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MATH/IV/04 Student’s Copy
2 0 1 8
( Pre-CBCS )
( 4th Semester )
MATHEMATICS
Paper : MATH–241
( Vector Calculus and Solid Geometry )
Full Marks : 75
Time : 3 hours
( PART : A—OBJECTIVE )
( Marks : 25 )
Answer all questions
SECTION—A
( Marks : 10 )
Each question carries 1 mark
Tick (3) the correct answer in the brackets provided :
1. If ra and
rb are two mutually perpendicular proper vectors, then
r r ra b a´ ´( ) is
parallel to
(a)ra ( )
(b)rb ( )
(c)r ra b´ ( )
(d) None of the above ( )
2. If | |ra = 4, | |
rb = 5 and
r ra b× = 0, then
r ra b´ is
(a) 20 $n ( )
(b) 9 $n ( )
(c) $n ( )
(d) 0 ( )
/409 1 [ Contd.
3. The vector rV x y z i x y z j x y az k= - - + + - + - + + +( ) $ ( ) $ ( ) $4 6 3 2 5 5 6 is solenoidal,
then the value of a is
(a) 5 ( )
(b) 8 ( )
(c) 3 ( )
(d) None of the above ( )
4. Suppose V be the volume bounded by a closed surface S, rr xi yj zk= + +$ $ $ and $n is
the unit vector normal (outward) to the surface S, thenrr ndS
S
×òò $
is
(a) 0 ( )
(b) 4V ( )
(c) 2V ( )
(d) 3V ( )
5. The equation of pair of straight lines through the origin perpendicular to the
pair ax hxy by2 22 0+ + = is
(a) ax hxy by2 22 0- + = ( )
(b) bx hxy ay2 22 0- + = ( )
(c) ax hxy by2 22 0+ + = ( )
(d) bx hxy ay2 22 0+ + = ( )
6. If the equation ax hxy by gx fy c2 22 2 2 0+ + + + + = represents a circle, if
(a) ab h- =2 0 ( )
(b) ab h- ¹2 0 ( )
(c) a b= and h = 0 ( )
(d) a b+ = 0 ( )
7. The intercepts made on the axes by the plane 3 4 6 12 0x y z- + - = are
(a) 4, -3 and 2 ( )
(b) -4, -3 and 5 ( )
(c) 5, 7 and -9 ( )
(d) None of the above ( )
MATH/IV/04/409 2 [ Contd.
8. The shortest distance between the line x y z-
=+
=-1
4
2
3
3
1 and z-axis is
(a)12
5 ( )
(b)11
5 ( )
(c)11
5 ( )
(d)12
7 ( )
9. The equation of sphere which passes through the origin and makes equal
intercepts of unit length of the axes is
(a) x y z2 2 2 1+ + = ( )
(b) ( ) ( ) ( )x y z- + - + - =1 1 1 02 2 2 ( )
(c) x y z x y z2 2 2 1+ + + + + = ( )
(d) x y z x y z2 2 2 0+ + - - - = ( )
10. The condition that the plane lx my nz+ + = 0 touches the cone
ax by cz2 2 2 0+ + = is
(a) bc l cam abn2 2 2 0+ + = ( )
(b) a l bm cn2 2 2 0+ + = ( )
(c) a l b m c n2 2 2 0+ + = ( )
(d) None of the above ( )
SECTION—B
( Marks : 15 )
Each question carries 3 marks
State True or False by putting a Tick (3) mark in the brackets provided and give a briefjustification :
1. If ra i j k= + +$ $ $2 3 ,
rb i j k= + +$ $ $3 5 and
rc i j k= + +$ $ $6 , then the value of
r r ra b c× ´( )
is 5.
True ( ) False ( )
Jus ti fi ca tion :
MATH/IV/04/409 3 [ Contd.
2. The value of Ñæèç
öø÷
1
r, where
rr xi yj zk= + +$ $ $ is -
rr
r 3.
