Mth 301

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MID Term Past Papers MTH 301 Past Papers1.Every real number corresponds to ____________ on the co-ordinate line.

Infinite number of points

Two points (one positive and one negative)

A unique point

None o these

2.There is one-to-one correspondence between the set o points on co-ordinate line and __________.

!et o real numbers

!et o integers

Set of natural numbers

!et o rational numbers

3."hich o the ollowing is associated to each point o three dimensional space#

A real number

An ordered pair

An ordered triple

A natural Number

4.All a$es are positive in __________octant.

First

!econd

%ourth

Eighth

5. The spherical co-ordinates o a point are &' '&

."hat are its cylindrical co-ordinates#

& &

' '

& cos ' & sin ' & &

& sin ' ' & cos& &

&' '&

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MID Term Past Papers MTH 301 Past Papers

6.!uppose ( ) ' & * f x y xy y where x t and y t= = + = . "hich one o the ollowing is true#

+ df

tdt

= +

*,df

t tdt

=

*- df

tdt

= +

* - *

dft t

dt= + +

7. et ( ) ( ) ( ) ( )' ' ' ' ' ' 'w f x y z and x g r s y h r s z t r s= = = = then by chain rulew

r

=

w x w y w z

x r y r z r

+ +

w x w y w z

r r r r r r

+ +

w x x w y y w z z

x r s y r s z r s

+ +

w r w r w r

r x r y r z

+ +

8. /agnitude o vector is ' magnitude o vector is &and angle between them when placed tail

to tail is +0 degrees. "hat is #

+.0

6.2

0.*

+.

9. 1s the unction ( )'x yf continuous at origin# 1 not' why#

( )'x yf is continuous at origin

( )'f is not deined

( )'f is deined but ( ) ( ) ( )' ''lim

x yf x y

does not e$ist

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( )'f is deined and ( ) ( ) ( )' ''lim

x yf x y

e$ist but these two numbers are not equal.

10. 1s the unction( )'f x y

continuous at origin# 1 not' "hy#

( )'f x y is continuous at origin

( )'f is not deined

( )'f is deined but ( ) ( ) ( )' ''lim

x yf x y

does not e$ist

( )'f is deined and ( ) ( ) ( )' ''lim

x yf x y

e$ist but these two numbers are not equal.

**. et 2 be a closed region in two dimensional space. "hat does the double integral over 2 calculates#

Area o 2

2adius o inscribed circle in 2.

3istance between two endpoints o 2.

None o these

*. "hich o the ollowing ormula can be used to ind the volume o a parallelepiped with ad4acent

edges ormed by the vectors ' and #

*&. Two suraces are said to be orthogonal at appoint o their intersection i their normals at that point

are_________.

5arallel

5erpendicular

1n opposite direction

!ame direction

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*+. 6y E$treme 7alue Theorem' i a unction ( )'x yf is continuous on a closed and bounded set 2' then

( )'x yf has both __________ on 2.

Absolute ma$imum and absolute minimum value

2elative ma$imum and relative minimum value

Absolute ma$imum and relative minimum value

2elative ma$imum and absolute minimum value

*0. et the unction ( )'x yf has continuous second-order partial derivatives ( )'xx yy xyf f and f in

some circle centered at a critical point ( ) 'x y and let ( ) ( ) ( )

' ' 'xx yy xyD f x y x y x yf f= i

( ) ' xx x yD and f then f has __________ .

2elative ma$imum at ( ) 'x y

2elative minimum at ( ) 'x y

!addle point at ( ) 'x y

No conclusion can be drawn.

*,.et the unction ( )'x yf has continuous second-order partial derivatives ( )'xx yy xyf f and f in

some circle centered at a critical point ( ) 'x y and let ( ) ( ) ( )

' ' 'xx yy xyD f x y x y x yf f= i

i D= then _________.

f has relative ma$imum at ( ) 'x y

fhas relative minimum at ( ) 'x y

fhas saddle point at ( ) 'x y


*8. * * 'R andIf R R where R R= are no over lapping regions then

( ) ( )*

' 'R R

f x y dA f x y dA+ =

( )'

R

f x y dA

( ) ( )*

' 'R R

f x y dA f x y dA

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( )'

R

f x y dV

( ) ( )

' 'R R

f x y dV f x y dA

*. ( ){ } ' 9 * + 'If R x y x and y then= ( ) &, +

R

x xy dA+ =

( )+

&

*

, +x xy dydx+

( ) +

&

*

, +x xy dydx+

( )+

&

*

, +x xy dydx+

( )+ *

&

, +x xy dydx+

*:. ( ){ } ' 9 + * 'If R x y x and y then= ( )+ y

R

xe dA =

( )* +

+ yxe dydx

( )

* +

+ yxe dxdy

( )+

*

+ yxe dxdy

( )+

*

+ yxe dydx

. ( ){ } ' 9 + : 'If R x y x and y then= ( )& +R

x x xy dA =

( ): +

& +x x xy dydx

( ): +

& +x x xy dxdy

( ):

+

& +x x xy dxdy

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( )+ :

& +x x xy dydx

*. !uppose that the surace ( )' 'f x y z has continuous partial derivatives at the point ( )' 'a b c

"rite down the equation o tangent plane at this point.

