Quantitative Aptitude - Question Bank · PDF fileAlgebra - (i) (a + b)2 = a 2 + b + 2ab ... 6 Quantitative Aptitude – Question Bank Regression Lines Coefficientof er ation r...

13 PROBABILITY AND EXPECTED VALUE BY MATHEMATICAL EXPECTATION 165-175

14 THEORITICAL DISTRIBUTION(BINOMIAL, POISSON, AND NORMAL DISTRIBUTION) 176-185

15 SAMPLING THEORY 186-194

16 INDEX NUMBERS 195-203

17 APPENDIX 204-209

( )!

! ( ) !. ( ). ( )

( ) ! ( ) ( ) ! ( ) ! ( ) ! ( ) ! ( )!

( ) !( )

i Cn

r n rnum ber of com binations of n things taken r at a tim e n r ii C C

iii n n n iv v vi n n n vii Pn

n rn r

nr

nr

nn r

nr

=−

= > =

= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − = = = − =−

>

−

1 2 3 1 0 1 1 1 1

CPT - QUANTITATIVE APTITUDE

Basic Concepts That One Must Know Algebra -

(i) (a + b)2 = a2 + b2 + 2ab (ii) (a – b)2 = a2 + b2 – 2ab (iii) (a + b)2 = (a – b)2 + 4ab (iv) a2 – b2 = (a – b) (a + b) (v) (a + b)3 = a3 + b3 + 3ab2 + 3a2b = a3 + b3 + 3ab (a + b) (vi) (a – b)3 = a3 – b3 – 3a2b + 3ab2 = a3 – b3 – 3ab (a – b) (vii) a3 + b3 = (a + b)3 – 3ab (a + b) (viii) a3 – b3 = (a – b)3 + 3ab (a – b) (ix) a2n – b2n = (an)2 – (bn)2 = (an – bn) (an + bn) (x) a6 – b6 = (a3)2 – (a3)2 = (a3 – b3) (a3 + b3) (xi) a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca) (xii) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (xiii) a3 + b3 = (a + b) (a2 – ab + b2) (xiv) (a3 – b3) = (a – b) (a2 + ab + b2)

Indices and Surds

Logarithms

( ) loglog

log log

( ) log

( ) log . ( . )

( ) log

vii aa

b b

viii

ix e

x

ba

a a

ee

e

= =

=

= ∴ =

=

1

1

0 4343 2 7183

1

10

1010

Combinations

( ) log log log

( ) log log log

( ) log log

( ) log , . . , log

( ) log , . . , log

( ) loglog

log[ ]

i m n m n

iimn

m n

iii m n m

iv a i e arithm of a num ber to the sam e base

v i e arithm of to any base

vi mm

aChange of base

a a a

a a a

an

a

a

a

ab

b

= +

= −

=

= =

= =

=

1 1

1 0 1 0

QA_QB_P0i-220A.indd 1 16/04/14 3:08 PM

2� Quantitative Aptitude - Question Bank2 Quantitative Aptitude – Question Bank

( ) '

( ) . log

( ). log . log .

. . int .

( )

( ) Re

i Sturge s Rule

a K N

b CH ighest value Low est value

N

Range

N

Range

N o of classes

w here K N o of classes, N N o of item s, C Class w idth or class erval

ii Frequency density of a classFrequency of that class

W idth of that class

iii lative frequency of a classFrequency of that class

Sum of the frequencies of all the classes

= +

=−

+=

+=

= = =

=

=

1 1 322

1 3 322 1 3 322

10

10 10

( ).

. .

, . . ,

, . .

.

( ) . . . .'

, '.

i A rithm etic m eanSum of the values

N o of values

For individual series, A Mx

nx

For frequency distribution A Mfxf

x w here f N

For a grouped frequency distribution A Mfxf

w here x are the m id values of different groups

ii A M Afdf

or A M Afdf

h

w here A A ssum ed m ean d s are deviations of the m id values of different classes fromthe assum ed m ean

=

=∑

=

=∑∑

= ∑ =

=∑∑

−

= +∑∑

= +∑∑

×

=

Concept of sigma (∑)

∑ = ∑ =+

∑ =+ +

∑ =+

11

21 2 16

14

2 32 2

N NN N

NN N N

NN N

,( )

,( ) ( )

,( )

Frequency Distribution

Measures of Central Tendency

ddh

h Som e com m on factor in the d values

iii Com bined A Mn x n x

n nx

iv The sum of the deviations of all the values from their A M is x x

' , .