True ( ) False ( )
Jus ti fi ca tion :
3. The equation of the diameter of the conic 4 6 5 12 2x xy y+ - = conjugate to the
diameter y x- =2 0 is 10 7 0y x- = .
True ( ) False ( )
Jus ti fi ca tion :
4. The equation of the plane through the line x y z+ + + =3 0, 2 3 1 0x y z- + + =
and parallel to the line x y z
1 2 3= = is x y z- + - =5 3 7 0.
True ( ) False ( )
Jus ti fi ca tion :
5. The equation of the orthogonal projection of the curve 2 3x y z+ - = ,
x y z2 2 22 3 1+ + = on the z-plane is
z = 0, 13 5 36 18 12 26 02 2x y x y xy+ - - + + =
True ( ) False ( )
Jus ti fi ca tion :
( PART : B—DESCRIPTIVE )
( Marks : 50 )
The figures in the margin indicate full marks for the questions
Answer one question from each Unit
Unit—I
1. (a) Find a unit vector perpendicular to the plane of rA i j k= - -2 6 3$ $ $ and
rB i j k= + -4 3$ $ $. 3
(b) If ABC be a triangle, then prove that
cos Ab c a
bc=
+ -2 2 2
2 3
MATH/IV/04/409 4 [ Contd.
(c) Find the set of vectors reciprocal to the set $ $ $i j k+ +2 3 , 5$ $ $i j k- - and $ $ $i j k+ - . 4
2. (a) If ra ,
rb and
rc are three non-coplanar vectors, then prove that
[ ] [ ]r r r r r r r r ra b b c c a a b c´ ´ ´ = . 5
(b) If three concurrent edges of a parallelepiped is given byra i j k= - +2 3 4$ $ $,
rb i j k= + -$ $ $2 and
rc i j k= - +3 2$ $ $, then find its volume. 5
Unit—II
3. (a) If f ( , , )x y z x yz xyz= -2 24 , then find the directional derivative of f in
the direction of rA i j k= - -2 2$ $ $ at ( , , )1 3 1. 4
(b) Prove that curl ( ) ( ) ( ) ( ) ( )r r r r r r r r r ra b b a b a a b a b´ = × Ñ - Ñ × - × Ñ + Ñ × . 6
4. (a) If rr xi yj zk= + +$ $ $, then show that Ñ = -r nr rn n 2
r. 4
(b) Show that r r rF ndS Fdv
S vòò òòò× = Ñ × where rF x z i y j y z k= - +4 2$ $ $ and S
is the surface of the cube bounded by x = 0, x =1, y = 0, y =1, z = 0,
z =1. 6
Unit—III
5. (a) Find the angle through which a set of rectangular axes must be turned
without the change of origin so that the expression 7 4 32 2x xy y+ +
will be transformed into the form ¢ + ¢a x b y2 2. 5
(b) For what value of k will the equation
3 3 29 3 18 02 2x kxy y x y+ - + - + = represent a pair of straight lines? 5
6. (a) Reduce the equation 144 120 25 243 448 113 02 2x xy y x y- + + + - = to
the standard form and hence show that it is the equation of parabola. 6
(b) Find the vertex and length of the latus rectum of the parabola
( ) ( )3 4 17 35 4 3 62x y x y+ - = - - . 4
MATH/IV/04/409 5 [ Contd.
