model paper-4.docx

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Contents

S.No. Particulars Page No.

12

3

4

5 Gist of Chapters

6 Marking Scheme

!mportant "uestions

# Sol$e% "uestions &ith $alue points

' (ips for scoring &ell


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2 M)*+!NG SC,-M- / C0)SS !! M)(,-M)(!CS

Course Structure

Unit Topic Marks

!. *elations an% /unctions 1

!!. )lgera 13

!!!. Calculus 44

!. ectors an% 3 Geometr7 1

. 0inear Programming 6

!. Proailit7 1

Total 100

Unit I: Relations and Functions

1. Relations and Functions

(7pes of relations8 re9e:i$e; s7mmetric; transiti$e an% e"ui$alence relations. ne to onean% onto functions; composite functions; in$erse of a function.


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Unit III: Calculus

1. Continuit# and "i$erentia!ilit#

Continuit7 an% %ierentiailit7; %eri$ati$e of composite functions; chain rule; %eri$ati$es of in$erse trigonometric functions; %eri$ati$e of implicit functions. Concept of e:ponentialan% logarithmic functions.

eri$ati$es of logarithmic an% e:ponential functions. 0ogarithmic %ierentiation;%eri$ati$e of functions e:presse% in parametric forms. Secon% or%er %eri$ati$es. *olleDsan% 0agrangeDs Mean alue (heorems ?&ithout proof@ an% their geometric interpretation.

2. pplications o% "erivatives

)pplications of %eri$ati$es8 rate of change of o%ies; increasingE%ecreasing functions;tangents an% normals; use of %eri$ati$es in appro:imation; ma:ima an% minima ?=rst%eri$ati$e test moti$ate% geometricall7 an% secon% %eri$ati$e test gi$en as a pro$aletool@. Simple prolems ?that illustrate asic principles an% un%erstan%ing of the suBect as

&ell as reallife situations@.

&. Integrals

!ntegration as in$erse process of %ierentiation. !ntegration of a $ariet7 of functions 7sustitution; 7 partial fractions an% 7 parts; -$aluation of simple integrals of thefollo&ing t7pes an% prolems ase% on them.

e=nite integrals as a limit of a sum; /un%amental (heorem of Calculus ?&ithout proof@.


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Unit I,: ,ectors and T(ree-"imensional eometr#

1. ,ectors

ectors an% scalars; magnitu%e an% %irection of a $ector. irection cosines an% %irectionratios of a $ector. (7pes of $ectors ?e"ual; unit; >ero; parallel an% collinear $ectors@;position $ector of a point; negati$e of a $ector; components of a $ector; a%%ition of $ectors; multiplication of a $ector 7 a scalar; position $ector of a point %i$i%ing a linesegment in a gi$en ratio. e=nition; Geometrical !nterpretation; properties an% applicationof scalar ?%ot@ pro%uct of $ectors; $ector ?cross@ pro%uct of $ectors; scalar triple pro%uct of $ectors.

2. T(ree - dimensional eometr#

irection cosines an% %irection ratios of a line Boining t&o points. Cartesian e"uation an%$ector e"uation of a line; coplanar an% ske& lines; shortest %istance et&een t&o lines.Cartesian an% $ector e"uation of a plane. )ngle et&een ?i@ t&o lines; ?ii@ t&o planes; ?iii@ aline an% a plane. istance of a point from a plane.

Unit ,: /inear rogramming

1. /inear rogramming

!ntro%uction; relate% terminolog7 such as constraints; oBecti$e function; optimi>ation;%ierent t7pes of linear programming ?0.P.@ prolems; mathematical formulation of 0.P.

prolems; graphical metho% of solution for prolems in t&o $ariales; feasile an%infeasile regions ?oun%e% an% unoun%e%@; feasile an% infeasile solutions; optimalfeasile solutions ?up to three nontri$ial constraints@.

Unit ,I: ro!a!ilit#

1. ro!a!ilit#

Con%itional proailit7; multiplication theorem on proailit7. in%epen%ent e$ents;total proailit7;


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G!S( / C,)P(-*S

Chapter1. *elations an% /unctions

Introduction8

)n7 set of or%ere% pairs ? x ; y @ is calle% a relation in x an% y . /urthermore;

H (he set of =rst components in the or%ere% pairs is calle% the %omain of the relation.

H (he set of secon% components in the or%ere% pairs is calle% the range of the relation.

Relation8 0et ) an% < e t&o sets. (hen a relation * from set ) to Set < is a suset of

A × B ..

T#pes o% relations: -

Empty relation: ) relation * in a set ) is calle% empty relation; if no element of ) is

relate% to an7 element of ) i

Universal relation 8) relation * in a set ) is calle% universal relation; if each element of )

is relate% to e$er7 element of ); i.e.; * ) I ).

*+uivalence relation.8 ) relation * in a set ) is sai% to e an equivalence relation if * is

re9e:i$e; s7mmetric an% transiti$e

) relation * in a set ) is calle%?i@ refexive; if ?a; a@ ∈ *; for e$er7 a ∈ );?ii@ symmetric if ?a; @ ∈ * implies that ?b, a@ ∈ *; for all a; ∈ ).?iii@ transitive if ?a; @ ∈ * an% ?; c@ ∈ * implies that ?a; c@ ∈ *; for all a, b, c ∈

).Function:

) %unction is a relation et&een a set of inputs an% a set of permissile outputs &ith the

propert7 that each input is relate% to e:actl7 one output. )n e:ample is the %unction that

relates each real numer : to its s"uare :2.

) %unction is a relation for &hich each $alue from the set the =rst components of the

or%ere% pairs is associate% &ith e:actl7 one $alue from the set of secon% components of

the or%ere% pair.


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6

Types o unctions:

One-one or injective3:

) function f 8 J is %e=ne% to e one-one ?or injective@; if the images

of %istinct elements of un%er f are %istinct; i.e.; for e$er7 a, b in ; f ?a@ f ?@

implies a b. ther&ise; f is calle% many-one.


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4ori5ontal line test:

(o check the inBecti$it7 of the function; f ?:@ 2:. ra& a hori>ontal line such that this linecuts the graph onl7 at one place. Such t7pes of functions are kno&n as oneone functions.

!n this case &here the line cuts the graph of a function at more than one place; thefunctions are not oneone.

Onto or surjective3:

) function f 8 J is sai% to e onto ?or surjective@; if e$er7 element of J is the image of

some element of un%er f ; i.e.; for e$er7 y in J; there e:ists an element x in such that f

? x @ y .

Bijective unction:

) function f 8 J is sai% to e one-one an% onto ?or bijective@; if f is oth oneone an% onto.


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Inverse o a Bijective Function

0et f8 ) K < e a function. !f; for an aritrar7 : L ) &e ha$e f?:@ 7 L


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Chapter 2. !n$erse (rigonometric /unctions

!n$erse trigonometric functions or c7clometric functions are the socalle% in$erse

functions of the trigonometric functions; &hen their %omain are restricte% to principal

$alue ranch to make the trigonometric functions iBecti$e. (he principal in$erses are

liste% in the follo&ing tale.

Name Rsual

notation

e=nitio

n

omain of x

for real result

*ange of usual

principal $alue

*ange of usual

principal $alue


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?radians@ ?degrees@

arcsine y sin1 x x

sin y 1 T x T 1 UE2 T y T UE2 'V T y T 'V

arccosine y cos1 x x

cos y 1 T x T 1 T y T U V T y T 1#V

arctangen

t y tan1 x

x

tan y

all real

numersUE2 W y W UE2 'V T y T 'V

arccotangent

y cot1 x x cot y

all realnumers

W y W U V W y W 1#V

arcsecant y sec1 x x

sec y

x T 1 or 1 T

x

T y W UE2 or

UE2 W y T U

V T y W 'V or 'V

W y T 1#V

arccoseca

nt y csc1 x

x

csc y

x T 1 or 1 T

x

UE2 T y W or

W y T UE2

'V T y T V or V

W y T 'V

Chapter 3. Matrices

e=nition8 ) matri: ) is %e=ne% as an or%ere% rectangular arra7 of numers in m ro&s

an% n columns.

http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Degree_(mathematics)http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Cotangenthttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Cosecanthttp://en.wikipedia.org/wiki/Degree_(mathematics)http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Cotangenthttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Cosecanthttp://en.wikipedia.org/wiki/Radian


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1. *o& matri:8 ) matri: can ha$e a single ro& ) X a11 a12 a13 Y a1nZ2. Column matri:8 ) matri: can ha$e a single column ) 3. [ero or null matri:8 \ ) matri: is calle% the >ero or null matri: if all the entries are .4. S"uare matri:8 ) matri: for &hich hori>ontal an% $ertical %imensions are the same

?i.e.; an matri:@.5. iagonal matri:8 ) s"uare matri: ) is calle% %iagonal matri: if aiB for .6. Scalar matri:8 ) %iagonal matri: ) is calle% the scalar matri: if all its %iagonal

elements are e"ual.. !%entit7 matri: 8\ ) %iagonal matri: ) is calle% the i%entit7 matri: if aiB 1 for i B ; it

is %enote% 7 !n.#. Rpper triangular matri:8 ) s"uare matri: ) is calle% upper triangular matri: if aiB

for'. 0o&er triangular matri: 8 ) s"uare matri: ) is calle% lo&er triangular matri: if aiB

for

• Matri: operations

1. e=nition8 (&o matrices ) an% < can e a%%e% or sutracte% if an% onl7 if their%imensions are the same ?i.e. oth matrices ha$e the i%entical amount of ro&s an%columns.

2. )%%ition!f ) an% < are matrices of the same t7pe then the sum is a matri:C otaine% 7 a%%ing the correspon%ing elements aijFbij i.e. )F< C if aijFbij =cij

1. Matri: a%%ition is commutati$e ; associati$e an% %istriuti$e o$er multiplication

) F < < F )

) F ?< F C@ ?)F


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correspon%ing entries in the jth column of < an% summing the n terms. (he elements of C

are8

Note8 ):< is not e"ual to


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Chapter 4 eterminants

-$er7 s"uare matri: ) is associate% &ith a numer; calle% its %eterminant an% it

is %enote% 7 %et ?)@ or ^)^ .

nl7 s"uare matrices ha$e %eterminants. (he matrices &hich are not s"uare

%o not ha$e %eterminants

?i@ First 6rder "eterminant !f ) XaZ; then %et ?)@ ^)^ a

?ii@ Second 6rder "eterminant

^)^ a11a22 \ a21a12

?iii@ T(ird 6rder "eterminant

*valuation o% "eterminant o% S+uare Matri7 o% 6rder & !# Sarrus Rule

then %eterminant can e forme% 7 enlarging the matri: 7 a%Boining

the first t&o columns on the right an% %ra& lines as sho& elo& parallel an%

perpen%icular to the %iagonal.