( ) . .

( ) . . . ( )

= =

=++

=

= ∑ − =

1 1 2 2

1 212

0 0algebraic

Quantitative Aptitude – Question Bank 3

For individual series, the m edian = size ofn + 1

2th item . (A fter arranging the data in ascending

or descending order. )

For grouped frequency distribution, M edian = L

N2+−

×c f

fi

.

( ) . .log

( ) . .log

/

( ) . .log . log .

( ) . [

( ) . [ /

v G M A Lx

Nfor individual series

vi G M A Lf x

Nfor discrete continuous series

vii G M A LN G M N G M

N N

viii H MN

x

for invividual series]

ix H MN

fx

for discrete continuous series]

=∑

=∑

=++

=∑

=∑

121 1 2 2

1 2

1

Median

W here L = Low er lim it of the m edian class

N2

N um ber of values

f = Frequency of the m edian class

i = Class interval of the m edian class

c. f = cum ulative frequency of the class preceding the m edian class.

=2

Quartiles

For individual series, item having maximum frequency is the mode.

For grouped frequency distribution, M ode = Lf f

2f f fi

LD

D Di

Em pirical relationship betw een M ean, M ode & M edian : M ode 3 M ed 2 M ean

1 0

1 0 2

1

1 2

+−

− −×

= ++

×

= −

QA_QB_P0i-220A.indd 2 16/04/14 3:08 PM

Quantitative Aptitude - Question Bank 32 Quantitative Aptitude – Question Bank

( ) '

( ) . log

( ). log . log .

. . int .

( )

( ) Re

i Sturge s Rule

a K N

b CH ighest value Low est value

N

Range

N

Range

N o of classes

w here K N o of classes, N N o of item s, C Class w idth or class erval

ii Frequency density of a classFrequency of that class

W idth of that class

iii lative frequency of a classFrequency of that class

Sum of the frequencies of all the classes

= +

=−

+=

+=

= = =

=

=

1 1 322

1 3 322 1 3 322

10

10 10

( ).

. .

, . . ,

, . .

.

( ) . . . .'

, '.

i A rithm etic m eanSum of the values

N o of values

For individual series, A Mx

nx

For frequency distribution A Mfxf

x w here f N

For a grouped frequency distribution A Mfxf

w here x are the m id values of different groups

ii A M Afdf

or A M Afdf

h

w here A A ssum ed m ean d s are deviations of the m id values of different classes fromthe assum ed m ean

=

=∑

=

=∑∑

= ∑ =

=∑∑

−

= +∑∑

= +∑∑

×

=

Concept of sigma (∑)

∑ = ∑ =+

∑ =+ +

∑ =+

11

21 2 16

14

2 32 2

N NN N

NN N N

NN N

,( )

,( ) ( )

,( )

Frequency Distribution

Measures of Central Tendency

ddh

h Som e com m on factor in the d values

iii Com bined A Mn x n x

n nx

iv The sum of the deviations of all the values from their A M is x x

' , .

( ) . .

( ) . . . ( )

= =

=++

=

= ∑ − =

1 1 2 2

1 212

0 0algebraic


For individual series, the m edian = size ofn + 1

2th item . (A fter arranging the data in ascending

or descending order. )

For grouped frequency distribution, M edian = L

N2+−

×c f

fi

.

( ) . .log

( ) . .log

/

( ) . .log . log .

( ) . [

( ) . [ /

v G M A Lx

Nfor individual series

vi G M A Lf x

Nfor discrete continuous series

vii G M A LN G M N G M

N N

viii H MN

x

for invividual series]

ix H MN

fx

for discrete continuous series]

=∑

=∑

=++

=∑

=∑

121 1 2 2

1 2

1

Median

W here L = Low er lim it of the m edian class

N2

N um ber of values

f = Frequency of the m edian class

i = Class interval of the m edian class

c. f = cum ulative frequency of the class preceding the m edian class.

=2

Quartiles

For individual series, item having maximum frequency is the mode.

