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