(he $alue of the %eterminant; thus &ill e the sum of the pro%uct of element. in line

parallel to the %iagonal minus the sum of the pro%uct of elements in line perpen%icular

to the line segment. (hus;

_ a11a22a33 F a12a23a31 F a13a21a32 \ a13a22a31 \ a11a23a32 \

a12a21a33.


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roperties o% "eterminants

?i@ (he $alue of the %eterminant remains unchange%; if ro&s are change% intocolumns an% columns are change% into ro&s e.g.; ^)Q^ ^)^

?ii@ !n a %eterminant; if an7 t&o ro&s ?columns@ are inter change%; then the $alue of the

%eterminant is multiplie% 7 1.

?iii@ !f t&o ro&s ?columns@ of a s"uare matri: ) are proportional; then ^)^ .

?i$@ ^ero; then its $alue

is >ero. ?i:@ !f an7 t&o ro&s ?columns@ of a %eterminant are i%entical; then its $alue

is >ero.


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?:@ !f each element of ro& ?column@ of a %eterminant is e:presse% as a sum of t&o

or more terms; then the %eterminant can e e:presse% as the sum of t&o or more

%eterminants.

Important Results on "eterminants

?i@ ^)


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?i:@ !n general; ^< F C^ ` ^ero


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1#?ii@ (he sum of the pro%uct of elements of an7 ro& ?or column@ of a %eterminant &ith

the

cofactors of the correspon%ing elements of the same ro& ?or column@ is _

pplications o% "eterminant in eometr#

0et three points in a plane e )?:1; 71@;


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1')1 ?)@ )1 <

ST*S

-$aluate . *efer the note gi$en elo&.

. -$aluate the cofactors of elements of ).

i. /orm the a%Boint of ) as the matri: of cofactorsiv. Calculate in$erse of ) an% .

N(- /rom the ao$e it is clear that the e:istence of a solution %epen%s on the

$alue of the %eterminant of ). (here are three cases8

1. !f the then the s7stem is consistent &ith uni"ue solution gi$en 7

2. !f ?) is singular@ an% a%B ) .< then the solution %oes not e:ist. (he s7stem is

inconsistent.

&. !f ?) is singular@ an% a%B ) .< then the s7stem is consistent &ith in=nitel7

man7 solutions.to =n% these solutions put > k in t&o of the e"uations an%

sol$e them 7 matri: metho%.

For (omogeneous s#stem o% linear e+uations < = 0 > = 03

1. !f the then the s7stem is consistent &ith tri$ial solution : ; 7 ; >

2. !f ?) is singular@ an% a%B ) .< then the solution %oes not e:ist. (he s7stem is

inconsistent.

&. !f ?) is singular@ an% a%B ) .< then the s7stem is consistent &ith in=nitel7

man7 solutions. to =n% these solutions put > k in t&o of the e"uations an%

sol$e them 7 matri: metho%.

Chapter 5 Continuit7 an% ierentiailit7


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2"e?nition

) function f is sai% to e continuous at : a if

) function is sai% to e continuous on the inter$al Xa, bZ if it is continuous at eachpoint in the inter$al.

e=nition of %eri$ati$e 8 !f 7 f?:@ then 71

) function f is %ierentiale if it is continuous.

hen 0, b *, oth e:ist an% are e"ual then f is sai% to e %eri$ale or

%ierentiale.


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Parametric %ierentiation8 if 7 f?t@; : g?t@ then;

0ogarithmic %ierentiation8 !f 7 f?:@g?:@ then take log on oth the si%es.

rite log7 g?:@ logXf?:@Z. ierentiate 7 appl7ing suitale rule for %ierentiation.

!f 7 is sum of t&o %ierent e:ponential function u an% $; i.e. 7 u F $. /in% 7 using

logarithmic %ierentiation separatel7 then e$aluate.

2

2

d y

dx ,igher eri$ati$es8 (he %eri$ati$e of =rst %eri$ati$e is calle% as the secon%

%eri$ati$e. !t is %enote% 7 72 or 7QQ or

/*MR0)S / -*!)(!-S

1.( ) 0

d c

dx=

2.( ) 1n n

d x nx

dx

−=

3.1

1n n

d n

dx x x +− = ÷

4.( ) 1

d x

dx=

5.2

1 1d

dx x x

−

= ÷ 6. ( )

1

2

d

xdx x=

7.( ) log x x

d a a a

dx=

8.( ) x x

d e e

dx=

9.( )

1log

d x

dx x=

10.( )sin cos

d x x

dx=

11.( )cos sin

d x x

dx= −

12.( ) 2tan sec

d x x

dx=

13.( )cos cos cotd ec x ec x x

dx= −

14.( )s tand ec x sec x x

dx=

15.( ) 2cot

d x cosec x

dx= −

16.( )1

2

1sin

1

d x

dx x

− =−

17.( )1

2

1cos

1

d x

dx x

− −=− 18.

( )1 21

tan1

d x

dx x

− =+

19.( )1 2

1cot

1

d x

dx x

− −=+ 20.

( )12

1sec

1

d x

dx x x

− =−


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. e =rst check &hether f?:@ satis=es con%itions of *olleQs (heorem or not.

. ierentiate the function

. -"uate the %eri$ati$es &ith [ero to get x .

. !f x elongs to gi$en inter$al then; *olleQs (heorem $eri=e%.

(he follo&ing rules ma7 e helpful &hile sol$ing the prolem.

. ) pol7nomial function is e$er7&here continuous an% %ierentiale

. (he e:ponential function; sine an% cosine function are e$er7&here continuous an%

%ierentiale.

. 0ogarithmic function is continuous an% %ierentiale in its %omain.

. (an : is not continuous

Chapter 6 )pplication of eri$ati$es

Motion in a straig(t /ine

Suppose a particle is mo$ing in a straight line

(ake OQ as origin in timeOtQ the position of the particle e at ) &here ) s an% in time tFdt the

position of the particle e at < &here


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sFdsf ?tFdt@

lgorit(m:

1. Sol$ing an7 prolem =rst; see &hat are gi$en %ata.

2. hat "uantit7 is to e foun%

3. /in% the relation et&een the point no. ?1@ b ?2@.

4. ierentiate an% calculate for the =nal result.

)PP*!M)(!NS; !//-*-N(!)0S )N -***S

• )solute error

• *elati$e error

• Percentage error )ppro:imation 1. (ake the "uantit7 gi$en in the "uestion as 7 Fk f?: F h@

2. (ake a suitale $alue of : nearest to the gi$en $alue. Calculate

3. Calculate 7 f?:@ at the assume% $alue of :.

4. calculate at the assume% $alue of :

5. Rsing %ierential calculate

6. =n% the appro:imate $alue of the "uantit7 aske% in the "uestion as 7 Fk ; from the

$alues of 7 an% e$aluate% in step 3 an% 5.

(angents an% normals \

• Slope of the tangent to the cur$e 7 f?:@ at the point ?:;7@ is gi$en 7 m.

• -"uation of the tangent to the cur$e 7 f?:@ at the point ?:;7@ is ?7 7@ m ?: \ :@

• Slope of the normal to the cur$e 7 f?:@ at the point ?:;7@ is gi$en 7 1 E m

• -"uation of the normal to the cur$e 7 f?:@ at the point ?:;7@ is ?7 7@ ? 1 Em@ ?:

\ :@


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!n this section &e inten% to see ho& &e can use %eri$ati$e of a function to %etermine &here it is

increasing an% &here it is %ecreasing.

Necessar7 con%ition8 if f?:@ is an increasing function on?a; @ then the tangent at e$er7 point on the

cur$e 7f?:@ makes an acute angle &ith the positi$e %irection of : a:is.

!t is e$i%ent from =g that if f?:@ is a %ecreasing function on ?a; @; then tangent at e$er7 point on

the cur$e 7f?:@ makes an% otuse angle &ith the positi$e %irection of : a:is.

(hus; fQ?:@ ?W@ for all is the necessar7 con%ition for a function f?:@ to e increasing

?%ecreasing@ on a gi$en inter$al ?a; @. in other &or%s; if it is gi$en f?:@ is increasing ?%ecreasing on

?a; @; then &e can sa7 that fQ?:@?W@.

SUFFICI*9T C69"ITI69

T(eorem Let f be a di$erentiable real function dened on an open interval %a, b&.

%a& "f f%x& + for all , then f%x& increasing on %a, b&

%b& "f f%x& * for all , then f%x& decreasing on %a, b&

/6RIT4M F6R FI9"I9 T4* I9T*R,/ I9 @4IC4 FU9CTI69 IS I9CR*SI9 6R "*CR*SI9.

0et f?:@ e a real function. (o =n% the inter$als of monotonicit7 of f?:@ &e procee% as follo&s8

Step 1=n% fQ?:@.

Step !!Put fQ?:@ an% sol$e this ine"uation./or the $alues of : otaine% in step !! f?:@ is increasing an% for the remaining points in its

%omain it is %ecreasing.r

!. Calculate fQ?:@ for critical points#. 0et c1; c2; c3 are the roots of fQ?:@ .'. Cut the no line at c1; c2; c3/. rite the inter$al in tale0. Consi%er an7 point in the inter$al1. See the sign of fQ?:@ in that inter$al an% accor%ingl7 %etermine the function is increasing or

%ecreasing in that inter$al.


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Ma7imum 0et f?:@ e the real function %e=ne on the inter$al !. then f?:@is sai% to ha$e the

ma:imum $alue in !; if there e:ists a point a in ! such that f?:@Wf?a@ for all :

in such a case; the numer f?a@ is calle% the ma:imum $alue of f?:@ in the inter$al !

an% the point a is calle% a point of ma:imum $alue of f in the inter$al !.

Minimum 0et f?:@ e the real function %e=ne on the inter$al !. then f?:@ is sai% to ha$e the

minimum $alue in !; if there e:ists a point a ! such that

f?:@f?a@ for all :

!n such a case; the numer f?a@ is calle% the minimum $alue of f?:@ in the inter$al ! an%

the point a is calle% a point of minimum $alue of f in the inter$al !.