For grouped frequency distribution, M ode = Lf f

2f f fi

LD

D Di

Em pirical relationship betw een M ean, M ode & M edian : M ode 3 M ed 2 M ean

1 0

1 0 2

1

1 2

+−

− −×

= ++

×

= −

QA_QB_P0i-220A.indd 3 16/04/14 3:08 PM


( ) var. .

( ) . . ( )

vi Coefficient of iationS D

M ean x

vii Com bined S D of tw o groupsn n n d n d

n n

= × = ×

=+ + +

+

100 100

121 1

22 2

212

12

2 22

1 2

σ

σσ σ

Measures of Dispersion

=

++

++

−n n

n n

n n

n nx x1 1

22 2

2

1 2

1 2

1 22 1 2

2σ σ( )

( )

w here n n num bers of item s in group I and II respectively

S D of group I and II respectively

d x x d x x

1 2

1 2

1 1 12 2 2 12

,

, . . .

,

=

=

= − = −

σ σ


Correlation and regression

1. Karl Pearson Coefficient of Correlation r = x yσ σ

Cov(x,y) (Covariance method)

2. Direct Method r =

( ) ( )2 2

2 2.

n

x yy

n n

∑ ∑∑

∑ ∑∑ ∑

x. yxy -

x - -

3. Short cut method r =

( ) ( )22

2 2

.

.

x y

yxy

d dd d

n

ddd d

n n

∑ ∑∑

∑∑∑ ∑

x y

x

.-

- -

4. Actual mean method r = 2 2, where x = X- X and y = Y - Y

. y

∑

∑ ∑

xyx

5. In case of bivariate frequency distribution, r = ( ) ( )22

2 2

.

.

x y

yxy

fd fdfd d

n

fdfdfd fd

n n

∑ ∑∑

∑∑∑ ∑

x y

x

.-

- -

Rank correlation coefficient

1. R = 2

2

61 , when ranks are unequal

( 1)

d

N N

∑−

−

2. R =

3 3 32

2

6 ...................12 12 12

1 , when ranks are equal( 1)

m m m m m md

N N

− − −∑ + + + −

−

QA_QB_P0i-220A.indd 4 16/04/14 3:08 PM


( ) var. .

( ) . . ( )

vi Coefficient of iationS D

M ean x

vii Com bined S D of tw o groupsn n n d n d

n n

= × = ×

=+ + +

+

100 100

121 1

22 2

212

12

2 22

1 2

σ

σσ σ

Measures of Dispersion

=

++

++

−n n

n n

n n

n nx x1 1

22 2

2

1 2

1 2

1 22 1 2

2σ σ( )

( )

w here n n num bers of item s in group I and II respectively

S D of group I and II respectively

d x x d x x

1 2

1 2

1 1 12 2 2 12

,

, . . .

,

=

=

= − = −

σ σ


Correlation and regression

1. Karl Pearson Coefficient of Correlation r = x yσ σ

Cov(x,y) (Covariance method)

2. Direct Method r =

( ) ( )2 2

2 2.

n

x yy

n n

∑ ∑∑

∑ ∑∑ ∑

x. yxy -

x - -

3. Short cut method r =

( ) ( )22

2 2

.

.

x y

yxy

d dd d

n

ddd d

n n

∑ ∑∑

∑∑∑ ∑

x y

x

.-

- -

4. Actual mean method r = 2 2, where x = X- X and y = Y - Y

. y

∑

∑ ∑

xyx

5. In case of bivariate frequency distribution, r = ( ) ( )22

2 2

.

.

x y

yxy

fd fdfd d

n

fdfdfd fd

n n

∑ ∑∑

∑∑∑ ∑

x y

x

.-

- -

Rank correlation coefficient

1. R = 2

2

61 , when ranks are unequal

( 1)

d

N N

∑−

−

2. R =

3 3 32

2

6 ...................12 12 12

1 , when ranks are equal( 1)

m m m m m md

N N

− − −∑ + + + −

−

QA_QB_P0i-220A.indd 5 16/04/14 3:08 PM


Regression Lines

Coefficient of er ation r coefficient of correlationExplained iance

Total iance

S dard error of estim ate of X values SX a X b XY

Nx

det m in ( )var

var

tan

= = =

= =∑ − ∑ − ∑

2 2

2

Index Numbers

(i)

