12th Std Business Math Formulae

8/18/2019 12th Std Business Math Formulae

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K. MANIMARAN. M.Sc.,B.Ed., P.G.Asst – GOLDEN GATES MHSS, SALEM – 8.

2 STD BUSINESS MATHEMATICS

FORMULAE

CHAPTER 1 . APPLICATIONS OF MATRICES AND DETERMINANTS

1. Adjoint of a matrix A is .T

c A AdjA

(where Ac is a cofactor matrix)

2. Inverse of a matrix A is .11 AdjA A

A

3. Results:

(i) .)( I A A AdjA AdjA A

(ii) .)()( AdjA AdjB AB Adj

(iii) .111 A B AB (iv) .11 I A A AA

(v) .

1

1 A A

4. The rank of a zero matrix (irrespective of its order) is 0.

5. Conditions for consistency of Simultaneous Linear Equations (Non – homogeneous):

(i) If ,)(),( n A B A then the equations are consistent and has unique solution.

(ii) If ,)(),( n A B A then the equations are consistent and has infinitely many solutions.

(iii) If ),(),( A B A then the equations are inconsistent and has no solution.

6. Conditions for consistency of Simultaneous Linear Equations (Homogeneous):

(i) If ,)(),( n A B A (OR) If 0 A then the equations have trivial solutions only.

(ii) If ,)(),( n A B A (OR) If 0 A then the equations have non trivial solutions also.

7. Cramer’s rule: .;;

z

y x z y x

8. Technology matrix .

2

22

1

21

2

12

1

11

x

a

x

a

x

a

x

a

B

9. Output matrix .1 D B I X

10. Transition Probability Matrix

BB BA

AB AA

P P

P P T (OR)

QQQP

PQ PP

P P

P P T

( depends on the name of the products A, B or P , Q)

11. For finding Equilibrium share of market A + B = 1 (OR) P + Q = 1

(This step carries 1 mark and it is compulsory)


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CHAPTER 2 . ANALYTICAL GEOMETRY

1. .e PM

SP

2. Eccentricity of parabola e = 1.

3. Eccentricity of ellipse e < 1.

4. Eccentricity of hyperbola e > 1.

5. Eccentricity of rectangular hyperbola .2e .

6.

Parabola:

y 2 = 4ax y

2 = -4ax x 2 = 4ay x 2 = - 4ay

Vertex (0,0) (0,0) (0,0) (0,0)

Focus 0,a 0,a a,0 a,0

Directrix a x a x a y a y

Latusrectum 4a 4a 4a 4a

Axis 0 y 0 y 0 x 0 x

7. Ellipse:

bab

y

a

x ,1

2

2

2

2

baa

y

b

x ,1

2

2

2

2

Centre (0,0) (0,0)

Eccentricity

222 1 eab

(OR)

2

2

1a

be

222 1 eab

(OR)

2

2

1a

be

Vertices 0,,0, aa aa ,0,,0

Directrixe

a x

e

a y

Latusrectuma

b22

a

b 22

Foci 0,,0, aeae aeae ,0,,0


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8. Hyperbola:

12

2

2

2

b

y

a

x 1

2

2

2

2

b

x

a

y

Centre (0,0) (0,0)

Eccentricity

1222 eab

(OR)

2

2

1a

be

1222 eab

(OR)

2

2

1a

be

Vertices 0,,0, aa aa ,0,,0

Directrixe

a x

e

a y

Latusrectumab

2

2 ab

2

2

Foci 0,,0, aeae aeae ,0,,0

9. The general equation of Rectangular Hyperbola (R.H) is xy = c2.

where

2

22 a

c ( useful for objectives)

10. The eccentricity of Rectangular Hyperbola (R.H) is 2e


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CHAPTER 3 . APPLICATIONS OF DIFFERENTIATION – I

1. Average cost (AC) = .)(

)( x

k x f or

x

C

2. Average variable cost (AVC) = .)(

x

x f

3. Average fixed cost (AFC) = . x

k

4. Marginal cost (MC) = .dx

dC

5. Marginal average cost (MAC) =

.dx

AC d

6. Total revenue R = px.

7. Average revenue (AR) = . x

R

(Average revenue = Demand function i.e, AR = p)

8. Marginal average revenue (MR) = .dx

dR

9. If x = f(p) is a demand function, then Elasticity of demanddp

dx

x

pd .

.

(Where x – quantity demanded ; p – price)

Note : For a demand function q = f(p) ,dp

dq

q

pd .

10. If x = f(p) is a supply function, then Elasticity of supply

dp

dx

x

p s .

