Time : 3 hours
Solutions 01  Mathematics 
March
2006
Marks : 100
Pg 
2
5.
Write
the
equation
of
the
auxiliary
circle
of
1
9
y
4
x
2
2

=
.
( a )
x
2
+
y
2
=
 5
( b )
x
2
+
y
2
=
5
( c )
x
2
+
y
2
=
4
( d )
x
2
+
y
2
=
9
Solution :
The
equation
of
the
auxiliary
circle
of
the
hyperbola
1
b
y
a
x
2
2
2
2
=

is
x
2
+
y
2
=
a
2
.
Here,
a
2
=
4
=>
the
equation
of
the
auxiliary circle
of
the
given
hyperbola
is
x
2
+
y
2
=
4.
6.
Obtain
l
cos
θ
cos
α
,
cos
θ
sin
α
,
sin
θ
l
( a )
 1
( b )
0
( c )
1
( d )
none
of
these
Solution :
l
cos
θ
cos
α
,
cos
θ
sin
α
,
sin
θ
l
=
θ
sin
α
sin
θ
cos
α
cos
θ
cos
2
2
2
2
2
+
+
=
θ
sin
)
α
sin
α
cos
(
θ
cos
2
2
2
2
+
+
=
θ
sin
θ
cos
2
2
+
=
1.
7.
Find
the
resultant
force
of
( 1,
2,
 1 )
and
( 1,
 2,
1 )
( a )
( 2,
0,
0 )
( b )
(  1,
4,
2 )
( c )
( 2,
4,
2 )
( d )
(  2,
0,
0 )
Solution :
The
resultant
of
( x,
y,
z )
=
( 1,
2,
 1 )
and
( x’,
y’,
z’ )
=
( 1,
 2,
1 )
is
( x + x’,
y + y’,
z + z’ )
=
( 1 + 1,
2  2,
 1 + 1 )
=
( 2,
0,
0
).
8.
Find
the
centre
of
the
sphere
x
2
+
y
2
+ z
2

2x

4y

6z

11
=
0.
( a )
(  1,
 2,
 3 )
( b )
( 3,
2,
1 )
( c )
( 1,
2,
3 )
( d )
( 1,
2,
 3 )
Solution :
Comparing
the
given
equation
of
the
sphere
x
2
+
y
2
+ z
2

2x

4y

6z

11
=
0
with
the
general
equation
of
the
sphere
x
2
+
y
2
+ z
2
+
2ux
+
2vy
+
2wz
+
c
=
0,
the
centre
of
the
sphere
is
(  u,
 v,
 w )
=
( 1,
2,
3 ).
9.
Find
x
1
e
lim

3x
0
x
→
( a )
3
( b )
3
1
( c )
log
e
e
( d )
log
e
3
Solution :
x
1
e
lim

3x
0
x
→
=
3x
1
e
lim
3

3x
0
x
→
=
3.