frd ft6dr i I
qR
qt fr vTft e eit
3o0o d. m 3rffift e++, sS fr1 TqTk
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+wfr.8tftHrfrffif,6T8%yRq{tqreIRg|TIT]IZIT
ir
stgrcn

vo qk I ty"YR sd q-m-gk efl-.,I so00 t' stm
rR d

etk Et 20 qtFR fr qrarcr ffi n erq€ 6-c{T
* r qR tt
fr sITfr 3{s t fi-{ e{ Erc gs fr urq A fu€ FF-{S *ft
qrBq t

(.r) rrq * E'w{rdr t zsooo d sslR ftrq crg s d 3i-{Fr
qa q{

n- *S +qr Tft ? qrqr | 3iir: €qrqrflI + Tq+ ss'eso
t' *
nfu fr | €qqrfrr * cm frT vR{rd rM
Er*.1E'qrs tn

frqtqrt

P.T.O.

598 24

n i ax vR sS fr qre qt qt 20000 d 3tIR frq + fr
3T?ffi*. tq i r B{+ A qd erq 4000 Q. erqq frq ft'
6000 d. i{q-a A s{ m s-dDI t frq etir aq * 3TRq t

ero eff * Tqrk qt E?6lzTI {RT A fi r Yq+ ffiq 3rflqrft
wrd?

(8500)