(ii) Unweighted Index Numbers

( ) Pra Sim ple A ggregate ice Index Pp

p= =

∑∑

×011

0

100

Pr Re . Reice lative PP

PQuantity lative Q

q

q= = × = = ×1

0

1

0

100 100


(iii) Weighted Index Numbers

(a) Weighted Aggregative Method:

Sim ple A ggregative Index N um ber w ith Fixed W eight = P

P

P

01

01

01

=∑∑

×

= =∑∑

×

= =∑∑

×

p w

p w

Laspeyre s ice Index N um berp q

p q

Paasche s ice Index N um berp q

p q

La

Pa

1

0

1 0

0 0

1 1

0 1

100

100

100

' Pr

' Pr

( )

( )

M arshal Edgew orth's Index N um ber Pp ( q q )

p ( q q )100

Fisher's Ideal Index N um ber Pp q

p q

p q

p q100

01(M E) 1 0 1

0 0 1

01(F ) 1 0

0 0

1 1

0 1

i

− = =∑ +∑ +

×

= =∑∑

×∑∑

×

−

(b) Weighted Average of Price Relatives:

(iv) Kelly’s Method

Pp q

p qw here q is a fixed quantity used in current as w ell as base year i e q

q q01

1

0

1 21002

=∑∑

× =+

, . .

(v) Dorbish and Bowley’s Method

P

p q

p q

p q

p qor P

L P

A M of Laspeyre s and Paasche s Index N um bers)

01

1 0

0 0

1 1

0 1012

1002

100

100

=

∑∑

+∑∑

× =+

×

= ×( . . ' ' .

QA_QB_P0i-220A.indd 6 16/04/14 3:08 PM


Regression Lines

Coefficient of er ation r coefficient of correlationExplained iance

Total iance

S dard error of estim ate of X values SX a X b XY

Nx

det m in ( )var

var

tan

= = =

= =∑ − ∑ − ∑

2 2

2

Index Numbers

(i)

(ii) Unweighted Index Numbers

( ) Pra Sim ple A ggregate ice Index Pp

p= =

∑∑

×011

0

100

Pr Re . Reice lative PP

PQuantity lative Q

q

q= = × = = ×1

0

1

0

100 100


(iii) Weighted Index Numbers

(a) Weighted Aggregative Method:

Sim ple A ggregative Index N um ber w ith Fixed W eight = P

P

P

01

01

01

=∑∑

×

= =∑∑

×

= =∑∑

×

p w

p w

Laspeyre s ice Index N um berp q

p q

Paasche s ice Index N um berp q

p q

La

Pa

1

0

1 0

0 0

1 1

0 1

100

100

100

' Pr

' Pr

( )

( )

M arshal Edgew orth's Index N um ber Pp ( q q )

p ( q q )100

Fisher's Ideal Index N um ber Pp q

p q

p q

p q100

01(M E) 1 0 1

0 0 1

01(F ) 1 0

0 0

1 1

0 1

i

− = =∑ +∑ +

×

= =∑∑

×∑∑

×

−

(b) Weighted Average of Price Relatives:

(iv) Kelly’s Method

Pp q

p qw here q is a fixed quantity used in current as w ell as base year i e q

q q01

1

0

1 21002

=∑∑

× =+

, . .

(v) Dorbish and Bowley’s Method

P

p q

p q

p q

p qor P

L P

A M of Laspeyre s and Paasche s Index N um bers)

01

1 0

0 0

1 1

0 1012

1002

100

100

=

∑∑

+∑∑

× =+

×

= ×( . . ' ' .

QA_QB_P0i-220A.indd 7 16/04/14 3:08 PM


Key Point

( ) RePr

( ) Re ( . ). ( . ) Re ( . )

( ) . . ( . ). . ( . ) . . . ( . )

i Link lativeCurrent year price

evious year price

ii Chain lative for C YA verage L R for C Y Chain lative for P Y

iii C B I for C YF B I for C Y C B I for P Y

= ×

=×

=×

100

100

100

Probability and Mathematical Expectation

1. Pr obability of the occurance of an event

N um ber of cases favourable tothe occurrence of the event

Total num ber of m utually exclusiveequally likely and exhaustive cases

=

=+

< <

=

=

∪ =

= + − ∩

AA B

w here B denotes the num ber of cases against the occurrence of the

the event and A denotes the num ber of cases in favour of the event

probability of the occurrence of an event

Odds in favour of the eventAB

Odd against the eventBA

A ddition thereonP A B obability of the occurrence of at least one of the events

P A P B P A B

.