Unit—IV
7. (a) Find the equation of the plane which passes through the point
( , , )2 3 1- and is perpendicular to the line joining the points ( , , )4 5 2-
and ( , , )2 1 6- . 4
(b) Find the equation of the plane which passes through the point ( , , )2 1 4
and is perpendicular to the planes 9 7 6 48 0x y z- + + = and
x y z+ - = 0. 4
(c) Find the perpendicular distance of the points ( , , )1 4 2- and ( , , )5 1 3 from
the plane 2 3 5x y z- + = . 2
8. (a) Prove that the lines
x y z-=
-=
-2
3
1
2
4
5
and 2 3 0x y z- + = , x y z+ + + =2 4 0 are coplanar. 5
(b) Prove that the shortest distance between the lines
x y z-
=-
=+
-
3
1
4
1
1
3 and
x y z-
-=
-=
-1
1
3
3
1
2 is
15
138
and the equations of the shortest distance are 7 37 10 117 0x y z- - + =
and 5 13 17 27 0x y z+ - - = . 5
Unit—V
9. (a) The plane x
a
y
b
z
c+ + =1 cuts the axes at A, B and C. Find the equation
of the cone whose vertex is the origin and the guiding curve is the
circle ABC. 5
(b) Find the equation of the sphere which passes through the origin and
touches the sphere x y z2 2 2 56+ + = at the point ( , , )2 4 6- . 5
MATH/IV/04/409 6 [ Contd.
10. (a) Determine the angle between the lines of intersection of the plane
x y z- + =3 0 and the quadric cone x y z2 2 25 0- + = . 5
(b) Find the equation of the cylinder generated by the lines parallel to the
line
x y z
1 2 1= =
and intersecting the guiding curve z = 3 and x y2 2 4+ = . 5
H H H
MATH/IV/04/409 7 8G—160
MATH/IV/04 Student’s Copy
2 0 1 8
( Pre-CBCS )
( 4th Semester )
MATHEMATICS
Paper : MATH–241
( Vector Calculus and Solid Geometry )
Full Marks : 75
Time : 3 hours
( PART : A—OBJECTIVE )
( Marks : 25 )
Answer all questions
SECTION—A
( Marks : 10 )
Each question carries 1 mark
Tick (3) the correct answer in the brackets provided :
1. If ra and
rb are two mutually perpendicular proper vectors, then
r r ra b a´ ´( ) is
parallel to
(a)ra ( )
(b)rb ( )
(c)r ra b´ ( )
(d) None of the above ( )
2. If | |ra = 4, | |
rb = 5 and
r ra b× = 0, then
r ra b´ is
(a) 20 $n ( )
(b) 9 $n ( )
(c) $n ( )
(d) 0 ( )
/409 1 [ Contd.
3. The vector rV x y z i x y z j x y az k= - - + + - + - + + +( ) $ ( ) $ ( ) $4 6 3 2 5 5 6 is solenoidal,
then the value of a is
(a) 5 ( )
(b) 8 ( )
(c) 3 ( )
(d) None of the above ( )
4. Suppose V be the volume bounded by a closed surface S, rr xi yj zk= + +$ $ $ and $n is
the unit vector normal (outward) to the surface S, thenrr ndS
S
×òò $
is
(a) 0 ( )
(b) 4V ( )
(c) 2V ( )
(d) 3V ( )
5. The equation of pair of straight lines through the origin perpendicular to the
pair ax hxy by2 22 0+ + = is
(a) ax hxy by2 22 0- + = ( )
(b) bx hxy ay2 22 0- + = ( )
(c) ax hxy by2 22 0+ + = ( )
(d) bx hxy ay2 22 0+ + = ( )
6. If the equation ax hxy by gx fy c2 22 2 2 0+ + + + + = represents a circle, if
(a) ab h- =2 0 ( )
(b) ab h- ¹2 0 ( )
(c) a b= and h = 0 ( )
(d) a b+ = 0 ( )
7. The intercepts made on the axes by the plane 3 4 6 12 0x y z- + - = are
(a) 4, -3 and 2 ( )
(b) -4, -3 and 5 ( )
(c) 5, 7 and -9 ( )
(d) None of the above ( )
MATH/IV/04/409 2 [ Contd.