/6C/ M


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2#

Case I. 0et f ha$e a local ma:imum $alue at :c; then f is an increasing function in the left nbd of

x=c i.e., for : slightl7 W c an% a %ecreasing function f ! the right nbd of x=c i.e. for : slightl7 c.

)lso for :c; the graph has a hori>ontal tangent.

(hus; f %x& changes continuousl7 from F$e to \$e as increases through c.

Case II. 0et f ha$e a local minimum $alue at :c; then f is an increasing function in the left nbd of

x=c i.e., for : slightl7 W c an% a %ecreasing function f ! the right nbd of x=c i.e. for : slightl7 c.

)lso for :c; the graph has a hori>ontal tangent.

(h fQ%x& changes continuousl7 from F$e to F$e as

increases through c.

@orking rule %or ?rst derivative test

/in% the sign of fQ?:@ &hen : is slightl7 W c an% &hen : is slightl7 c.

!f fQ?:@ changes sign from F$e to \$e as : increases through c; then f has a local ma:imum $alue at

:c.

!f fQ?:@ changes sign from $e to F$e as : increases through c; then f has a local minimum $alue at

:c.

Second "erivative Test %or Finding /ocal Ma7ima and /ocal Minima

Suppose a "uantit7 7 %epen%s on the another "uantit7 : in a manner sho&n in =g. it ecomes

ma:imum at :1 an% minimum at :2. )t these points the tangent to the cur$e is parallel to the :

a:is an% hence its slope is .


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2'

Ma7ima and minima !# using second derivative test

ust efore the ma:imum the slope is positi$e; at the minimum it is >ero an% Bust after the

ma:imum it is negati$e. (hus %ecreases at a ma:imum an% hence the rate of change of is

negati$e at a ma:imum.

Minima

Similarl7; at a minimum the slope changes from negati$e to positi$e.

,ence &ith increases of :. the slope is increasing that means the rate of change of slope &.r.t : is

positi$e;

@orking Rule:

. /in% fQ?:@.

. Sol$e fQ?:@ &ithin the %omain to get critical point let one of the $alue of : c.

. Calculate fQQ?:@ at : c.

.!f fQQ?c@ then; f?:@ is minimum at : c; an% if fQQ?c@ W ; then f?:@ is ma:imum at : c.

Ma7ima

Similarl7; at a ma:imum the slope changes from positi$e to negati$e.

@orking Rule:

. /in% fQ?:@.

1.Sol$e fQ?:@ &ithin the %omain to get critical point let one of the $alue of : c.

2.Calculate fQQ?:@ at : c.

3.!f fQQ?c@ W then; f?:@ is minimum at : c; an% if fQQ?c@ W ; then f?:@ is ma:imum at : c.

pplication o% Ma7ima and Minima to pro!lems

@orking rule

?i@ !n or%er to illustrate the prolem; %ra& a %iagram; if possile. istinguish clearl7

et&een the $ariale an% constants.

?ii@ !f 7 is the "uantit7 to e ma:imi>e% or minimi>e%; e:press 7 in terms of a single

in%epen%ent $ariale &ith the help of gi$en %ata.

?iii@ Get %7E%: an% %27E%:2; then e"uate %7E%: get the $alue of :.


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3

?i$@ etermine the sign of %27E%:2 at : an% procee%.


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31

Chapter !ntegration

Integration is a &a7 of a%%ing slices to =n% the &hole. Integration can

e use% to =n% areas; $olumes; central points an% man7 useful things.

(he Chain *ule tells us that %eri$ati$e of g?f?:@@ gD?f?:@@fD?:@.


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32 ( \ (rigonometric function

- \ -:ponential function

Note that if 7ou are gi$en onl7 one function; then set the secon%

one to e the constant function g?:@1. integrate the gi$en function

7 using the formula

-/!N!(- !N(-G*)0 )S 0!M!( / ) SRM

e=nite integral is closel7 relate% to concepts like anti%eri$ati$e an%

in%e=nite integrals. !n this section; &e shall %iscuss this relationship in

%etail.

e=nite integral consists of a function f?:@ &hich is continuous in a close%

inter$al Xa; Z an% the meaning of %e=nite integral is assume% to e in

conte:t of area co$ere% 7 the function f from ?sa7@ OaQ to OQ.

)n alternati$e &a7 of %escriing is that the %e=nite integral

is a limiting case of the summation of an in=nite series; pro$i%e%

f?:@ is continuous on Xa; Z

(he con$erse is also true i.e.; if &e ha$e an in=nite series of the ao$e

form; it can e e:presse% as a %e=nite integral.


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33

(he /un%amental (heorem of Calculus Busti=es our proce%ure of

e$aluating an anti%eri$ati$e at the upper an% lo&er limits of integration

an% taking the %ierence.

Fundamental T(eorem o% Calculus

0et f e continuous on Xa bZ. !f 2 is an7 anti%eri$ati$e for f on Xa bZ; then

2 ?b@2 ?a@

roperties

ropert# 1:

ropert# 2:

Propert7 38


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34

ropert#':

ropert#):

ropert#A:

ropert# B:

Propert7 #8

1.( )

1

, 11

nn x x dx n

n

+

= ≠ −+∫ 2. ( )

1

1 1

1n ndx

x n x −−

=−∫

3. dx x=∫ 4.

1logdx x

x=∫

5.log

x x aa dx

a=∫

6. x xe dx e=∫

7. sin cos x dx x= −∫ 8. cos sin x dx x=∫

9.2sec tan x dx x=∫ 10.

2cos cotec x dx x= −∫

11. sec tan sec x x dx x=∫ 12. cos cot cosec x x dx ec x= −∫

13. tan log sec x dx x=∫ 14. cot log sin x dx x=∫

15. sec log sec tan x dx x x= +∫ 16. cos log cosec cotecx dx x x= −∫

17.

1

2 2

1tan

dx x

a x a a

− = ÷+ ∫ 18. 2 21

log2

dx a x

a x a a x

+=

− −∫

19.2 2

1log

2

dx x a

x a a x a−=− +∫ 20.1

2 2 sindx x

aa x− = ÷ −∫


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35

21.

2 2

2 2log

dx x x a

x a= + −

−∫

22.

2 2

2 2log

dx x x a

x a= + +

+∫

23.

22 2 2 2 1sin

2 2

x a xa x dx a x

a

− − = − + ÷ ∫

24.

2

2 2 2 2 2 2log2 2 x a x a dx x a x x a− = − − + −∫

25.

22 2 2 2 2 2log

2 2

x a x a dx x a x x a+ = + + + +∫

Chapter #. )pplication of !ntegration

) plane cur$e is a cur$e that lies in a single plane. ) plane cur$e ma7 e

close% or open.

(he area et&een the cur$e %e=ne% 7 a positi$e function an% the :

a:is et&een t&o $alues of 7 is calle% the %e=nite integral of f et&een

these $alues. (he area of a rectangle is the pro%uct of length an% rea%th

the area un%er a cur$e &ill e generate%. (his metho% is generalise% to

%e=ne a $ariet7 of integrals that %o not %escrie area.

(he stan%ar% approach %eals &ith a more general class of functions 7

imagining that &e %i$i%e the inter$al et&een a an% into a large numer

of small strips an% estimate the area of each strip to e the pro%uct of its

&i%th an% some $alue of f?:@ for : &ithin the strip.

(he area &ill then e something like the sum of the areas of the strips. !f

&e then let the ma:imum strip length go to >ero; &e can hope to =n% the

resulting sum of strip areas approach the true area.

http://mathworld.wolfram.com/Curve.htmlhttp://mathworld.wolfram.com/Plane.htmlhttp://mathworld.wolfram.com/Curve.htmlhttp://mathworld.wolfram.com/Plane.html


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36

(he stan%ar% &a7 to %o this is to let the ith strip egin at :i1 an% en% at

:iA the area of that strip is estimate% as ?: i:i1@f?:Di@ &ith :Di an7&here in

the strip.

) *iemann sum is a sum of the form Bust in%icate%8 it is a sum o$er strips

of the &i%th of the strip times a $alue of the f?:@ &ithin the strip. (hefunction is sai% to e *iemann integrale if the sum of the area of the

strips approaches a constant in%epen%ent of &hich arguments are use%

&ithin each strip to estimate the area of the strip; as ma:imum strip &i%th

goes to >ero.

(he notation use% ao$e can e un%erstoo% from this approachA &e are

summing the area of the in%i$i%ual strips; &hich for a $er7 small inter$al

aroun% : of si>e %: is estimate% to e f?:@%:; an% summing this o$er allsuch strips. (he integral sign represents the sum &hich is not an or%inar7

sum; ut the limit of or%inar7 sums as the si>e of the inter$als goes to

>ero.

Suppose &e ha$e a nonnegati$e function f of the $ariale :;

%e=ne% in some %omain that inclu%es the inter$al Xa; Z &ith a W . !f f is

sucientl7 &ell eha$e%; there is a &ell %e=ne% area enclose% et&een

the lines : a; : ; 7 an% the cur$e 7 f?:@.


37/122

3 (hat area is calle% the %e=nite integral of f %: et&een : a an% : ?of

course onl7 for those functions for &hich it makes sense@.

!t is usuall7 &ritten as

!f c lies et&een a an% &e o$iousl7 ha$e

rea !eteen to curves8

(here are actuall7 t&o cases that &e are going to e looking

at.

!n the =rst case &e &ant to %etermine the area et&een

an% on the inter$al Xa,bZ. e are also going to assume

that .

13

(he secon% case is almost i%entical to the =rst case. ,ere &e are going to

%etermine the area et&een an% on the inter$al

Xc,dZ &ith .


38/122

3#

!n this case the formula is;

(he area of a triangle is an e:tension of area of a plane cur$e.

Chapter ' !//-*-N(!)0 -]R)(!NS


39/122

3'

e=nition8 )n e"uation containing an in%epen%ent $ariale; a %epen%ent

$ariale an% %ierential coecients of the %epen%ent $ariale &ith

respect to the in%epen%ent $ariale is calle% a %ierential e"uation -.

(he *-* of a %ierential e"uation is the highest %eri$ati$e that appears

in the e"uation.

(he -G*-- of a %ierential e"uation is the po&er or e:ponent of the

highest %eri$ati$e that appears in the e"uation.

Rnlike algeraic e"uations; the solutions of %ierential e"uations are

functions an% not Bust numers.

