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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$al.
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2
Ma7imum 0et f?:@ e the real function %e=ne on the inter$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 $alue of :.
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?i$@ etermine the sign of %27E%:2 at : an% procee%.
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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 $alue of f?:@ for : &ithin the strip.
(he area &ill then e something like the sum of the areas of the strips. !f
&e then let the ma:imum strip length go to >ero; &e can hope to =n% the
resulting sum of strip areas approach the true area.
http://mathworld.wolfram.com/Curve.htmlhttp://mathworld.wolfram.com/Plane.htmlhttp://mathworld.wolfram.com/Curve.htmlhttp://mathworld.wolfram.com/Plane.html
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(he stan%ar% &a7 to %o this is to let the ith strip egin at :i1 an% en% at
:iA the area of that strip is estimate% as ?: i:i1@f?:Di@ &ith :Di an7&here in
the strip.
) *iemann sum is a sum of the form Bust in%icate%8 it is a sum o$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?:@.
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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 .
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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
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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 $ectors; the %iagonal C of the parallelogram
&ill e resultant ; such that
MR0(!P0!C)(!N P-*)(!NS /* -C(*S8
1. MR0(!P0!C)(!N / ) -C(*
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43 (he %ot pro%uct is %istriuti$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.
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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$en points are .
(&o lines in space ma7 e parallel lines or intersecting lines or ske& lines.
(&o parallel lines or intersecting lines are in the same plane. !f t&o lines
%o not intersect then there is no angle et&een them.
)ngle et&een t&o lines
/in%ing the angle et&een t&o lines in 2 is eas7; Bust =n% the angle of
each line &ith the :a:is from the slope of the line an% take the
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46%ierence. !n 3 it is not so o$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.
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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
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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$ial step; ut one 7ou shoul% not forget;
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51is &riting out the nonnegati$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
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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.
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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 $ariale :.
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5 (o un%erstan% inomial %istriutions an% inomial proailit7; it helps to
un%erstan% inomial e:periments an% some associate% notationA so &e
co$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.
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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 !
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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$e.
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65. 0et ( e the set of all triangles in a plane &ith * as relation in ( gi$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$ertile &ith .
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61
1.0et ) an% e a inar7 operation on ) %e=ne% 7 ?a; @ ?c; %@
?aFc; F%@. Sho& that is commutati$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. -$aluate
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62
4. -$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$e that tan1 .
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63
1'.Pro$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
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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.
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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:
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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$ind the derivative of the following functions wrt %:&! 2in&'% '! Tan4&*%
! 2in4& '& x − +!
'&sin'&
x x
− ÷ ÷ +
*!( )& )sin ) + x x − −
-!
&&sec'' & x
− ÷ ÷ −
.!
&
&s&
x x co
x x
− ÷− ÷ ÷+ ÷ /!
'&&s'
x co
+−
0!
( )&tan sec tan x x − +&,!
cos&tan& sin
x
x
− ÷+
&&!
& cos&tan& cos
x
x
−−+ &'!
& sin&tan& sin
x
x
+−−
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6
&!
&tan&
a x
ax
−− ÷ ÷+ &+!
'&&tan'&
x
x
+− ÷ ÷−
&*!
&&t&
x co
x
+ − ÷− &-!
' '& &&tan a x
ax
+ −− ÷
÷ ÷
&.!
&tan'& &
x
x
− ÷ ÷
+ − &/!
& 'tan & x x − + − ÷
&0!
& &&tan& &
x x
x x
+ + −− ÷ ÷+ − − ',!
& sin & sin&t& sin & sin
x x co
x x
+ + −− ÷ ÷+ − −
'&!
& 'sin & & x x x x − − − − ÷ ''!
( ) ( )& ) & )sin ) + cos + ) x x x x − −− + −
'! 3f # =
&sin
'&
x
x
−
− then prove that( )'& &dy x xy dx − − = !
"ifferentiation of log and e%ponential functions$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:
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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 +
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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
÷= ÷
&! $ind the second derivative of %' 1 log % wrt %!
'! 3f # = e% (sin % 1 cos %) then prove that #' '#& 1 '# = ,!
! 3f # = sec % 1 tan % then prove that( )
' cos
' '& sin
d y x
dx x
=−
+! 3f # = cosec % 1 cot % then prove that( )
' sin
' '& s
d y x
dx co x
=−
*! 3f # = a sin '% 1 b cos '% then prove that+ ,
'y y + =
-! 3f # =&nta x − then prove that
( )'& ' ,&' x y xy + + =
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.! 3f # =( )
'&nta x −
then prove that( ) ( )
'' '& ' & ' ,
&' x y x x y + + + − =
/! 3f # =( )
'&nsi x −
then prove that( )'& ' ,&' x y xy − − − =
0! 3f # =&sin x e
−then prove that
( )'& ,&' x y xy y − − − =
&,!3f # =&nta x e
−then prove that
( ) ( )'' '& ' & ,
&' x y x x y y + + + − =
&&! 3f # = sin (log %) 1 cos (log %) then prove that
' ,&'
x y xy y + + =
&'!3f % = a ( sin )6 # = a (& cos ) then find 'y
at = '
π
!
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213./in% the points on the cur$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.-$aluate
( )2sec x dx −∫
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3
1.-$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.-$aluate
( ) ( )
2 3
1 2
x x dx
x x
−− −∫
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4
1. /in% the area of the region inclu%e% et&een 72 4a: an% :2 4a7; a .
2. /in% the area l7ing ao$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 .
22./in% the area of the region .
23./in% the area of the region
24./in% the area of the region .
25./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 .
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'
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 .
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#
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 > .
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#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 )
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#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@;
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#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 $ector e"uations of the plane passing through
the points ?; ; @ b ?3; 1; 2@ an% parallel to the line .
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#4
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#51. ) factor7 o&ner purchases t&o t7pes of machines; ) an% < for his factor7.
(he re"uirements an% the limitations for the machines are as follo&s8
Machine )rea occupie% 0aourforce
ail7 output ?inunits@
) 1 m2 12 men 6
< 2 m2 # men 4
2.
,e has ma:imum area of ' m2 a$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$e it graphicall7.
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#66. ) small =rm manufactures gol% rings an% chains. (he total numer of rings
an% chains manufacture% per %a7 is atmost 24. !t takes 1 hour to make a
ring an% 3 minutes to make a chain. (he ma:imum numer of hours
a$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.
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#
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