8. The shortest distance between the line x y z-
=+
=-1
4
2
3
3
1 and z-axis is
(a)12
5 ( )
(b)11
5 ( )
(c)11
5 ( )
(d)12
7 ( )
9. The equation of sphere which passes through the origin and makes equal
intercepts of unit length of the axes is
(a) x y z2 2 2 1+ + = ( )
(b) ( ) ( ) ( )x y z- + - + - =1 1 1 02 2 2 ( )
(c) x y z x y z2 2 2 1+ + + + + = ( )
(d) x y z x y z2 2 2 0+ + - - - = ( )
10. The condition that the plane lx my nz+ + = 0 touches the cone
ax by cz2 2 2 0+ + = is
(a) bc l cam abn2 2 2 0+ + = ( )
(b) a l bm cn2 2 2 0+ + = ( )
(c) a l b m c n2 2 2 0+ + = ( )
(d) None of the above ( )
SECTION—B
( Marks : 15 )
Each question carries 3 marks
State True or False by putting a Tick (3) mark in the brackets provided and give a briefjustification :
1. If ra i j k= + +$ $ $2 3 ,
rb i j k= + +$ $ $3 5 and
rc i j k= + +$ $ $6 , then the value of
r r ra b c× ´( )
is 5.
True ( ) False ( )
Jus ti fi ca tion :
MATH/IV/04/409 3 [ Contd.
2. The value of Ñæèç
öø÷
1
r, where
rr xi yj zk= + +$ $ $ is -
rr
r 3.
True ( ) False ( )
Jus ti fi ca tion :
3. The equation of the diameter of the conic 4 6 5 12 2x xy y+ - = conjugate to the
diameter y x- =2 0 is 10 7 0y x- = .
True ( ) False ( )
Jus ti fi ca tion :
4. The equation of the plane through the line x y z+ + + =3 0, 2 3 1 0x y z- + + =
and parallel to the line x y z
1 2 3= = is x y z- + - =5 3 7 0.
True ( ) False ( )
Jus ti fi ca tion :
5. The equation of the orthogonal projection of the curve 2 3x y z+ - = ,
x y z2 2 22 3 1+ + = on the z-plane is
z = 0, 13 5 36 18 12 26 02 2x y x y xy+ - - + + =
True ( ) False ( )
Jus ti fi ca tion :
( PART : B—DESCRIPTIVE )
( Marks : 50 )
The figures in the margin indicate full marks for the questions
Answer one question from each Unit
Unit—I
1. (a) Find a unit vector perpendicular to the plane of rA i j k= - -2 6 3$ $ $ and
rB i j k= + -4 3$ $ $. 3
(b) If ABC be a triangle, then prove that
cos Ab c a
bc=
+ -2 2 2
2 3
MATH/IV/04/409 4 [ Contd.
(c) Find the set of vectors reciprocal to the set $ $ $i j k+ +2 3 , 5$ $ $i j k- - and $ $ $i j k+ - . 4
2. (a) If ra ,
rb and
rc are three non-coplanar vectors, then prove that
[ ] [ ]r r r r r r r r ra b b c c a a b c´ ´ ´ = . 5
(b) If three concurrent edges of a parallelepiped is given byra i j k= - +2 3 4$ $ $,
rb i j k= + -$ $ $2 and
rc i j k= - +3 2$ $ $, then find its volume. 5
Unit—II
3. (a) If f ( , , )x y z x yz xyz= -2 24 , then find the directional derivative of f in
the direction of rA i j k= - -2 2$ $ $ at ( , , )1 3 1. 4
(b) Prove that curl ( ) ( ) ( ) ( ) ( )r r r r r r r r r ra b b a b a a b a b´ = × Ñ - Ñ × - × Ñ + Ñ × . 6
4. (a) If rr xi yj zk= + +$ $ $, then show that Ñ = -r nr rn n 2
r. 4
(b) Show that r r rF ndS Fdv
S vòò òòò× = Ñ × where rF x z i y j y z k= - +4 2$ $ $ and S
is the surface of the cube bounded by x = 0, x =1, y = 0, y =1, z = 0,
z =1. 6
Unit—III
5. (a) Find the angle through which a set of rectangular axes must be turned
without the change of origin so that the expression 7 4 32 2x xy y+ +
will be transformed into the form ¢ + ¢a x b y2 2. 5
(b) For what value of k will the equation
3 3 29 3 18 02 2x kxy y x y+ - + - + = represent a pair of straight lines? 5
6. (a) Reduce the equation 144 120 25 243 448 113 02 2x xy y x y- + + + - = to
the standard form and hence show that it is the equation of parabola. 6
(b) Find the vertex and length of the latus rectum of the parabola
( ) ( )3 4 17 35 4 3 62x y x y+ - = - - . 4
MATH/IV/04/409 5 [ Contd.