Initial value pro!lem is one in (ic( some initial conditions are

given to solve a "*.

(o form a - from a gi$en e"uation in : an% 7 containing aritrar7

constants ?parameters@ \

1. ierentiate the gi$en e"uation as man7 times as the

numer of aritrar7 constants in$ol$e% in it .

2. -liminate the aritrar7 constant from the e"uations of 7; 7Q; 7QQetc.

) =rst or%er linear %ierential e"uation has the follo&ing form8

(he general solution is gi$en 7 &here

calle% the integrating factor. !f an initial con%ition is gi$en; use it to =n%

the constant 3.

,ere are some practical steps to follo&8

1. !f the %ierential e"uation is gi$en as ; re&rite it in

the form ; &here

2. /in% the integrating factor .


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43. -$aluate the integral

4. rite %o&n the general solution .

5. !f 7ou are gi$en an !P; use the initial con%ition to =n% the constant 3.

)*!)


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413. (hen use the pro%uct rule to get

4. Sustitute to re&rite the %ierential e"uation in terms of $ an% : or $

an% 7 onl7

5. /ollo& the steps for sol$ing separale %ierential e"uations.

6. *esustitute $ 7E: or $ : E 7.

Chapter 1 -C(*S

• ) "uantit7 that has magnitu%e as &ell as %irection is calle% a $ector.

!t is %enote% 7 %irecte% line segment; &here ) is the initial point

an% < is the terminal point . (he %istance )< is calle% the magnitu%e

%enote% 7 an% the $ector is %irecte% from ) to


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421. (riangle la& of $ectors for a%%ition of t&o $ectors . !f t&o $ectors can

e represente% oth in magnitu%e an% %irection 7 the t&o si%es of

a triangle taken in the same or%er; then the resultant is represente%

completel7; oth in magnitu%e an% %irection; 7 the thir% si%e of the

triangle taken in the opposite or%er.

Corollar78 1@ !f three $ectors are represente% 7 the three si%es of a

triangle taken in or%er; then their resultant is >ero.

2@ !f the resultant of three $ectors is >ero; then these can e represente%

completel7 7 the three si%es of a triangle taken in or%er.

2. Parallelogram la& of $ectors for a%%ition of t&o $ectors. !f t&o

$ectors are completel7 represente% 7 the t&o si%es )

an% < respecti$el7 of a parallelogram. (hen; accor%ing to the la&

of parallelogram of $ectors; the %iagonal C of the parallelogram

&ill e resultant ; such that

MR0(!P0!C)(!N P-*)(!NS /* -C(*S8

1. MR0(!P0!C)(!N / ) -C(*


43/122

43 (he %ot pro%uct is %istriuti$e o$er $ector a%%ition8

(he %ot pro%uct is ilinear8

hen multiplie% 7 a scalar $alue; %ot pro%uct satis=es8

(&o non>ero $ectors a an% are perpen%icular if an% onl7 if a H

.

(he %ot pro%uct %oes not oe7 the cancellation la&8 !f a H a H c

an% a ` 8 then a H ? c@ 8

!f a is perpen%icular to ? c@; &e can ha$e ? c@ ` an%

therefore ` c.

Vector product:

The Cross roduct a × b of two vectors is another vector that is at right

angles to both!

a × b = |a| |b| sin(θ) n

• |a| is the magnitude (length) of vector a

• |b| is the magnitude (length) of vector b

• θ is the angle between a and b

• n is the unit vector at right angles to both a and b

(he cross pro%uct is anticommutati$e8 a : : a

^ a : ^ is the area of the parallelogram forme% 7 a an%

istriuti$e o$er a%%ition. (his means that a I ? F c@ a I F a I c

/or t&o parallel $ectors a I

http://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Perpendicularhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Cancellation_lawhttps://www.mathsisfun.com/algebra/vector-unit.htmlhttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Perpendicularhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Cancellation_lawhttps://www.mathsisfun.com/algebra/vector-unit.html


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44

i % i " & % & " ' % ' " 0.

#he cross ro!$ct o( a vector )ith itsel( is the n$ll vector

*( a % + " a % c an! a 0 then: a % + / a % c " 0 an!, + the !istri+$tive la) a+ove:

a % + / c " 0 o), i( a is arallel to + / c, then even i( a 0 it is ossi+le that + / c 0

an! there(ore that + c.

SC/R TRI/* R6"UCT

!t is %e=ne% for three $ectors in that

or%er as the scalar . ? I @!t %enotes the

$olume of the parallelopipe% forme% 7

taking a; ; c as the coterminus e%ges.

i.e. magnitu%e of I . ̂ I . ^

(he $alue of scalar triple pro%uct %epen%s on the c7clic or%er of the

$ectors an% is in%epen%ent of the position of the %ot an% cross. (hese ma7

e interchange at pleasure. ,o&e$er an% antic7clic permutation of the

$ectors changes the $alue of triple pro%uct in sign ut not a magnitu%e.

Properties of Scalar (riple Pro%uct8

!f; then their scalar triple pro%uct is gi$en 7

• i.e. position of %ot an% cross can e interchange% &ithout altering

the pro%uct.


45/122

45

•

; ; in that or%er form a right han%e% s7stem if . .

• i.e. scalar triple pro%uct is if an7 t&o $ectors are e"ual.

• i.e. scalar triple pro%uct is if an7 t&o $ectors are parallel.

• Sho& that

• !f three $ectors are coplanar then .

Chapter 11. (hree imensional Geometr7

!n space; an7 point can e e:presse% as P?:; 7; >@. (he e"uations of a line

passing through a gi$en point an% parallel to a gi$en %irection are gi$en

7 . (he e"uations of a line &hich

passes through t&o gi$en points are .

(&o lines in space ma7 e parallel lines or intersecting lines or ske& lines.

(&o parallel lines or intersecting lines are in the same plane. !f t&o lines

%o not intersect then there is no angle et&een them.

)ngle et&een t&o lines

/in%ing the angle et&een t&o lines in 2 is eas7; Bust =n% the angle of

each line &ith the :a:is from the slope of the line an% take the


46/122

46%ierence. !n 3 it is not so o$ious; ut it can e sho&n ?using the

Cosine *ule@ that the angle et&een t&o $ectors a an% is gi$en 78

Cos

Rnfortunatel7 this gi$es poor accurac7 for angles close to >eroA for

instance an angle of 1.- ra%ians e$aluates &ith an error e:cee%ing

1j; an% 1.-# ra%ians e$aluates as >ero. ) similar formula using the

sine of the angle Sin has similar prolems &ith angles close to

' %egrees.


47/122

4

(hus the $ector representing the shortest %istance et&een )< an% C

&ill e in the same %irection as ?)< C@; &hich can e &ritten as a

constant times ?)< C@. e can then e"uate this $ector to a general

$ector et&een t&o points on )< an% C; &hich can e otaine% from the

$ector e"uations of the t&o lines. Sol$ing the three e"uations otaine%

simultaneousl7; &e can =n% the constant an% the shortest %istance.

Plane8

) plane is a 9at ?not cur$e%@ t&o %imensional space eme%%e% in a higher

numer of %imensions.

!n t&o %imensional space there is onl7 one plane that can e containe%

&ithin it an% that is the &hole 2 space.


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4# (hree %imensional space is a special case ecause there is a %ualit7

et&een points ?&hich can e represente% 7 3 $ectors@ an% planes ?also

represente% 7 3 $ectors@. e can $isualise the $ector representing a

plane as a normal to the plane.

(he angle et&een t&o planes is %e=ne% as the angle et&een their

normals. !t is gi$en 7

Coplanarit7 of four points8

Coplanar points are three or more points &hich lie in the same plane&here a plane is a 9at surface &hich e:ten%s &ithout en% in all %irections.

!tDs usuall7 sho&n in math te:tooks as a 4si%e% =gure. Jou can see that

points );


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4'coplanar. (he7 form a triangle; an% 7ou can $isuali>e that. !f 7ou &ere to

cut through the tissue o: an% pass through these points; 7ou &oul% ha$e

a piece of the tissue o: that &oul% ha$e a plane =gure; a triangle; as its

ase.

Chapter 12 0inear Programming

0inear programming is the process of taking $arious linear ine"ualities

relating to some situation; an% =n%ing the est $alue otainale un%er

those con%itions. ) t7pical e:ample &oul% e taking the limitations of

materials an% laor; an% then %etermining the est pro%uction le$els for

ma:imal pro=ts un%er those con%itions.

!n real life; linear programming is part of a $er7 important area of

mathematics calle% optimi>ation techni"ues. (his =el% of stu%7 ?or atleast the applie% results of it@ are use% e$er7 %a7 in the organi>ation an%

allocation of resources. (hese real life s7stems can ha$e %o>ens or


50/122

5hun%re%s of $ariales; or more. !n algera; though; 7ouDll onl7 &ork &ith

the simple ?an% graphale@ t&o$ariale linear case.

(he general process for sol$ing linearprogramming e:ercises is to graph

the ine"ualities ?calle% the constraints@ to form a &alle%o area on

the x,y plane ?calle% the feasiilit7 region@. (hen 7ou =gure out thecoor%inates of the corners of this feasiilit7 region ?that is; 7ou =n% the

intersection points of the $arious pairs of lines@; an% test these corner

points in the formula ?calle% the optimi>ation e"uation@ for &hich 7ouDre

tr7ing to =n% the highest or lo&est $alue.

(he ecision ariales8 (he $ariales in a linear program are a set of

"uantities that nee% to e %etermine% in or%er to sol$e the prolemA i.e.;

the prolem is sol$e% &hen the est $alues of the $ariales ha$e eeni%enti=e%. (he $ariales are sometimes calle% %ecision $ariales ecause

the prolem is to %eci%e &hat $alue each $ariale shoul% take. (7picall7;

the $ariales represent the amount of a resource to use or the le$el of

some acti$it7.

(he Becti$e /unction8 (he oBecti$e of a linear programming prolem

&ill e to ma:imi>e or to minimi>e some numerical $alue. (his $alue ma7

e the e:pecte% net present $alue of a proBect or a forest propert7A or it

ma7 e the cost of a proBectA

(he Constraints8 Constraints %e=ne the possile $alues that the $ariales

of a linear programming prolem ma7 take. (he7 t7picall7 represent

resource constraints; or the minimum or ma:imum le$el of some acti$it7

or con%ition.