.

.

.

.( ) Pr

( ) ( ) ( )

2 0 1

3

4

5

6.

( ) ( ) . ( | ) ( ) . ( | )

Pr int

Pr tan

M ultiplication theorem

P A B P A P B A P B P A B

obability of the jo occurrence of the events A and B

obability of sim ul eous occurrence of the events A and B

∩ = =

=

=


11. Bay’s theorem

P E AP E P A E

P E P A Ew here P Ei

i i

i ii( | )

( ) ( | )

( ) ( | )( )=

∑∑ =1

12 1

1

2 22 2

. ( ) var

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) [ ( )] ( ) . . ( ) [ ( )]

( ) ( )

i Expected value of a random iable x P x w here P

ii E a a iii E ax aE x

iv E x y E x E y v E xy E x E y

iv E x p x vii E a bx E a E bx a bE x

viii Variance of x E x E x ix S D of x E x E x

x E x x

i i i

ri i

r

i

n

= ∑ ∑ =

= =

+ = + =

= + = + = +

= − = −

−

=∑

=

=∩

=∩

=∩

=∩

=∩

=∩

0

13. ( ) ( / )( )

( )( ) ( / )

( )( )

( ) ( / )( )

( )( ) ( / )

( )

( )

( ) ( / )( )

( )( ) ( / )

( )

( )

i P A BP A B

P Bii P B A

P A BP A

iii P A BP A B

P Biv P B A

P B A

P A

v P A BP A B

P Bvi P B A

P B A

P A

QA_QB_P0i-220A.indd 8 16/04/14 3:08 PM


Key Point

( ) RePr

( ) Re ( . ). ( . ) Re ( . )

( ) . . ( . ). . ( . ) . . . ( . )

i Link lativeCurrent year price

evious year price

ii Chain lative for C YA verage L R for C Y Chain lative for P Y

iii C B I for C YF B I for C Y C B I for P Y

= ×

=×

=×

100

100

100

Probability and Mathematical Expectation

1. Pr obability of the occurance of an event

N um ber of cases favourable tothe occurrence of the event

Total num ber of m utually exclusiveequally likely and exhaustive cases

=

=+

< <

=

=

∪ =

= + − ∩

AA B

w here B denotes the num ber of cases against the occurrence of the

the event and A denotes the num ber of cases in favour of the event

probability of the occurrence of an event

Odds in favour of the eventAB

Odd against the eventBA

A ddition thereonP A B obability of the occurrence of at least one of the events

P A P B P A B

.

.

.

.

.( ) Pr

( ) ( ) ( )

2 0 1

3

4

5

6.

( ) ( ) . ( | ) ( ) . ( | )

Pr int

Pr tan

M ultiplication theorem

P A B P A P B A P B P A B

obability of the jo occurrence of the events A and B

obability of sim ul eous occurrence of the events A and B

∩ = =

=

=


11. Bay’s theorem

P E AP E P A E

P E P A Ew here P Ei

i i

i ii( | )

( ) ( | )

( ) ( | )( )=

∑∑ =1

12 1

1

2 22 2

. ( ) var

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) [ ( )] ( ) . . ( ) [ ( )]

( ) ( )

i Expected value of a random iable x P x w here P

ii E a a iii E ax aE x

iv E x y E x E y v E xy E x E y

iv E x p x vii E a bx E a E bx a bE x

viii Variance of x E x E x ix S D of x E x E x

x E x x

i i i

ri i

r

i

n

= ∑ ∑ =

= =

+ = + =

= + = + = +

= − = −

−

=∑

=

=∩

=∩

=∩

=∩

=∩

=∩

0

13. ( ) ( / )( )

( )( ) ( / )

( )( )

( ) ( / )( )

( )( ) ( / )

( )

( )

( ) ( / )( )

( )( ) ( / )

( )

( )

i P A BP A B

P Bii P B A

P A BP A

iii P A BP A B

P Biv P B A

P B A

P A

v P A BP A B

P Bvi P B A

P B A

P A

QA_QB_P0i-220A.indd 9 16/04/14 3:08 PM


Probability Distributions

1

1

2

3

4

5

. ( ) Pr

;

Pr

Pr

.

