Chapter 1 . Applications Ofmatrices and Determinants

12–STD– BUSINESSMATHEMATICS FORMULAE

CHAPTER 1 . APPLICATIONS OFMATRICES AND DETERMINANTS

1. AdjointofamatrixAis

AdjAAT.

(whereAc isa cofactor matrix)

2. InverseofamatrixAis

3. Results:

A1 

1AdjA.

A

(i)

AAdjA(AdjA)A

AI.

(ii)

Adj(AB)(AdjB)AdjA.

(iii)AB1 B1 A1.

(iv)

AA1 A1AI.

(v) A11 A.

4. Therankofa zeromatrix(irrespectiveofits order)is0.

5. Conditionsfor consistencyofSimultaneousLinearEquations(Non–homogeneous):

(i)If

(ii)If

(iii) If

(A,B)(A)n,thenthe equationsare consistentandhasunique solution.

(A,B)(A)n,thenthe equationsare consistentandhasinfinitelymanysolutions.

(A,B)(A),thenthe equationsare inconsistentandhasno solution.

6. Conditionsfor consistencyofSimultaneousLinearEquations(Homogeneous):

(i)If

(A,B)(A)n,(OR) If

A0thenthe equationshavetrivial solutionsonly.

(ii)If

(A,B)(A)n,(OR) If

A0

thenthe equationshavenontrivial solutions also.

7. Cramer’srule:



xx ;





yy ;



zz .



aa

 11

12 

8. Technologymatrix

Bx1

x2 .

a21



a22 



x1

x2 

9. Outputmatrix

XIB1D.

PP

10.TransitionProbability MatrixTPAA

PAB



(OR)

T PP

PQ



PBA

PBB

PQP

PQQ

(dependsonthenameofthe productsA, BorP, Q)

11.ForfindingEquilibriumshareof market A+B=1(OR) P+Q=1

(Thisstepcarries1markandit iscompulsory)

CHAPTER 2 . ANALYTICALGEOMETRY

1.SP e.

PM

2.Eccentricityofparabola e =1.

3.Eccentricityofellipse e1.

4.Eccentricityofhyperbolae >1.

5.Eccentricityofrectangularhyperbolae

2..

6.Parabola:

y2 =4ax (opens rightward) / y2 =-4ax (opens leftward) / x2 =4ay (opens upward) / x2 =- 4ay (opens downward)
Vertex / (0,0) / (0,0) / (0,0) / (0,0)
Focus / a,0 / a,0 / 0,a / 0,a
Directrix / xa / xa / y a / y a
Latusrectum / 4a / 4a / 4a / 4a
Axis / y0 / y0 / x0 / x0

7.Ellipse:

22
x y 1,ab
a2b2 / x2y2
1,ab
b2a2
Centre / (0,0) / (0,0)
Eccentricity / b2 a21e2
(OR)
b2
e 1
a2 / b2 a21e2
(OR)
b2
e 1
a2
Vertices / a,0,a,0 / 0,a,0,a
Directrix / a x
e / ya e
Latusrectum / 2
2b
a / 2b2
a
Foci / ae,0,ae,0 / 0,ae,0,ae

8.Hyperbola:

x2y2
1
a2b2 / y2x2
1
a2b2
Centre / (0,0) / (0,0)
Eccentricity / b2 a2e2 1
(OR)
b2
e 1
a2 / b2 a2e2 1
(OR)
b2
e 1
a2
Vertices / a,0,a,0 / 0,a,0,a
Directrix / xa e / ya e
Latusrectum / 2
2b
a / 2b2
a
Foci / ae,0,ae,0 / 0,ae,0,ae

9. Thegeneralequationof RectangularHyperbola (R.H)isxy= c2.

a2 

 wherec



(usefulfor objectives)

2

10.TheeccentricityofRectangularHyperbola(R.H) ise2

CHAPTER3 . APPLICATIONS OFDIFFERENTIATION–I

1. Averagecost(AC)=

C(or)f(x)k.

x

2. Averagevariablecost(AVC)=

x

f(x).

x

3. Averagefixedcost(AFC)=k.

x

4. Marginalcost(MC)=

dC.

dx

5. Marginalaveragecost(MAC)=

6. Total revenueR=px.

dAC

.

dx

7. Averagerevenue(AR)= R.

x

(Averagerevenue= Demandfunctioni.e,AR=p)

8. Marginalaveragerevenue (MR)=

dR.

dx

9. Ifx =f(p) isa demandfunction,thenElasticityofdemand

p.dx .

(Wherex –quantitydemanded;p–price)

Note: Fora demandfunction q=f(p),

p.dq

dxdp

dqdp

10.Ifx =f(p) isasupply function,thenElasticityofsupply

p.dx

(Wherex –quantitysupplied; p–price)

11.RelationbetweenMRandElasticityofdemandis

sx dp

1

MRp1.

12.Atequilibriumlevel,Qd =Qs.

13.Equationoftangentisyy1mxx1.