(Where x – quantity supplied ; p – price)

11. Relation between MR and Elasticity of demand is .1

1

d

p MR

12. At equilibrium level, Qd = Q s.

13. Equation of tangent is .11 x xm y y

14. Equation of normal is .1

11 x xm

y y


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CHAPTER 4 . APPLICATIONS OF DIFFERENTIATION – II

1. Euler’s theorem : If u is a homogeneous function of x and y with degree n then, .nu y

u y

x

u x

( f or z can be used in the place of u depends on the name of the function)

2.

Partial Elasticities

1

1

1

1

1

1 . p

q

q

p

Ep

Eq

and

2

1

1

2

2

1 . p

q

q

p

Ep

Eq

3. Economic order quantity .2

)(1

3

0C

RC q

(where R – Requirement ; C3 – ordering cost ; C1 – carrying cost)

4. If unit price and percentage of inventory are given then carrying cost .100

%1 unitpriceC

5. Time between two consecutive orders .)( 00 R

qt

6. Number of orders = .0

q

R

7. Minimum average variable cost = .2 13C RC

8. Total ordering cost = .30

C

q

R

9. Total carrying cost = .2

1

0 C q


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CHAPTER 5 . APPLICATIONS OF INTEGRAL CALCULUS

Properties of Definite integrals:

1. ..)()(

b

a

a

b

dx x f dx x f

2. If f(x) is an odd function, i.e, if f(-x) = -f(x) then ..0)(

a

a

dx x f

3. If f(x) is an even function, i.e, if f(-x) = f(x) then ..)(2)(0

a

a

a

dx x f dx x f

4. ..)()( b

a

b

a

dx xba f dx x f

5. a a

dx xa f dx x f 0 0

.)()(

6. The area under the curve ),( x f y the x-axis and the ordinates at a x and b x is

b

a

ydx Area

7. The area under the curve x = g (y), the y-axis and the lines y = c and y = d is

d

c

xdy Area .

8. If MC is the marginal cost function then total cost function is given by . k dx MC C

9. If MR is the marginal revenue function then total revenue function is given by

. k dx MR R

10. The producers’ surplus for the supply function ) x( g p for the quantity 0 x and price 0 p is

.)(.0

0

00 x

dx x g x pS P

11. The consumers’ surplus for the demand function p = f (x) for the quantity x0 and price p

0 is

.)(.0

0

00 x

x pdx x f S C


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CHAPTER 6 . DIFFERENTIAL EQUATIONS

1. The General form of Homogeneous differential equations is

.,

,

y x g

y x f

dx

dy

2. Working rule for finding the solution of linear differential equations

(i) Extract P and Q.

(ii) Find . dx P (iii) Find Integrating Factor (I.F) = dx P e

3. The solution to linear differential equations of type Q Pydx

dy (Where P and Q are functions of

x only) is C dx F I Q F I y .. (OR) C dxQe ye Pdx Pdx

4. The solution to linear differential equations of type Q Pxdy

dx (Where P and Q are functions of

y only) is C dy F I Q F I x .. (OR) . C dyQe xe Pdy Pdy

5. Second order linear differential Equations

If m1 and m2 are the roots of the Auxilliary equation is of the type ax2 + bx + c = 0

(Quadratic equation)

(i) If the roots m1 and m2 are real and distinct, C.F = .21 xm xm Be Ae

(ii) If the roots m1 and m2 are real and equal(m1 = m2), C.F = .mxe B Ax

(iii) If the roots m1 and m2 are unreal, i.e, if im , C.F = .sincos x B x Ae x

(C.F – Complementary Function)

CHAPTER 7 . INTERPOLATION

1. Forward operator (delta) 010 )( y y y (or) ).()())(( x f h x f x f

2. Backward operator (nabla) 011 )( y y y (or) ).()())(( x f h x f h x f

3. The Shifting operator .........)(,)(,)( 303

20

2

10 y y E y y E y y E and so on.

4. The relation between forward operator (delta) and shifting operator E is

1 E (or) .1 E

5.

(For missing term problems)


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(a) .1331 02303

y E E E y E

(b) .14641 023404

y E E E E y E

(c) .15101051 0234505

y E E E E E y E

6. Gregory – Newton’s forward formula :

.

!

1.......21...........

!3

21

!2

1

!1 00

3

0

2

00 yn

nuuuu y

uuu y

uu y

u y y n

Where .0h

x xu

and h – equal interval between the x - values

(number of terms in the formula depends on the number of terms in the problem)

7. Gregory – Newton’s backward formula :

.!