. . .

.

.

p r obability of exactly r successes in Binom ial D istribution C q p

w here n N um ber of trials r num ber of successes

p obability of success in one trial

q obabillity of failure in one trial p

M ean of distribution np

S D npq

Variance npq

Skew nessq p

npq

nr

n r r= =

= =

=

= = −

=

=

=

=−

−


( ) [ ]

( ) . [ ]

( ) . ,

( ) . ,

iv SEn

w hen sam ple is draw n from population

v S En

N nN

w hen sam ple is draw n from finite population

vi IfnN

then N is called finite size of population

vii IfnN

then N is called ize of population

X

X

=

= ×−−

>

<

σ

σ

infinite

infinite s

1

0 05

0 05

G uru M an tra:

( ) ( ) ( )

. ( )

( ) ( )

( ) ( ) ( )

.

( ) ( )

( )

( ) ( )( )

i X X W hy

A ns X X X XX N X

N X N X XX

Nii X X least alw ays

iii X X X X X X

X N X X X

X NX

N

X

NX

XX

N

X

N

XX

N

iv Y Y YY

N

Σ

Σ Σ ΣΣ

Σ

Σ

Σ Σ

Σ Σ

ΣΣ Σ

Σ

ΣΣ Σ

ΣΣ

Σ ΣΣ

− =

− = −= −

= − = =

− =

− = + −

= + −

= + −

= + −

= −

− = −

0

0

2

2 2 2 2

2 2 2

22

2

22

22

22

2 22

( ) ( ) ( ) ( )

. .. .

.

v X X Y Y XY X Y X Y X Y

XY Y X X Y X Y N

XYX Y

N

X Y

N

X

N

Y

NN

XYX Y

N

Σ Σ

Σ Σ Σ

ΣΣ Σ Σ Σ Σ Σ

ΣΣ Σ

− − = − − +

= − − +

= − − +

= −

QA_QB_P0i-220A.indd 10 16/04/14 3:08 PM


Probability Distributions

1

1

2

3

4

5

. ( ) Pr

;

Pr

Pr

.

. . .

.

.

p r obability of exactly r successes in Binom ial D istribution C q p

w here n N um ber of trials r num ber of successes

p obability of success in one trial

q obabillity of failure in one trial p

M ean of distribution np

S D npq

Variance npq

Skew nessq p

npq

nr

n r r= =

= =

=

= = −

=

=

=

=−

−


( ) [ ]

( ) . [ ]

( ) . ,

( ) . ,

iv SEn

w hen sam ple is draw n from population

v S En

N nN

w hen sam ple is draw n from finite population

vi IfnN

then N is called finite size of population

vii IfnN

then N is called ize of population

X

X

=

= ×−−

>

<

σ

σ

infinite

infinite s

1

0 05

0 05

G uru M an tra:

( ) ( ) ( )

. ( )

( ) ( )

( ) ( ) ( )

.

( ) ( )

( )

( ) ( )( )

i X X W hy

A ns X X X XX N X

N X N X XX

Nii X X least alw ays

iii X X X X X X

X N X X X

X NX

N

X

NX

XX

N

X

N

XX

N

iv Y Y YY

N

Σ

Σ Σ ΣΣ

Σ

Σ

Σ Σ

Σ Σ

ΣΣ Σ

Σ

ΣΣ Σ

ΣΣ

Σ ΣΣ

− =

− = −= −

= − = =

− =

− = + −

= + −

= + −

= + −

= −

− = −

0

0

2

2 2 2 2

2 2 2

22

2

22

22

22

2 22

( ) ( ) ( ) ( )

. .. .

.

v X X Y Y XY X Y X Y X Y

XY Y X X Y X Y N

XYX Y

N

X Y

N

X

N

Y

NN

XYX Y

N

Σ Σ

Σ Σ Σ

ΣΣ Σ Σ Σ Σ Σ

ΣΣ Σ

− − = − − +

= − − +

= − − +

= −

QA_QB_P0i-220A.indd 11 16/04/14 3:08 PM

Quantitative Aptitude - Question Bank · PDF fileAlgebra - (i) (a + b)2 = a 2 + b + 2ab ... 6 Quantitative Aptitude – Question Bank Regression Lines Coefficientof er ation r...

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