Unit—IV
7. (a) Find the equation of the plane which passes through the point
( , , )2 3 1- and is perpendicular to the line joining the points ( , , )4 5 2-
and ( , , )2 1 6- . 4
(b) Find the equation of the plane which passes through the point ( , , )2 1 4
and is perpendicular to the planes 9 7 6 48 0x y z- + + = and
x y z+ - = 0. 4
(c) Find the perpendicular distance of the points ( , , )1 4 2- and ( , , )5 1 3 from
the plane 2 3 5x y z- + = . 2
8. (a) Prove that the lines
x y z-=
-=
-2
3
1
2
4
5
and 2 3 0x y z- + = , x y z+ + + =2 4 0 are coplanar. 5
(b) Prove that the shortest distance between the lines
x y z-
=-
=+
-
3
1
4
1
1
3 and
x y z-
-=
-=
-1
1
3
3
1
2 is
15
138
and the equations of the shortest distance are 7 37 10 117 0x y z- - + =
and 5 13 17 27 0x y z+ - - = . 5
Unit—V
9. (a) The plane x
a
y
b
z
c+ + =1 cuts the axes at A, B and C. Find the equation
of the cone whose vertex is the origin and the guiding curve is the
circle ABC. 5
(b) Find the equation of the sphere which passes through the origin and
touches the sphere x y z2 2 2 56+ + = at the point ( , , )2 4 6- . 5
MATH/IV/04/409 6 [ Contd.
10. (a) Determine the angle between the lines of intersection of the plane
x y z- + =3 0 and the quadric cone x y z2 2 25 0- + = . 5
(b) Find the equation of the cylinder generated by the lines parallel to the
line
x y z
1 2 1= =
and intersecting the guiding curve z = 3 and x y2 2 4+ = . 5
H H H
MATH/IV/04/409 7 8G—160
MATH/IV/EC/04 Student’s Copy
2 0 1 8
( CBCS )
( 4th Semester )
MATHEMATICS
FOURTH PAPER
( Vector Calculus and Solid Geometry )
Full Marks : 75
Time : 3 hours
( PART : A—OBJECTIVE )
( Marks : 25 )
The figures in the margin indicate full marks for the questions
SECTION—A
( Marks : 10 )
Each question carries 1 mark
Tick (3) the correct answer in the brackets provided :
1. The component of ra i j k= - +2$ $ $ on
rb i j k= - +$ $ $2 is
(a)1
62($ $ $)i j k- + ( )
(b)5
62($ $ $)i j k- + ( )
(c)1
62( $ $ $)i j k- + ( )
(d)5
62( $ $ $)i j k- + ( )
/342 1 [ Contd.
2. If ra ,
rb and
rc be any three vectors, then
r r r r r r r r ra b c b c a c a b´ ´ + ´ ´ + ´ ´( ) ( ) ( ) is
(a)r0 ( )
(b)ra ( )
(c)rb ( )
(d)rc ( )
3. If f is a scalar point function, then grad f is
(a) both solenoidal and irrotational ( )
(b) solenoidal ( )
(c) irrotational ( )
(d) neither solenoidal nor irrotational ( )
4. If r rF dr
p
p×ò
1
2 is independent of the path joining the two points a and b in a given
region, then for all closed paths in the region, r rF dr×ò is
(a) p2 ( )
(b) p1 ( )
(c) 0 ( )
(d) None of the above ( )
5. The equation ax hxy by gx fy c2 22 2 2 0+ + + + + = represents a hyperbola, if
(a) ab h- =2 0 ( )
(b) ab h- <2 0 ( )
(c) ab h- >2 0 ( )
(d) a b= and h = 0 ( )
MATH/IV/EC/04/342 2 [ Contd.