1.......21

...........!3

21

!2

1

!1

32

n

n

nnnn yn

nuuuu

y

uuu

y

uu

y

u

y y

Where .h

x xu n

and h – equal interval between the x - values

(number of terms in the formula depends on the number of terms in the problem)

8. Lagrange’s formula:

110

110

12101

20

1

02010

21

0

..............

..............

.........................................................................

..............

..............

..............

nnnn

n

n

n

n

n

n

x x x x x x

x x x x x x y

x x x x x x

x x x x x x y

x x x x x x

x x x x x x y y

(depends on the number of terms given in the problem)

9. Line Of Best Fit:

Normal equations are

ynb xa xy xb xa

2

The line of best fit is

y = ax + b


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CHAPTER 8 . PROBABILITY DISTRIBUTION

1. If X is a continuous random variable, then b

a

dx x f b X a P .)()(

2. For a discrete random variable X ,

Mean .)( ii p x X E

.)( 22 ii p x X E

.)()()( 22 X E X E X Var 3. For a continuous random variable X ,

Mean .)()(

dx x xf X E

.)()( 22

dx x f x X E

.)()()( 22 X E X E X Var

4. If the discrete random variable X follows Binomial distribution then

..,.........2,1,0,)( n xq pnC x X P xn x x

5. Results related to Binomial distribution:

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

6.

If the discrete random variable X follows Poisson distribution then

..,.........2,1,0,!

)(

x x

e x X P

x

7. Results related to Poisson distribution:

Mean np ; Variance = .

In Poisson distribution Mean = Variance

8. If the continuous random X follows Normal distribution, then its p.d.f is given by

.,2

12

2

1

xe x f

x

9. To convert Normal variate X to standard Normal variate z we use,

X z .


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CHAPTER 9 . SAMPLING DISTRIBUTION

1.

Notations:

(a) N – Population size

(b) n – Sample size

(c)

X Mean of the sample(d) Mean of the population

(e) s - Standard deviation (S.D) of sample

(f) - Standard deviation (S.D) of population

2. Confidence limits for = .n

s Z X c (If N is not given)

= .1 N n N

n s Z X c (If N is given)

3. Confidence intervals for proportion = .n

pq Z p c (If N is not given)

= .1

N

n N

n

pq Z p c (If N is given)

Note : For 95% confidence interval Z c = 1.96

For 99% confidence interval Z c = 2.58

4.

Testing of Hypothesis Formulae:

Test statistic .

n

X Z

Test statistic .

n

pq

P p Z

5. For 5% level of significance : Acceptance region .96.1 Z

Critical region .96.1 Z

6. For 1% level of significance : Acceptance region .58.2 Z

Critical region .58.2 Z


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CHAPTER 10 . APPLIED STATISTICS

1. Correlation coefficient formulae:

(a)

2222

),(

Y Y N X X N

Y X XY N Y X r

(If Y X , are integers or non-integers)

(b)

22),(

y x

xy y xr Where X X x and .Y Y y

(If Y X , are integers) andn

X X

and

n

Y Y

(c)

2222

),(

dydy N dxdx N

dydxdxdy N Y X r

(If Y X , are integers or non - integers)

Where A X dx and . BY dy (A, B are arbitrary values of X and Y)

(Note: Correlation coefficient should lie between -1 and 1)

2. Regression Formulae:

(a) Regression line of X on Y is

).()( Y Y b X X xy

(b) Regression line of Y on X is

).()( X X bY Y yx Wheren

X X

and

n

Y Y

Where

22 Y Y

Y X XY N b xy and

22 X X

Y X XY N b yx

(If Y X , are integers or non - integers)

Where

2

y

xyb xy and

2

x

xyb yx (If Y X , are integers)

(Note: Regression lines will intersect at .,Y X )

3. Seasonal Index = .100averageGrand

averageQuaterly

4.

Index Numbers:


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(a) Las peyre’s price Index number .10000

01

01

q p

q p P

L

(b) Paasche’s price index number .10010

11

01

q p

q p P

P

(c) Fisher’s price index number .10010

11

00

01

01

q p

q p

q p

q p P

F

(OR) .010101 P L F P P P

(d) Cost of Living Index numbers:

(i) Aggregate Expenditure method (C.L.I) .100

00

01

q p

q p

(ii) Family Budget method (C.L.I) .

V

PV

Where 1000

1 p

p P and .00q pV

5. Statistical Quality Control (SQC) Formulae:

Range chart ( R Chart): C.L = .n

R R

U.C.L = R D4

L.C.L = R D3

X Chart : C.L = .n

X X

U.C.L = .2 R A X

L.C.L = .2 R A X

(Where C.L – Central Line ; U.C.L - Upper Control Line ; L.C.L - Lower Control Line)

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12th Std Business Math Formulae