0inear Programming Prolem /ormulation8 e are not going to e

concerne% &ith the "uestion of ho& 0P prolems are sol$e%. !nstea%; &e

&ill focus on prolem formulation translating real&orl% prolems into

the mathematical e"uations of a linear program an% interpreting the

solutions to linear programs. e &ill let the computer sol$e the prolems

for us. (his section intro%uces 7ou to the process of formulating linear

programs. (he asic steps in formulation are8 1. !%entif7 the %ecision

$arialesA 2. /ormulate the oBecti$e functionA an% 3. !%entif7 an%

formulate the constraints. 4. ) tri$ial step; ut one 7ou shoul% not forget;


51/122

51is &riting out the nonnegati$it7 constraints. (he onl7 &a7 to learn ho& to

formulate linear programming prolems is to %o it.

(he corner points are the $ertices of the feasile region. nce 7ou ha$e

the graph of the s7stem of linear ine"ualities; then 7ou can look at the

graph an% easil7 tell &here the corner points are. Jou ma7 nee% to sol$e

a s7stem of linear e"uations to =n% some of the coor%inates of

the points in the mi%%le.

Chapter 13 Probability


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52e=nition8 (&o e$ents are %epen%ent if the outcome or occurrence of the

=rst aects the outcome or occurrence of the secon% so that the

proailit7 is change%.

(he con%itional proailit7 of an e$ent < in relationship to an e$ent ) is

the proailit7 that e$ent < occurs gi$en that e$ent ) has alrea%7

occurre%. (he notation for con%itional proailit7 is P?


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53

Summar78 (he proailit7 of t&o or more in%epen%ent e$ents

occurring in se"uence can e foun% 7 computing the

proailit7 of each e$ent separatel7; an% then

multipl7ing the results together.

Summar78 (he proailit7 of t&o or more in%epen%ent e$entsoccurring in se"uence can e foun% 7 computing the

proailit7 of each e$ent separatel7; an% then

multipl7ing the results together.

Summar78 (he proailit7 of t&o or more in%epen%ent e$ents occurring

in se"uence can e foun% 7 computing the proailit7 of each e$entseparatel7; an% then multipl7ing the results together.


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54@(en to ppl# >a#esE T(eorem

Part of the challenge in appl7ing ing the

t7pes of prolems that &arrant its use. Jou shoul% consi%er ero an% plus in=nit7. ithin that range; though; the

numer of hea%s can e onl7 certain $alues. /or e:ample; the

numer of hea%s can onl7 e a &hole numer; not a fraction.

(herefore; the numer of hea%s is a %iscrete $ariale. )n% ecause

the numer of hea%s results from a ran%om process 9ipping a coin

it is a %iscrete ran%om $ariale.

Continuous. Continuous $ariales; in contrast; can take on an7

$alue &ithin a range of $alues. /or e:ample; suppose &e ran%oml7

select an in%i$i%ual from a population. (hen; &e measure the age of

http://stattrek.com/Help/Glossary.aspx?Target=Sample_spacehttp://stattrek.com/Help/Glossary.aspx?Target=Sethttp://stattrek.com/Help/Glossary.aspx?Target=Mutually_exclusivehttp://stattrek.com/Help/Glossary.aspx?Target=Mutually_exclusivehttp://stattrek.com/Help/Glossary.aspx?Target=Eventhttp://stattrek.com/Help/Glossary.aspx?Target=Variablehttp://stattrek.com/Help/Glossary.aspx?Target=Discrete%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Continuous%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Sample_spacehttp://stattrek.com/Help/Glossary.aspx?Target=Sethttp://stattrek.com/Help/Glossary.aspx?Target=Mutually_exclusivehttp://stattrek.com/Help/Glossary.aspx?Target=Mutually_exclusivehttp://stattrek.com/Help/Glossary.aspx?Target=Eventhttp://stattrek.com/Help/Glossary.aspx?Target=Variablehttp://stattrek.com/Help/Glossary.aspx?Target=Discrete%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Continuous%2520variable


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55that person. !n theor7; hisEher age can take on an7 $alue et&een

>ero an% plus in=nit7; so age is a continuous $ariale. !n this

e:ample; the age of the person selecte% is %etermine% 7 a chance

e$entA so; in this e:ample; age is a continuous ran%om $ariale.

ro!a!ilit# "istri!ution

) pro!a!ilit# distri!ution is a tale or an e"uation that links each

possile $alue that a ran%om $ariale can assume &ith its proailit7 of

occurrence.

"iscrete ro!a!ilit# "istri!utions

(he proailit7 %istriution of a %iscrete ran%om $ariale can al&a7s e

represente% 7 a tale. /or e:ample; suppose 7ou 9ip a coin t&o times.

(his simple e:ercise can ha$e four possile outcomes8 ,,; ,(; (,; an% ((.

No&; let the $ariale represent the numer of hea%s that result from the

coin 9ips. (he $ariale can take on the $alues ; 1; or 2A an% is a

%iscrete ran%om $ariale.

(he tale elo& sho&s the proailities associate% &ith each possile

$alue of . (he proailit7 of getting hea%s is .25A 1 hea%; .5A an% 2

hea%s; .25. (hus; the tale is an e:ample of a proailit7 %istriution for

a %iscrete ran%om $ariale.

9um!er o%

(eads 7

ro!a!ilit

# 73

.25

1 .5

2 .25

9ote: Gi$en a proailit7 %istriution; 7ou can =n% cumulati$e

proailities. /or e:ample; the proailit7 of getting 1 or fe&er hea%s

X P? W 1@ Z is P? @ F P? 1@; &hich is e"ual to .25 F .5 or .5.

http://stattrek.com/Help/Glossary.aspx?Target=Random%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Discrete%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Cumulative%2520probabilityhttp://stattrek.com/Help/Glossary.aspx?Target=Cumulative%2520probabilityhttp://stattrek.com/Help/Glossary.aspx?Target=Random%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Discrete%2520variablehttp://stattrek.com/Help/Glossary.aspx?Target=Cumulative%2520probabilityhttp://stattrek.com/Help/Glossary.aspx?Target=Cumulative%2520probability


56/122

56Continuous ro!a!ilit# "istri!utions

(he proailit7 %istriution of a continuous ran%om $ariale is represente%

7 an e"uation; calle% the pro!a!ilit# densit# %unction ?p%f@. )ll

proailit7 %ensit7 functions satisf7 the follo&ing con%itions8

(he ran%om $ariale J is a function of A that is; 7 f?:@.

(he $alue of 7 is greater than or e"ual to >ero for all $alues of :.

(he total area un%er the cur$e of the function is e"ual to one.

Mean an% ariance

ust like $ariales from a %ata set; ran%om $ariales are %escrie% 7

measures of central ten%enc7 ?like the mean@ an% measures of $ariailit7

?like $ariance@. (his lesson sho&s ho& to compute these measures

for %iscrete ran%om $ariales.

Mean o% a "iscrete Random ,aria!le

(he mean of the %iscrete ran%om $ariale is also calle% the e7pected

value of . Notationall7; the e:pecte% $alue of is %enote% 7 -?@. Rse

the follo&ing formula to compute the mean of a %iscrete ran%om $ariale.

-?@ q: X :i P?:i@ Z

&here :i is the $alue of the ran%om $ariale for outcome i; q: is the mean

of ran%om $ariale ; an% P?:i@ is the proailit7 that the ran%om $ariale

&ill e outcome i.

,aria!ilit# o% a "iscrete Random ,aria!le

(he e"uation for computing the $ariance of a %iscrete ran%om $ariale issho&n elo&.

2 X :i -?:@ Z2 P?:i@

&here :i is the $alue of the ran%om $ariale for outcome i; P?: i@ is the

proailit7 that the ran%om $ariale &ill e outcome i; -?:@ is the

e:pecte% $alue of the %iscrete ran%om $ariale :.


57/122

5 (o un%erstan% inomial %istriutions an% inomial proailit7; it helps to

un%erstan% inomial e:periments an% some associate% notationA so &e

co$er those topics =rst.

>inomial *7periment

) !inomial e7periment is a e:periment that has the follo&ing

properties8

(he e:periment consists of n repeate% trials.

-ach trial can result in Bust t&o possile outcomes. e call one of

these outcomes a success an% the other; a failure.

(he proailit7 of success; %enote% 7 6; is the same on e$er7 trial.

(he trials are in%epen%ent that is; the outcome on one trial %oes not

aect the outcome on other trials.

Consi%er the follo&ing statistical e:periment. Jou 9ip a coin 2 times an%

count the numer of times the coin lan%s on hea%s. (his is a inomial

e:periment ecause8

(he e:periment consists of repeate% trials. e 9ip a coin 2 times.

-ach trial can result in Bust t&o possile outcomes hea%s or tails.

(he proailit7 of success is constant .5 on e$er7 trial.

(he trials are in%epen%entA that is; getting hea%s on one trial %oes

not aect &hether &e get hea%s on other trials. >inomial

"istri!ution

) !inomial random varia!le is the numer of successes x in n repeate%

trials of a inomial e:periment. (he proailit7 %istriution of a inomialran%om $ariale is calle% a !inomial distri!ution.

(he inomial distriution %escries the eha$iour of a count

$ariale 7 if the follo&ing con%itions appl78

!: (he number of observations n is xed.

#: 8ach observation is independent.

$: 8ach observation represents one of t9o outcomes %:success: or

:failure:&.

%: (he probability of :success: p is the same for each outcome.


58/122

5#Mean and ,ariance o% t(e >inomial "istri!ution

(he inomial %istriution for a ran%om $ariale 7 &ith

parameters n an% p represents the sum of n in%epen%ent

$ariales ; &hich ma7 assume the $alues or 1. !f the proailit7 that

each ; $ariale assumes the $alue 1 is e"ual to p; then the mean of each$ariale is e"ual to !


59/122

5'

5 Continuit7 an% ierentiailit7 12?3@ 12?3@

6 )pplication of eri$ati$es 6?1@ 6?1@

!ntegrals 12?3@ 12?3@

# )pplication of !ntegrals 6?1@ 6?1@

' ierential e"uations 2?2@ 6?1@ #?3@

1 ectors 2?2@ 2?2@

11 3%imensional Geometr7 1?1@ #?2@ 6?1@ 15?4@

12 0inear Programming 6?1@ 6?1@

13 Proailit7 4?1@ 6?1@ 1?2@

(()0 6?6@ 52?13@ 42?@ 1?26@

1. !f f?:@ : F an% g?:@ : \ then =n% ?f g@ ?@.