14.Equationofnormalisyy1xx.

d 

1m1

CHAPTER4 . APPLICATIONS OFDIFFERENTIATION–II

1.Euler’stheorem:If uisa homogeneousfunctionofxandy withdegreenthen,

xuyunu.

(forzcanbeusedinthe placeofudependsonthenameofthefunction)

xy

2.PartialElasticities

Eq1

p1 .q1

and

Eq1

p2 .q1

Ep1

q1p1

Ep2

q1p2

3.Economicorderquantity(q)

2RC3 .

0

1

(whereR–Requirement; C3 –orderingcost ; C1 –carryingcost)

4.Ifunitprice andpercentageofinventoryare giventhencarryingcostC1

%

100

unitprice.

5.Time betweentwoconsecutiveorders(t

)q0 .

0R

6.Numberoforders= R.

q0

7.Minimumaveragevariablecost=

R

2RC3C1.

8.Total orderingcost=

C3.

q0

9.Total carryingcost=

q0 C.

21

CHAPTER5 . APPLICATIONS OF INTEGRAL CALCULUS

PropertiesofDefiniteintegrals:

ba

1.f(x)dx f(x)dx..

ab

a

2. Iff(x)isanodd function,i.e, if f(-x)=-f(x)thenf(x)dx0..

a

aa

3. Iff(x)isanevenfunction,i.e, if f(-x)=f(x)thenf(x)dx 2f(x)dx..

a0

bb

4.f(x)dxf(abx)dx..

aa

aa

5.f(x)dx f(ax)dx.

00

6. Theareaunderthe curve

yf(x),thex-axisandthe ordinatesat

xa

and

xbis

b

Areaydx

a

7. Theareaunderthe curvex =g(y),they-axisandthe linesy=candy = dis

d

Areaxdy.

c

8. IfMCisthemarginalcost functionthentotalcost functionisgivenbyCMCdxk.

9. IfMRisthemarginalrevenuefunctionthentotal revenuefunctionis givenby

RMRdxk.

10.Theproducers’surplus for the supply function

pg(x)for the quantityx0

andprice

p0 is

x0

P.Sp0x0 g(x)dx.

0

11.Theconsumers’ surplus for the demandfunction p=f(x)for the quantityx0 andpricep0 is

x0

C.Sf(x)dxp0x0.

0

CHAPTER6 . DIFFERENTIAL EQUATIONS

1. TheGeneral formofHomogeneous differential equationsis

dy

dx

fx, y

gx, y.

2. Working ruleforfinding the solution oflinear differentialequations

(i)ExtractPandQ.

(ii)FindP

dx.

(iii)FindIntegratingFactor(I.F) =ePdx

3. Thesolutiontolinear differentialequationsoftype

dyPyQ

dx

(WherePandQarefunctionsof

xonly)is

yI.FQI.FdxC

(OR)

yePdx QePdxdxC

4. Thesolutiontolinear differentialequationsoftype

dxPxQ

dy

(WherePandQarefunctionsof

yonly)is

xI.FQI.FdyC

(OR)

xePdy QePdydyC.

5. Secondorderlinear differentialEquations

Ifm1 andm2 arethe rootsoftheAuxilliaryequationisofthe typeax2 +bx+c=0

(Quadraticequation)

(i)Ifthe rootsm1 andm2 arereal anddistinct,C.F=

Aem1x Bem2x.

(ii) Ifthe rootsm1 andm2 arereal andequal(m1 =m2),C.F=AxBemx.

(iii)Ifthe rootsm1 andm2 areunreal,i.e, ifmi, C.F=exAcosxBsinx.

(C.F– ComplementaryFunction)

CHAPTER7 . INTERPOLATION

1. Forwardoperator(delta)(y0) y1 y0

(or)(f(x))

f(xh)f(x).

2. Backwardoperator(nabla)(y1)y1 y0

(or)(f(xh))

f(xh)f(x).

3. TheShiftingoperator

E(y0)y1,

E (y0)y2,

E (y0)y3...... andsoon.

4. Therelation betweenforwardoperator(delta)andshiftingoperatorEis

E1

(or)

E1.

5. (Formissing termproblems)

(a)E13y

E3 3E2 3E1y .

(b)E14 y

(c)E15 y

E4 4E3 6E2 4E1y .

E5 5E4 10E3 10E2 5E1y .

00

6. Gregory–Newton’sforwardformula:

yy0

u y

1!0

uu1 2

y0

2!

uu1u2



3!

3y

...... 

uu1u2...... un1

n!

ny.

Whereu xx0 .

h

andh–equalinterval betweenthex- values

(numberoftermsintheformuladepends onthenumberoftermsintheproblem)

7. Gregory–Newton’sbackwardformula:

y yn 

u

yn 

1!

uu12



2!

yn 

uu1u23



3!

yn...... 

uu1u2...... un1n



n!

yn.

