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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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MID Term Past Papers MTH 301 Past Papers
( )'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$ist but these two numbers are not equal.
**. et 2 be a closed region in two dimensional space. "hat does the double integral over 2 calculates#
Area o 2
2adius o inscribed circle in 2.
3istance between two endpoints o 2.
None o these
*. "hich o the ollowing ormula can be used to ind the volume o a parallelepiped with ad4acent
edges ormed by the vectors ' and #
*&. Two suraces are said to be orthogonal at appoint o their intersection i their normals at that point
are_________.
5arallel
5erpendicular
1n opposite direction
!ame direction
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MID Term Past Papers MTH 301 Past Papers
*+. 6y E$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
No conclusion can be drawn.
*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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MID Term Past Papers MTH 301 Past Papers
( )'
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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MID Term Past Papers MTH 301 Past Papers
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$imum at
( ' )x y
fhas relative minimum at ( ' )x y
fhas saddle point at
( ' )x y
No conclusion can be drawn.
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MID Term Past Papers MTH 301 Past Papers
&+ , ,df
t tdt
= +
, * df
t tdt
= +
&et i + j an( kbe unit )e"tors in t!e (ire"tion of /-a/is+ -a/is an( -a/is respe"ti)el.
Suppose t!at 0a i j k
= + . !at is t!e ma$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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MID Term Past Papers MTH 301 Past Papers
!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$onall if t!eir normals at e)er point "ommon to
t!em are ----------perpendicularparallelin opposite direction
&et t!e fun"tion( ' )f x y
!as "ontinuous se"on(-or(er partial (eri)ati)es
( )'xx yy xyf f and fin some "ir"le "entere( at a "riti"al point
( ' )x yan( let
( ' ) ( ' ) ( ' )
xx yy xyD f x y f x y f x y=
1 D> and ( ' ) xxf x y < then fhas ---------------
2elative ma$imum at ( ' )x y
2elative minimum at ( ' )x y
!addle point at ( ' )x y
No conclusion can be drawn.
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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 ) *(+
%ues&in N# 2 ( Mar"s# 1 ! - 'lease cse ne
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= + +
%ues&in N# 3 ( Mar"s# 1 ! - 'lease cse ne
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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MID Term Past Papers MTH 301 Past Papers
%
+ 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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MID Term Past Papers MTH 301 Past Papers
&( ' )
( ' )
( ' )
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!
%ues&in N# 10 ( Mar"s# 1 ! - 'lease cse ne
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!
%ues&in N# 11 ( Mar"s# 1 ! - 'lease cse ne
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
( ' ) ( ' ) ( ' )
xx yy xyD f x y f x y f x y=
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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!
%ues&in N# 12 ( Mar"s# 1 ! - 'lease cse ne
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
( ' ) ( ' ) ( ' )
xx yy xyD f x y f x y f x y=
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!
%ues&in N# 13 ( Mar"s# 1 ! - 'lease cse ne
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
( ' ) ( ' ) ( ' )
xx yy xyD f x y f x y f x y=
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
'uesin N# 1 ( Mar%s# 1 ! - )lease c*se ne
&
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+ *
&
(, + )x xy dxdy+
'uesin N# 1 ( Mar%s# 1 ! - )lease c*se ne
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MID Term Past Papers MTH 301 Past Papers
&
(+ )yxe dydx
'uesin N# 20 ( Mar%s# 1 ! - )lease c*se ne
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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