6. The centre of the conic given by the equation 3 8 7 4 2 7 02 2x xy y x y- + - + - = is
(a) (2, –1) ( )
(b) (1, –2) ( )
(c) (2, 1) ( )
(d) (1, 2) ( )
7. The intercepts on z-axis by the plane x y z+ + =2 2 is
(a) 1 ( )
(b) 2 ( )
(c) 3 ( )
(d) 4 ( )
8. The angle between the planes x y z+ + =1 and x y- = 2 is
(a) 0 ( )
(b)p
2 ( )
(c)p
3 ( )
(d)p
4 ( )
9. The equation of sphere which passes through the origin and makes equalintercepts of unit length of the axes is
(a) x y z2 2 2 1+ + = ( )
(b) x y z x y z2 2 2 0+ + - - - = ( )
(c) x y z x y z2 2 2 0+ + + + + = ( )
(d) ( ) ( ) ( )x y z- + - + - =1 1 1 02 2 2 ( )
MATH/IV/EC/04/342 3 [ Contd.
10. The reciprocal cone of the cone ax by cz2 2 2 0+ + = is
(a) x y z2 2 2 0+ + = ( )
(b) bcx cay abz2 2 2 0+ + = ( )
(c) x y z a b c2 2 2 2 2 2+ + = + + ( )
(d) Does not exist ( )
SECTION—B
( Marks : 15 )
Each question carries 3 marks
1. (a) For any vector ra , prove that $ ( $) $ ( $) $ ( $)i a i j a j k a k a´ ´ + ´ ´ + ´ ´ =
r r r r2 .
OR
(b) A particle moves along a curve whose parametric equations are x e t= - , y t= 2 3cos , z t= 2 3sin , where t is the time. Determine its velocity andacceleration at any time.
2. (a) Prove that Ñ × Ñ ´ =( )rF 0.
OR
(b) Find divrf at ( , , )1 1 1- , if
rf x zi y z j xy zk= - +2 3 2 22$ $ $.
MATH/IV/EC/04/342 4 [ Contd.
3. (a) Prove that the diameter of the conic 15 20 16 12 2x xy y- + = conjugate to the
diameter y x+ =2 0 is 5 6x y= .
OR
(b) Show that the equation of the asymptotes of the hyperbola
2 5 3 5 3 21 02 2x xy y x y- - - - - = is 2 5 3 5 318
4902 2x xy y x y- - - - - = .
4. (a) Prove that the lines
x y z+=
+=
-
-
3
2
5
3
7
3,
x y z+=
+=
+
-
1
4
1
5
1
1
are coplanar.
OR
(b) Prove that the length of the perpendicular drawn from the point ( , , )x y z1 1 1to the plane ax by cz d+ + + = 0 is
| |ax by cz d
a b c
1 1 1
2 2 2
+ + +
+ +
5. (a) Show that the general equation of a cone which touches the three
co-ordinate planes is fx gy hz± ± = 0.
OR
(b) Find the equation of the sphere through the circle x y z2 2 2 9+ + = ,
2 3 4 5x y z+ + = and the point (1, 2, 3).
MATH/IV/EC/04/342 5 [ Contd.
( PART : B—DESCRIPTIVE )
( Marks : 50 )
The figures in the margin indicate full marks for the questions
Answer five questions, selecting one from each Unit
UNIT—I
1. (a) Prove that for a triangle ABC, sin sin sinA
a
B
b
C
c= = , where AB c= ,
BC a= and CA b= . 5
(b) A particle moves along the curve x t= 2 2, y t t= -2 4 , z t= -3 5, where t
is the time. Find the components of its velocity and acceleration at
time t =1 in the direction $ $ $i j k- +3 2 . 5
2. (a) Prove that a vector function rf t( ) will be of constant magnitude, if and
only if r
r
fdf
dt× = 0.