Whereu xxn .

h

andh–equalinterval betweenthex- values

(numberoftermsintheformuladepends onthenumberoftermsintheproblem)

8. Lagrange’s formula:

yy0

xx1xx2...... xxn

x0 x1x0 x2...... x0 xn

y1

xx0xx2...... xxn

x1 x0x1 x2...... x1 x 

n

......

yn

xx0xx1...... xxn1

xn x0xn x1...... xn xn1

(dependsonthenumberof terms giveninthe problem)

9. LineOf Best Fit:

Normalequationsare

axnby

2

ax

bxxy

Theline ofbestfitis y=ax+b

CHAPTER8 . PROBABILITY DISTRIBUTION

1. IfX isa continuousrandomvariable,then

2. Fora discreterandomvariable X,

b

P(aXb)f(x)dx.

a

Mean

E(X)xi pi.

E(X2)x2p.

ii

Var(X)E(X2)E(X)2.

3. Fora continuousrandomvariableX,



Mean

E(X)

xf(x)dx.



E(X2)



x2 f(x)dx.



Var(X)E(X2)E(X)2.

4. Ifthe discreterandomvariableXfollowsBinomialdistributionthen

P(X

x)nCx

pxqnx,

x0,1,2,...... n.

5. Resultsrelatedto Binomialdistribution:

Mean= np; Variance= npq ;and p+q= 1

6. Ifthe discreterandomvariableXfollowsPoissondistributionthen



P(Xx)

x

,

x0,1,2,......

x!

7. ResultsrelatedtoPoissondistribution:

Meannp

; Variance=.

InPoissondistributionMean= Variance

8. Ifthe continuousrandomXfollowsNormaldistribution,thenitsp.d.fisgivenby

1x2

fx



1 

e2   ,



x.

2

9. ToconvertNormal variate XtostandardNormalvariatezweuse,

zX.



CHAPTER9 . SAMPLING DISTRIBUTION

1. Notations:

(a)N–Populationsize

(b) n–Sample size

(c) X

Meanofthesample

(d)

Meanofthepopulation

(e)s- Standarddeviation(S.D) ofsample

(f) -Standarddeviation(S.D)ofpopulation

2. Confidencelimitsfor=

XZc

s .(IfNisnotgiven)

n

=XZc

n

Nn.(IfNisgiven)

N1

3. Confidenceintervals for proportion=

pZc

pq.(IfNisnotgiven)

n

=pZc

n

Nn.(IfNisgiven)

N1

Note: For95%confidenceintervalZc =1.96

For99%confidenceintervalZc =2.58

4. TestingofHypothesisFormulae:

Test statistic

Test statistic

ZX.



n

Z pP.

pq

n

5. For5%level ofsignificance: Acceptanceregion

Z1.96.

Criticalregion

Z1.96.

6. For1%level ofsignificance: Acceptanceregion

Z2.58.

Criticalregion

Z2.58.

CHAPTER10 .APPLIED STATISTICS

1.Correlationcoefficientformulae:

(a)

r(X,Y)

NXYXY

NX2 X2

NY2 Y2

(If

X,Yareintegersornon-integers)

(b)r(x,y)xy

Where

xXX

and

yYY.

x2

y2

(If

X,Yareintegers)and

XXandYY

(c)

r(X,Y)

nn

Ndxdydxdy

Ndx2 dx2

Ndy2 dy2

(If

X,Yareintegersornon-integers)

Where

dxXAand

dyYB.

(A,Bare arbitraryvaluesofXandY)

(Note:Correlationcoefficient shouldlie between -1and1)

2.RegressionFormulae:

(a)Regressionline ofXon Yis

(XX)bxy(YY).

(b)Regressionline ofYonXis

(YY)byx(XX).

Where

XX

n

andYY

n

Wherebxy 

NXYXY

Y2 Y2

andbyx 

NXYXY

X2 X2

(If

X,Yareintegersornon- integers)

Wherebxy

 xy

y2

andbyx

xy

x2

(If

X,Yareintegers)

(Note: RegressionlineswillintersectatX,Y.)

3.SeasonalIndex=

Quaterly

Grand

average100.

average

4.IndexNumbers:

(a)Laspeyre’spriceIndexnumberP Lp1q0 100.

p0q0

(b)Paasche’spriceindexnumberP Pp1q1 100.

p0q1

(c)Fisher’spriceindexnumberP F 

p1q0 p1q1 100.

p0q0

p0q1

(OR)P

F 

LP

0101

(d)Costof Living Indexnumbers:

(i)AggregateExpendituremethod(C.L.I) p1q0 100.

p0 q0

(ii)FamilyBudgetmethod (C.L.I)PV.

V

WhereP

p1 100

andVpq.

0 0

0

5.Statistical Quality Control(SQC)Formulae:

Rangechart(RChart):C.L=

RR.

U.C.L=

L.C.L=

n

D4 R

D3R

XChart:C.L=

XX.

U.C.L=

L.C.L=

n

XA2R.

XA2R.

(WhereC.L–Central Line ; U.C.L- UpperControlLine ; L.C.L- LowerControlLine)