. Evaluate the ollowing double integral ( ) &

* xy x dydx .&.Evaluate the ollowing double integral ( )& &x y dxdy+

+. et ( ) ' ' zf x y z xy e= %ind the gradient o f.

0. %ind' Equation o Tangent plane to the surace ( ) ' ' :f x y z x y z= + + at the point ( )*''+ .

,. ;se the double integral in rectangular co-ordinates to compute area o the region bounded by the

curves y x and y x= = .

Suppose&

( ' ) xy

f x y x e= . !i"! one of t!e follo#in$ is "orre"t% &

& xy xyf

x e x yex

= +

& xyf x ye

x

=

+&

xy xyfx e x e

x

= +

& xyf

x ex

=

&et ' be a "lose( re$ion in t#o (imensional spa"e. !at (oes t!e (ouble inte$ral o)er '

"al"ulates%Area o 2.2adius o inscribed circle in 2.

3istance between two endpoints o 2.

None o these

!at is t!e (istan"e bet#een points *3+ 2+ 4, an( *6+ 10+ -1,%

8

,

&+

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8 &

-------------------- planes interse"t at ri$!t an$le to form t!ree (imensional spa"e.

!ree

4

8

12

!ere is one-to-one "orrespon(en"e bet#een t!e set of points on "o-or(inate line an(

------------

!et o real numbers

!et o integers

!et o natural numbers!et o rational numbers

&et t!e fun"tion( ' )f x y

!as "ontinuous se"on(-or(er partial (eri)ati)es

( )'xx yy xyf f and fin some "ir"le "entere( at a "riti"al point

( ' )x yan( let

( ' ) ( ' ) ( ' )

xx yy xyD f x y f x y f x y=

1D=

then ---------------

fhas relative ma$imum at

( ' )x y

fhas relative minimum at ( ' )x y

fhas saddle point at

( ' )x y


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&+ , ,df

t tdt

= +

, * df

t tdt

= +

&et i + j an( kbe unit )e"tors in t!e (ire"tion of /-a/is+ -a/is an( -a/is respe"ti)el.

Suppose t!at 0a i j k

= + . !at is t!e ma$nitu(e of )e"tor a

%

630

&

-

A strai$!t line is --------------- $eometri" fi$ure.

>ne-dimensional

Two-dimensionalThree-dimensional

3imensionless

&

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!at is t!e relation bet#een t!e (ire"tion of $ra(ient at an point on t!e surfa"e to t!e

tan$ent plane at t!at point %

parallelperpendicularopposite directionNo relation between them.

Suppose&

( ' ) xy

f x y x e= . !i"! one of t!e statements is "orre"t%&& xy

fx e

y

=

& xyf x ey

=

+ xyf x e

y

=

& xyf x ye

y

=

#o surfa"es are sai( to interse"t ort!o$onall if t!eir normals at e)er point "ommon to

t!em are ----------perpendicularparallelin opposite direction

&et t!e fun"tion( ' )f x y

!as "ontinuous se"on(-or(er partial (eri)ati)es

( )'xx yy xyf f and fin some "ir"le "entere( at a "riti"al point

( ' )x yan( let

( ' ) ( ' ) ( ' )


1 D> and ( ' ) xxf x y < then fhas ---------------

2elative ma$imum at ( ' )x y

2elative minimum at ( ' )x y

!addle point at ( ' )x y


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If

( ' ' ) x y

f x y z xyzz

= +

t!en #!at is t!e )alue of (*' *' *)f %(*' *' *) *f =

(*' *' *) f =

(*' *' *) &f =(*' *' *) +f =

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MIDTERM EXAMINATION

Spring 2010

MTH301- Calculus II (Sessin - 2!