5
(b) Find the value of l so that the four points with position vectors
A i j k( $ $ $)- + +6 3 2 , B i j k( $ $ $)3 4+ +l , C i j k( $ $ $)5 7 3+ + and D i j k( $ $ $)- + -13 17 2
are coplanar. 5
UNIT—II
3. (a) Let f( , , )x y z x y z= + +3 3 2. Find the directional derivative of f at
( , , )1 1 2- in the direction of the vector $ $ $i j k+ +2 . 5
(b) Suppose Ñ ´ =rA 0. Evaluate Ñ × ´ =( )
r rA r 0, where
rr xi yj zk= + +$ $ $ and
rA A i A j A k= + +1 2 3
$ $ $. 5
MATH/IV/EC/04/342 6 [ Contd.
4. (a) Evaluate r rA ndS
S
×òò , where rA yi xj zk= + -$ $ $2 and S is the surface of the
plane 2 6x y+ = in the first octant cut off by the plane z = 4. 5
(b) Find the work done in moving a particle in the force field rF x i xz y j zk= + - +3 22$ ( )$ $ along—
(i) a straight line from (0, 0, 0) to (2, 1, 3);
(ii) the curve defined by x y2 4= , 3 83x z= from x = 0 to x = 2. 5
UNIT—III
5. (a) If, by a rotation of the rectangular axes about the origin, the
expression ax hxy by2 22+ + changes to a x h x y b y1 12
1 1 1 1 122+ + , then
show that a b a b+ = +1 1 and ab h a b h- = -21 1 1
2. 5
(b) Find the equations of the parabolas passing through the points
of intersection of x xy y x y2 26 2 3 5 0+ - + - - = and
2 8 3 2 1 02 2x xy y y- + + - = . 5
6. (a) Reduce the equation of 144 120 25 243 448 113 02 2x xy y x y- + + + - =
to the standard form and hence show that it is the equation of a
parabola. 5
(b) Prove that the straight lines represented by the equation
ax hxy by gx fy c2 22 2 2 0+ + + + + = will be equidistant from the origin,
if f g c bf ag4 4 2 2- + = -( ). 5
UNIT—IV
7. (a) A variable plane is at a constant distance 3p from the origin and meet
the axes in A, B and C. Show that the locus of the centroid of the
triangle ABC is x y z p- - - -+ + =2 2 2 2. 5
MATH/IV/EC/04/342 7 [ Contd.
(b) Show that the lines x a d y a z a d- +
-=
-=
- -
+a d a a d and
x b c y b z b c- +
-=
-=
- -
+b g b b g are coplanar.
5
8. (a) Find the equation of the plane passing through the line of intersection
of the planes x y z- + =2 1 and 2 8x y z+ + = , and parallel to the line
with direction ratios 1, 2, 1. Find also the perpendicular distance of
(1, 1, 1) from this plane. 5
(b) Show that the lines x y z-
=-
=-1
2
2
3
3
4 and
x yz
-=
-=
4
5
1
2 intersect
each other. Find the point of their intersection. 5
UNIT—V
9. (a) Show that the condition for the plane lx my nz p+ + = to be a tangent
plane to x y z a2 2 2 2+ + = is a l m n p2 2 2 2 2( )+ + = . 5
(b) Find the radius of the circle, where the plane x y z- + =2 2 3 intersects
the sphere x y z x y z2 2 2 8 4 8 45+ + - + + = . 5
10. (a) Find the equation of a right circular cylinder of radius 5, whose axis
passes through (1, 2, 3) and is parallel to x y z-
=-
-=
-4
2
3
1
2
2.
5
(b) Find the equation of the cone whose vertex is ( , , )a b g and base is
ax by2 2 1+ = , z = 0. 5
H H H
MATH/IV/EC/04/342 8 8G—300