2. !f the function f is an in$ertile function =n% the in$erse of f?:@ .

3. !f the function f is an in$ertile function %e=ne% as f?:@ then &rite

f 1?:@.

4. Sho& that the relation * in the set of real numers %e=ne% as * ?a; @8

a is neither re9e:i$e; s7mmetric nor transiti$e.


60/122

65. 0et ( e the set of all triangles in a plane &ith * as relation in ( gi$en 7 *

?(1; (2@8 (1 (2. Sho& that * is an e"ui$alence relation.

6. 0et [ e the set of all integers an% * e the relation on [ gi$en 7 * ?a;

@8 a; [; a \ is %i$isile 7 5. Sho& that * is an e"ui$alence relation.

. !f f8 * * %e=ne% 7 f?:@ ?3 \ :3@ then =n% ?f f@ ?:@.

#. Sho& that the relation * in the set ) 1; 2; 3; 4; 5 gi$en 7 * ?a; @8

is e$en is an e"ui$alence relation.

'. Sho& that the relation S in the set ) : 8 : [; gi$en 7 S

?a;@8 a; [; is %i$isile 7 4 is an e"ui$alence relation.

1.!s the inar7 operation %e=ne% on N gi$en 7 for all a; N;

commutati$e ii@ !s associati$e

11.!f the inar7 operation on [ is %e=ne% 7 then =n% .

12.0et e the inar7 operation on N gi$en 7 . /in%

.

13.0et f8N N %e=ne% 7 f?n@ for all n N. /in%

&hether the function is iBecti$e.

14.0et e the inar7 operation on ] gi$en 7 2a F a. /in% 3 4.

15.hat is the range of f?:@

16.Consi%er f8 * %e=ne% 7 f?:@ ':2 F 6: \ 5. Sho& that f is

in$ertile &ith .


61/122

61

1.0et ) an% e a inar7 operation on ) %e=ne% 7 ?a; @ ?c; %@

?aFc; F%@. Sho& that is commutati$e an% associati$e. )lso =n% the

i%entit7 element for on ) if an7.

1#.!f f8 * * an% g8 * * are gi$en 7 f?:@ sin : an% g?:@ 5 :2 =n% ? @

?:@.

1'.!f f?:@ 2:3 an% g?:@ =n% ? @ ?:@.

1. -$aluate

2. -$aluate

3. -$aluate


62/122

62

4. -$aluate

5. -$aluate

6. -$aluate

. -$aluate

#. -$aluate

'. -$aluate

1.-$aluate

11.hat is the %omain of sin1 :

12.Pro$e that tan11 F tan12 F tan13 .

13.Pro$e that tan1 F tan1 F tan1 Ftan1 .

14.Pro$e that tan1 F tan1 .

15.Pro$e that .

16.Pro$e that .

1.Pro$e that tan1:Ftan1 .

1#.Pro$e that tan1 .


63/122

63

1'.Pro$e that .

2.Pro$e that .

21.Sol$e tan12: F tan13: .

22.Sol$e tan1?:F1@ F tan1?:1@ tan1 .

23.Sol$e .

24.Sol$e .

25.Sol$e 2tan1?Cos :@ tan1?2 Cosec:@

26.Sol$e

2.Pro$e that .

1. /in% : an% 7 if

2. /in% : an% 7 if


64/122

64

3. /in% : if

4. /in% 7 if

5. /in% 7 if

6. /in% 7 if

. Rsing elementar7 transformations =n% the in$erse of the matri:

.

#. Rsing elementar7 transformations =n% the in$erse of the matri:

.

'. Rsing elementar7 transformations =n% the in$erse of the matri: .

1.Rsing elementar7 transformations =n% the in$erse of the matri:.

11.Rsing elementar7 transformations =n% the in$erse of the matri: .

12.Rsing elementar7 transformations =n% the in$erse of the matri: .

13.Sol$e 2: \ 7 F > 3; : F 27 \ > 1 an% : \ 7 F 2> 1.

14.Sol$e 3: \ 27 F3> #; 2: F 7 \ > 1 an% 4: \37F2> 4.


65/122

6515.Sol$e : F 7 F > 4; 2: F 7 \ 3> ' an% 2: \ 7 F > 1.

16.Sol$e 2: \ 37 F 5> 11; 3: F 27\4> 5 an% :F72>3.

1.Sol$e : F 7 F > 6; : F 2> an% 3: F 7 F > 12.

1#.Sol$e :F27 3>4; 2:F 37 F2> 2 an% 3: \ 37 4> 11.

1'.!f ) =n% )1. Rsing )1 sol$e 2: \ 37 F 5> 16; 3:

F 27 \ 4> 4 an% : F 7 2> 3.

2.!f ) =n% )1. Rsing )1 sol$e 2: F 7 F 3> '; : F 37

\ > 2an% 2: F 7 F > .

"ifferentiation of composite functions:"ifferentiation b# chain rule:


66/122

66$ind the derivative of the following functions wrt %:

&! (%' )*! '!

+&')

x x

+ ÷ ÷

!

&,&

x x

+ ÷

+!( )

-) x x −

*! '& x − -!

') + x −

.!

&

&

x

x

+− /!

' &

' &

x

x

−

+

0!' ) * x x − &,! (%'1 &) (% &)+!

&&!

)+

'&

x

x − &'!

& &

& &

x x

x x

+ + −

+ − −

"ifferentiation of trigonometric functions$ind the derivative of the following functions wrt %:&! 2in *% '! Cos ('% )

! 2in (*% 1 +) +! Tan&,+%

*!

& sin '

& '

x

sin x

+− -!

& s

&

co x

cosx

−+

.!

& tan )

& tan )

x

x

−+

/! 3f # = % tan % then prove that % sin'%

dy

dx = #' 1 # sin'%!

0! 3f # = a sin % 1 b cos % then prove that #' 1

'dy

dx

÷ = a' 1 b'!

&,! 3f # =

& sin'

& sin'

x

x

−+ then prove that

dy

dx 1 sec' ( +

π

4%) = ,!

"ifferentiation of inverse trigonometric functions$ind the derivative of the following functions wrt %:&! 2in&'% '! Tan4&*%

! 2in4& '& x − +!

'&sin'&

x x

− ÷ ÷ +

*!( )& )sin ) + x x − −

-!

&&sec'' & x

− ÷ ÷ −

.!

&

&s&

x x co

x x

− ÷− ÷ ÷+ ÷ /!

'&&s'

x co

+−

0!

( )&tan sec tan x x − +&,!

cos&tan& sin

x

x

− ÷+

&&!

& cos&tan& cos

x

x

−−+ &'!

& sin&tan& sin

x

x

+−−


67/122

6

&!

&tan&

a x

ax

−− ÷ ÷+ &+!

'&&tan'&

x

x

+− ÷ ÷−

&*!

&&t&

x co

x

+ − ÷− &-!

' '& &&tan a x

ax

+ −− ÷

÷ ÷

&.!

&tan'& &

x

x

− ÷ ÷

+ − &/!

& 'tan & x x − + − ÷

&0!

& &&tan& &

x x

x x

+ + −− ÷ ÷+ − − ',!

& sin & sin&t& sin & sin

x x co

x x

+ + −− ÷ ÷+ − −

'&!

& 'sin & & x x x x − − − − ÷ ''!

( ) ( )& ) & )sin ) + cos + ) x x x x − −− + −

'! 3f # =

&sin

'&

x

x

−

− then prove that( )'& &dy x xy dx − − = !

"ifferentiation of log and e%ponential functions$ind the derivative of the following wrt %:

&!( )' x

e '!' ) x e +

!'& x e + +!

' x e

*!sin x e -!

sin x e

.!tan x x

e /!

x x e e

x x e e

−−−

+0! cos

x e x &,!

( )log sec x

&&!( )'log ) x −

&'!

&log x

x

+ ÷

&!( )log log x

&+!

'& &log

'& &

x

x

+ − ÷

÷ ÷+ +

&*!

& cos 'log

& cos '

x

x

− ÷+ &-!

& sinlog

& sin

x

x

+−

&.! ( )log x a x b− + − &/! ( )log cos cotecx x −

&0!( )log s tanecx x +

',!

log s tan' '

x x ec

+ ÷ ÷ ÷

'&!

'log & x x

+ + ÷ ''!

'log & x x

− − ÷

'!( )log sinax e bx

"ifferentiation of 3mplicit functions

$ind

dy

dx from the following:


68/122

6#

&!' ' '* x y + = '!

' '&

' '

x y

a b+ =

!) ) ) x y xy + = +!

( )) ) ' ') x y x y xy + = +

*!+ + + x y xy + = -!

) ) x y y x − =

.! x y a+ =

/! 3f

' '& log &y x x x

+ = + − ÷ then prove that

( )'& & ,!dy x xy dx + + + =

0! 3f& & , x y y x + + + =

then prove that( )

&

'&

dy

dx x

−=

+!

&,! 3f' '& & & x y y x − + − =

then prove that

'&,!

'&

dy y

dx x

−+ =

−

&&! 3f( )' '& & x y a x y − + − = −

then prove that

'& ,!

'&dy y dx x

−− =−

&'! 3f % 1 # = sin(% 1 #) then prove that

dy

dx = 4 &!

&! 3f # log % = % # then prove that( )

log!

'& log

dy x

dx x

=+

&+!!!!!!! x x x x + + + +

&*!sin sin sin sin !!!!!! x x x x + + + +

&-!log log log log !!!!!! x x x x + + + +

&.!

&

&

&

!!!!!

x

x

x x

++

++

5ogarithmic differentiation"ifferentiate the following wrt %:

&! x x '!

log x x

! sin x x +!

( )sin x

x

*!( )

cossin

x x

-!( )

sinn

x ta x

.!( )log

x x

/!( )sin x x

0!

'' )cot' '

x x x x x

−+

+ + &,!

' &sin cos' &

x x x x x

−− ++

&&!( ) ( )' ' x x x x +

&'!( ) ( )sin sin x x x x +

&!( ) ( )

s sinsin s

co x x x co x +

&+!( ) ( )

tan sinsin tan

x x x x +


69/122

6'

&*!( ) ( )

tan sinsin tan

x x x x +

&-! 3f( ) ( )

y x x y c + =

then find

dy

dx !

&.! 3f( ) ( ) ,

y x x y − =

then find

dy

dx !

&/! 3f( )

y x y x e −=then prove that

( )

log

'& log

dy x

dx x = +!

