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