Mar"s# $0

%ues&in N# 1 ( Mar"s# 1 ! - 'lease cse ne

Which of the folloi!" is the i!ter#al !otatio! of real li!e$

% &'( ) *(+

% &'( ) 0+

% &0 ) *(+


What is the "e!eral e,-atio! of para.ola hose a/is of smmetr is parallel to 'a/is$

% ( )y ax b a= +

%

( )x ay b a= +

% ( )y ax bx c a= + +

% ( )x ay by c a= + +


Which of the folloi!" is "eometrical represe!tatio! of the e,-atio! +y= ) i! three ime!sio!alspace$

% A poi!t o! 'a/is

% Pla!e parallel to /'pla!e

% Pla!e parallel to 'a/is

% Pla!e parallel to /'pla!e

%ues&in N# $ ( Mar"s# 1 ! - 'lease cse ne

S-ppose&( ' ) xyf x y x e= Which o!e of the stateme!ts is correct$

%

&& xyf

x ey

=

%

& xyf x ey

=

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%

+ xyf x ey

=

%

& xyf x yey

=

%ues&in N# ) ( Mar"s# 1 ! - 'lease cse ne

If &( ' ) lnf x y x y y x= +

the!

f

x

%

*xy

x+

%

*y

x+

%

*xy

x

%

*y

x

%ues&in N# * ( Mar"s# 1 ! - 'lease cse ne

5et f&/) ) + a! / "&r) s+) h&r) s+) t&r) s+ the! . chai! r-le

w

r

=

%

w x w y w z

x r y r z r

+ +

%w x w y w z

r r r r r r

+ +

%w x x w y y w z z

x r s y r s z r s

+ +

%w r w r w r

r x r y r z

+ +

%ues&in N# + ( Mar"s# 1 ! - 'lease cse ne

Is the f-!ctio! ( ' )f x y co!ti!-o-s at ori"i!$ If !ot) h$

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&( ' )

( ' )

( ' )

x yif x y

f x y x y

if x y

= + =

% ( ' )f x y is co!ti!-o-s at ori"i!

% (' )f is !ot efi!e

% (' )f is efi!e .-t( ' ) ( ' )

lim ( ' )x y

f x y oes !ot e/ist

% (' )f is efi!e a!( ' ) ( ' )

lim ( ' )x y

f x y e/ists .-t these to !-m.ers are !ot e,-al

%ues&in N# , ( Mar"s# 1 ! - 'lease cse ne

5et R .e a close re"io! i! to ime!sio!al space What oes the o-.le i!te"ral o#er Rcalc-lates$

% Area of R

% Rai-s of i!scri.e circle i! R

% Dista!ce .etee! to e!poi!ts of R

% 6o!e of these

%ues&in N# ( Mar"s# 1 ! - 'lease cse ne

To s-rfaces are sai to .e ortho"o!al at a poi!t of their i!tersectio! if their !ormals at thatpoi!t are '''''''''

% Parallel

% Perpe!ic-lar

% I! opposite irectio!


To s-rfaces are sai to i!tersect ortho"o!all if their !ormals at e#er poi!t commo! to themare ''''''''''

% perpe!ic-lar

% parallel

% i! opposite irectio!


5et the f-!ctio!( ' )f x y

has co!ti!-o-s seco!'orer partial eri#ati#es

( )'xx yy xyf f and fi! some circle ce!tere at a critical poi!t

( ' )x ya! let

( ' ) ( ' ) ( ' )


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If D> a! ( ' ) xxf x y > the! fhas '''''''''''''''

% Relati#e ma/im-m at ( ' )x y

% Relati#e mi!im-m at ( ' )x y

% Sale poi!t at ( ' )x y

% 6o co!cl-sio! ca! .e ra!





( ' )x ya! let

( ' ) ( ' ) ( ' )


If D> a! ( ' ) xxf x y < the! fhas '''''''''''''''

% Relati#e ma/im-m at ( ' )x y

% Relati#e mi!im-m at ( ' )x y

% Sale poi!t at ( ' )x y

% 6o co!cl-sio! ca! .e ra!





( ' )x ya! let

( ' ) ( ' ) ( ' )


IfD

'uesin N# 1 ( Mar%s# 1 ! - )lease c*se ne

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MID Term Past Papers MTH 301 Past Papers"lane is an e%ample of ---------------------

/ur!e

+ Sur8ace

'phere

/one

'uesin N# 1$ ( Mar%s# 1 ! - )lease c*se ne

If * R R R= where *R and R are no o!erlapping regions then

*

( ' ) ( ' )R R

f x y dA f x y dA+ =

( ' )

R

f x y dA

*

( ' ) ( ' )R R

f x y dA f x y dA

( ' )

R

f x y dV

*

( ' ) ( ' )R R

f x y dA f x y dA


&

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+ *

&

(, + )x xy dxdy+


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&

(+ )yxe dydx


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MID Term Past Papers MTH 301 Past Papers'uesin N# 2 ( Mar%s# !

ind ,)uation of a *angent plane to the surface &( ' ' ) & :f x y z x y z= + + at the point

(' *' )

'uesin N# 2$ ( Mar%s# !

,!aluate the iterated integral

( )+

x

x xy dydx

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