&0! 3f

x y y x x e

+

= then find

dy

dx !

',! 3fy x y x + =

then find

dy

dx !

'&!

!!! x x x

''!

!!!sinsinsin x x x

"ifferentiation of parametric functions

$ind

dy

dx from the following:&! %= at'6 # = 'at

'!

'&

'&

t x

t

−=

+ 6

'

'&

t y

t =

+ !! % = a cos Ө6 #= b sin Ө+! % = a (& cos Ө)6 # = a (Ө 1 sin Ө)

*! % = a (cos Ө 1 Ө sin Ө)6 # = a (sin Ө Ө cos Ө)-! % = 'cos Ө cos 'Ө6 # = ' sin Ө sin 'Ө!.!

7igher "erivativesThe derivative of the derivative of a function is called as the second derivative!

'88

' '

d y d dy y or y or

dx dx dx

= ÷

) '888

) ) '

d y d d y y or y or

dx dx dx

÷= ÷

) '

) '

d y d d y

dx dx dx

÷= ÷

&! $ind the second derivative of %' 1 log % wrt %!

'! 3f # = e% (sin % 1 cos %) then prove that #' '#& 1 '# = ,!

! 3f # = sec % 1 tan % then prove that( )

' cos

' '& sin

d y x

dx x

=−

+! 3f # = cosec % 1 cot % then prove that( )

' sin

' '& s

d y x

dx co x

=−

*! 3f # = a sin '% 1 b cos '% then prove that+ ,

'y y + =

-! 3f # =&nta x − then prove that

( )'& ' ,&' x y xy + + =


70/122

.! 3f # =( )

'&nta x −

then prove that( ) ( )

'' '& ' & ' ,

&' x y x x y + + + − =

/! 3f # =( )

'&nsi x −

then prove that( )'& ' ,&' x y xy − − − =

0! 3f # =&sin x e

−then prove that

( )'& ,&' x y xy y − − − =

&,!3f # =&nta x e

−then prove that

( ) ( )'' '& ' & ,

&' x y x x y y + + + − =

&&! 3f # = sin (log %) 1 cos (log %) then prove that

' ,&'

x y xy y + + =

&'!3f % = a ( sin )6 # = a (& cos ) then find 'y

at = '

π

!


71/122


72/122

213./in% the points on the cur$e 7 :3 at &hich the slope of the tangent is

e"ual to the 7 coor%inate of the point.

14./in% the e"uations of the normal to the cur$e 7 :3F2:F6 &hich are

parallel to the line :F147F4 .

15./in% the e"uation of the tangent to the cur$e 7 at the point

&here it cuts the : \ a:is.

16./in% the e"uations of the tangent an% the normal to the cur$e : 1 \ cos

; 7 at

1. -$aluate

2

31

xdx

x+∫ 1 a. -$aluate

1

2

01

dx

x+∫

2. -$aluate2

0

sin

1 cos

x xdx

x

π

+∫ 2a. -$a.

a

a

a xdx

a x−

−+∫

3. -$a. 0

tan

cos

x xdx

secx ecx

π

∫ 3a. -$a.

2

0

sin

1 cos

x xdx

x

π

+∫

4. -$a. 0

tan

sec tan

x xdx

x x

π

+∫ 4a. -$a.

2

cos6

3 sin 6

x xdx

x x

++∫

5. -$a.

2

0tan cot x x dx

π

+∫ 5a.-$a.

2

0logsin x dx

π

∫

6. -$a.

2sec

3 tan

xdx

x+∫ 6a. -$a. 25 4 x

x x

e dx

e e− −∫

. !f

( )1

2

3 2 x x > dx + +∫ then =n% the $alue of k.

#. -$aluate

( )

( )

:

3

4 e

2

x dx

x

−

−∫ # a .-$aluate

cos

cos cos

x

x x

edx

e e

π

−+∫

'. -$aluate

( ) ( )( )2

2log sin log sin2 x x dx

π

−∫ 'a.-$aluate

( )2sec x dx −∫


73/122

3

1.-$aluate

( )2

1 log x dx

x

+∫

1a.-$aluate

1

2

2 1

dx

x −∫

11.-$aluate

cos x dx

x ∫

11a.-$aluate25 4 2

dx

x x − −∫

12.-$aluate1sin x xdx −∫ 12a.-$aluate

sin x dx x ∫

13.-$aluate

2sec x dx

x ∫

13a.-$aluate ( ) ( )

cos

2 sin 3 4sin

x dx

x x + +∫

14.-$aluate2 1s x co xdx −∫ 14a.-$aluate ( ) ( )

4

21 1

x dx

x x − +∫

15.-$aluate( )

4

1

1 2 4 x x x dx − + − + −∫ 15a.-$aluate log

dx

x x +∫

16.-$aluate2 2 2 2

cos sin

x dx

a x b x

π

+∫ 16a.-$aluate

2 2

2 2

x x

x x

e edx

e e

−

−

−+∫

1.-$aluate

2 2

2 2

x x

x x

e edx

e e

−

−

−+∫ 1a.-$aluate

log x dx

x ∫

1#.-$aluate

( )sin4 4

1 cos4

x e x dx

x

−− 1#a.-$aluate ( )

21

1 2

x dx

x x

−−∫

1'.-$aluate

3

6

sin cos

sin2

x x dx

x

π

π

+

∫ 1'a.-$aluate x

dx x sinx +∫

2.-$aluate( )

3

2

1

3 2 x x dx +∫ as the limit of sum.

21.-$aluate( )

42

1

x x dx −∫ as the limit of sum.

22.-$aluate

2

5

2

sin xdx

π

π −

∫

22a.-$aluate

( ) ( )

2

2 3

x dx

x x

+

− −∫

23.-$aluate

2 2

21

5

4 3

x dx

x x + +∫ 23a.-$aluate

1

2

1

1 dx

x +∫

24.-$aluate

3

2

cos x x dx π ∫ 24a.-$aluate

3

2

1

sin x x dx π −∫

25.-$aluate

( )1

2

1 n

x x dx −∫ 25a.-$aluate

( ) ( )

2 3

1 2

x x dx

x x

−− −∫


74/122

4

1. /in% the area of the region inclu%e% et&een 72 4a: an% :2 4a7; a .

2. /in% the area l7ing ao$e the : a:is an% inclu%e% et&een :2 F 72 #:

an% 72 4:.

3. /in% the area of the region enclose% 7 the cur$es 7 :2 an% 7 :.

4. /in% the area of the region enclose% 7 the cur$es 72 : an% 7 : F 2.

5. /in% the area of the region oun%e% 7 the cur$es 7 ; : 3; : 3

an% 7 .

6. /in% the area of the smaller region foun%e% 7 the ellipse an%

the line .

. /in% the area of the region in the =rst "ua%rant enclose% 7 the :a:is; the

line : 7 an% the circle :2 F72 4.

#. /in% the area oun%e% 7 the lines 7 4: F 5; 7 5 \ : an% 47 : F 5.

'. /in% the area of the region oun%e% 7 72 4: an% 4:2 F 472 '.

1./in% the area of the region enclose% et&een :2 F 72 4 an% ?: \ 2@2 F72

4.

11./in% the area of that part of the circle :2 F72 16 &hich is e:terior to the

paraola 72 6:.

12./in% the area oun%e% 7 the lines : F 27 2; 7 \ : 1 an% 2: F 7 .

13./in% the area oun%e% 7 the lines 2:F 7 4; 3: 27 6 an% : 37 F 5

.

14./in% the area oun%e% 7 the lines 4: 7 F 5 ; : F 7 \ 5 an% : 47

F 5 .

15./in% the area oun%e% 7 the cur$es :2 F 72 1 an% ?: \ 1@2 F72 1.


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516.Pro$e that the cur$es 72 4: an% :2 47 %i$i%e the area of the s"uare

oun%e% 7 : ; : 4; 7 4 an% 7 into three e"ual parts.

1./in% the area of the region inclu%e% et&een the paraola 72 : an% :F7

2.

1#./in% the area of the region inclu%e% et&een the paraola :2 47 an% :

47 2.

1'./in% the area of the region inclu%e% et&een the paraola 47 3:2 an% 3:

27F12 .

2./in% the area of the region oun%e% 7 the cur$es 72 4a: an% :2

4a7.

21./in% the area of the region .


23./in% the area of the region



26./in% the area of the triangle )


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6

1. hat is the %egree of the - .

2. /orm the - representing the paraolas ha$ing the $erte: at the origin

an% the a:is along the positi$e %irection of :a:is.

3. /orm the - representing the famil7 of ellipses &hose foci on : a:is an%

the centre at the origin.

4. Sol$e the follo&ing8

5. Sol$e 8 gi$en that 71 &hen : 1.

6. Sol$e8 gi$en that 7 1 &hen : 1.

. Sol$e

#. Sol$e

'. Sol$e

1.Sol$e

11.Sol$e


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12.Sol$e ; 7 &hen : 1.

13.Sol$e

14.Sol$e

15.Sol$e

16.Sol$e ; 7 &hen :

1.Sol$e

1#.Sol$e

1'.Sol$e . J 1 &hen : .

2.Sol$e

21.Sol$e

22.Sol$e

23.Sol$e

24.Sol$e

25.Sol$e

26.Sol$e gi$en that : 1 an% 7 1.


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#

2./or the - =n% the solution cur$e passing through the

point ?1; 1@.

1. /in% a unit $ector in the %irection of 3 2 6a i j > = − +

.

2. /in% a unit $ector in the %irection 2a i j= −r

of &hose magnitu%e is .

3. /in% the angle et&een a i j > = − +an%

b i j > = + − .

4. /or &hat $alue of v are the $ectors 2a i j > λ = + +

an% 2 3b i j > = − +

perpen%icular to each other.

5. !f a i j > = + +an%

b j > = −=n% a $ector c such that

a b c× =an%

3a b• =.

6. !fa b c+ + =

r r

an%a

3;b

5 b c

sho& that the angle et&eenba b

r

is

6o

.

. !fa

3; b

2 an% the angle et&een them is 6othen =n%

a b•r

.

#. !f i j > + + ;

2 5i j+ ; 3 2 3i j > + − b

6i j > − − are the position $ectors of ); = + + an%

3b i pj > = + + are parallel$ectors.

1.!f a b c d× = × an% a c b d× = × then sho& that a d b c− −P &here ba d b c≠ ≠ .

11.rite a unit $ector in the %irection of

2 2b i j > = + +.

12.rite a unit $ector in the %irection of 2 3 6b i j > = − + .

13./in% the angle et&een ba b&ith magnitu%es 1 b 2 an%3a b× =

.

14. rite the $alue of p for &hich( ) ( ) 2 6 14 i j > i j > λ + + × − +

.

15.( ) ( ) 2 6 2 3i j > i j p> + + × + +

.

16.!f pis a unit $ector an%( ) ( ) x p x p− • +

# then =n% x

.

1.(he scalar pro%uct of

i j > + + &ith the unit $ector along the sum of 2 4 5i j > + − an%

2 3i j > λ + + is e"ual to one. /in% the $alue of λ .


79/122

'

1#.!f3a =

;2b =

an% a b• 3 then =n% the angle et&een ba b.

1'.!f ; ba b care three $ectors such that a b a c• = • an% a b a c× = × ;a ≠ then

pro$e that b c=r

.

2.rite a $ector of magnitu%e 15 units in the %irection of 2 2i j > − + .

21./in% the position $ector of a point * &hich %i$i%es the line hoining t&opoints P an% ] &hose position $ectors 2a b+ an% 3a b− e:ternall7 in theratio 182. )lso sho& that P is the mi%point of *].

22.rite a $ector of magnitu%e ' units in the %irection of 2 2i j > − + + .

23./in% λ if( ) ( ) 2 6 14 i j > i j > λ + + × − +

.

24.!f a i j > = + + ;

4 2 3b i j > = − + an% 2c i j > = − + =n% a $ector of magnitu%e 6

units &hich is parallel to 2 3a b c− + .

25.0et

4 2a i j > = + +;

3 2 b i j > = − +an%

2 4c i j > = − +. /in% a $ector d &hich is

perpen%icular to oth ba ban% c d• 1#.

26.!f ba bare t&o $ectors such thata b a b• = ×

then &hat is the angle

et&een ba b.

2.ectors ba bare such that3a =

;

2

3b =r

an%( )a b×

r

is a unit $ector. rite

the angle et&een ba br

.

2#./in% x

if for a unit $ector a ;( ) ( ) x a x a− • +

.

2'./or &hat $alue of p;( ) i j > p+ +

is a unit $ector

3.0et 4 2a i j > = + + ;

3 2 b i j > = − + an% 2 4c i j > = − + . /in% a $ector d &hich is

perpen%icular to oth ba ban% c d• 1.

31. (he scalar pro%uct of 2 4i j > + + &ith a unit $ector along the sum of the

$ectors 2 3i j > + + an%

4 5i j > λ + − is e"ual to one. /in% the $alue of λ .


80/122

#

1. /in% the shortest %istance et&een the lines an%

.

2. /in% the point on the line at a %istance from the

point ?1; 2;3@.

3. /in% the e"uation of the plane passing through the point ?1; 1; 2@ an%

perpen%icular to the planes 2: F 37 \ 3> 2 an% 5: \ 47 F > 6.

4. /in% the e"uation of the plane passing through the point ?1; 3; 2@ an%

perpen%icular to the planes : F 27 F 3> 5 an% 3: F 37 F > .


81/122

#15. /in% the e"uation of the plane passing through the point ?3; 4; 1@ an%

parallel to the line .

6. !f the e"uation of the line )< is =n% the %irection ratios

of a line parallel to the line )


82/122

#2

14./in% the shortest %istance et&een the lines an%

15./in% the shortest %istance et&een the lines

an%

16./in% the e"uation of the plane %etermine% 7 the points ?3;1; 2@; ?5; 2; 4@

an% ?1; 1; 6@. )lso =n% the %istance of the point ?6; 5; '@ from the

plane.

1./in% the %irection cosines of the line passing through the points ?2; 4; 5@

an% ?1; 2; 3@.

1#./in% the angle et&een the line an% the plane 1: F 27

\ 11> 3.

1'.Sho& that the lines an% are coplanar.

)lso =n% te e"uation of the plane containing the lines.

2./in% the angle et&een the line an% the plane : F 27 F

2> \ 5 .

21.hat is the cosine of the angle &hich the $ector makes &ith the

7 \ a:is

22./in% the e"uation of the line passing through the points ?; ; @ b ?3; 1;

2@ an% parallel to the line .

23.(he points ) ?4; 5; 1@;


83/122

#3

24./in% the $ector e"uations of the lines an%

an% hence =n% the %istance et&een them.

25.rite the %istance of the plane 2: \ 7 F 2> F 1 from the origin.

26./in% the points on the line at a %istance of 5 units from

the point ?1; 3; 3@.

2./in% the %istance of the point ?6; 5; '@ from the plane %etermine% 7 the

points ?3; 1; 2@; ?5; 2; 4@ an% ?1; 1; 6@.

2#./in% the coor%inates of the foot of the perpen%icular an% the

perpen%icular %istance of the point ?3; 2; 1@ from the plane 2: \ 7 F > F 1

. /in% also the image of the point in the plane.

2'./in% the e"uation of the plane passing through ?1; 1; 1@ an% containing the

line . )lso sho& that the plane contains the line

.

3.rite the Cartesian e"uation of the line .

31./in% the shortest %istance et&een the lines an%

. ,ence &rite &hether the lines intersect or not.

32./in% the coor%inates of the point &here the line through ?3; 4; 5@ b ?2; 2;

1@ crosses the plane %etermine% 7 ?1; 2; 3@; ?2; 2; 1@ b ?1; 3; 6@.

33. /in% the Cartesian an% the $ector e"uations of the plane passing through

the points ?; ; @ b ?3; 1; 2@ an% parallel to the line .


84/122

#4


85/122

#51. ) factor7 o&ner purchases t&o t7pes of machines; ) an% < for his factor7.

(he re"uirements an% the limitations for the machines are as follo&s8

Machine )rea occupie% 0aourforce

ail7 output ?inunits@

) 1 m2 12 men 6

< 2 m2 # men 4

2.

,e has ma:imum area of ' m2 a$ailale an% 2 skille% laourers &ho

can operate oth the machines. ,o& man7 machines of each t7pe shoul%

he u7 to ma:imi>e the %ail7 output

1 )n aeroplane can carr7 a ma:imum of 2 passengers. ) pro=t of *s 1

is ma%e on each e:ecuti$e class ticket an% a pro=t of *s 6 is ma%e on

each econom7 class ticket. (he airline reser$es at least 2 seats for

e:ecuti$e class. ,o&e$er; at least 4 times as man7 passengers prefer to

tra$el 7 econom7 class than 7 the e:ecuti$e class. etermine ho& man7

tickets of each t7pe must e sol% in or%er to ma:imi>e the pro=t for the

airline. hat is the ma:imum pro=t

3. ) %iet is to contain at least # units of $itamin ) an% 1 units of minerals.

(&o foo%s /1an% /2 are a$ailale. /oo% /1 costs *s 4 per unit foo% an% /2

costs *s 6 per unit. ne unit of foo% /1 contains 3 units of $itamin ) an% 4

units of minerals. ne unit of foo% /2 contains 6 units of $itamin ) an% 3

units of minerals. /ormulate this as a linear programming prolem. /in%

the minimum cost for %iet that consists of mi:ture of these t&o foo%s an%

also meets the minimal nutritional re"uirements.

4. ) %ealer &ishes to purchase a numer of fans an% se&ing machines. ,e

has onl7 *s 56 to in$est an% has a space for atmost 2 items. ) fan

costs him *s 36 an% a se&ing machine *s 24. ,is e:pectation is that he

can sell a fan at a pro=t of *s 22 an% a se&ing machine at a pro=t of *s

1#. )ssuming that he can sell all the items that he can u7 ho& shoul% he

in$est his mone7 in or%er to ma:imi>e the pro=t /ormulate this as a 0PP

an% sol$e it graphicall7.

5. ne kin% of cake re"uires 2g 9our an% 25g of fat; an% another kin% of

cake re"uires 1g of 9our an% 5g of fat. /in% the ma:imum numer of

cakes &hich can e ma%e from 5 kg of 9our an% 1 kg of fat assuming that

there is no shortage of the other ingre%ients use% in making the cakes

/ormulate this as a 0PP an% sol$e it graphicall7.


86/122

#66. ) small =rm manufactures gol% rings an% chains. (he total numer of rings

an% chains manufacture% per %a7 is atmost 24. !t takes 1 hour to make a

ring an% 3 minutes to make a chain. (he ma:imum numer of hours

a$ailale per %a7 is 16. !f the pro=t on a ring is *s 3 an% that on a chain

is *s 1' =n% the numer of rings an% chains that shoul% e manufacture%

per %a7 so as to earn the ma:imum pro=t. Make it 0PP an% sol$e it

graphicall7.

. ne kin% of cake re"uires 3g 9our an% 15g of fat; an% another kin% of

cake re"uires 15g of 9our an% 3g of fat. /in% the ma:imum numer of

cakes &hich can e ma%e from .5 kg of 9our an% 6 g of fat assuming

that there is no shortage of the other ingre%ients use% in making the

cakes /ormulate this as a 0PP an% sol$e it graphicall7.

#. ) factor7 makes t&o t7pes of items ) an% < ma%e of pl7&oo%. ne piece

of item ) re"uires 5 minutes for cutting an% 1 minutes for assemling.

ne piece of item < re"uires # minutes for cutting an% # minutes for

assemling. (here are 3 hours an% 2 minutes a$ailale for cutting an% 4

hours for assemling. (he pro=t on one piece of item ) is *s 5 an% that on

item < is *s 6. ,o& man7 pieces of each t7pe shoul% the factor7 make so

as to ma:imi>e the pro=t Make it as a 0PP an% sol$e it graphicall7.


87/122

#

1. ) pair of %ice is thro&n 4 times. !f getting a %oulet is consi%ere% as a

success; =n% the proailit7 %istriution of numer of successes.

2. ) an% < thro& a pair of %ie turn 7 turn. (he =rst to thro& ' is a&ar%e% a

pri>e. !f ) starts the game; sho& that the proailit7 of ) getting the pri>e

is ' E 1.

3. ) man is kno&n to speak 3 out of 4 times. ,e thro&s a %ie an% report it is

a 6. /in% the proailit7 that it is actuall7 a 6.

4. ) man is kno&n to speak 3 out of 5 times. ,e thro&s a %ie an% report it is

a numer greater than 4. /in% the proailit7 that it is actuall7 a numer

greater than 4.

5. )n

model paper-4.docx

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