DECEMBER—2021
BA/B.Sc (III Semester) Examination
MATHEMATICS
Paper : MATH–231 (New Course)
( Coordinate Geometry and Vector Analysis )
Full Marks : 80
Pass Marks : 35%
Time : Three Hours
Note : 1. Answer all questions as per the instructions given.
2. The figures in the margin indicate full marks for the questions.
1. Answer any five questions : 2×5=10
2. The figures in the margin indicate full marks for the questions.
1. Answer any five questions : 2×5=10
(a) Write the condition that the general equation of the second degree
(b) What is the general equation of a plane parallel to -axis? Write down the equation of the -axis in symmetrical form.
(c) If are four vectors, then express the product
in terms of the vectors and .
in terms of the vectors and .
(d) Find the angle between the planes 2x + 4y - z = 5 and 3x - y + 7z - 2 = 0.
(e) Define pole and polar of a conic section.
(f) Find the equation of the right circular cone whose vertex is P(2, - 3, 5), axis PQ makes equal angles with the axes and semi-vertical angle is 30°.
2. Answer any four questions : 5×4=20
(a) If by a rotation of the rectangular axes about the origin, the expression
bnbvrf 1 qaX1QA ~~~~~~~!Q`1!1!aaaaaaaaaaaaaaaaaA QZaaaaaaaaaaaaaaaaaaaaaaaaaaaax hxy by
2 2
+ 2 +
changes to
a¢x ¢ + h ¢x ¢y¢ + b¢y¢
2 2 2
then prove that
a + b = a¢ + b¢ and ab - h = a¢b¢ - h ¢
2 2
22ZP/145 ( Continued )
( 3 )
(b) Find the length and the equations of
the shortest distance between the
lines
x - y z
=
-
=
1 +
2
2
3
2
4
and x + y z
=
-
=
1 -
1
1
2
2
3
(c) Find the polar equation of the conic
section in the general form
l
r
= 1 + e cosq
(d) Find the equation of the right
circular cylinder whose axis is the
line
x - y z
=
-
=
2 -
5
3
6
4
7
and radius equal to 10.
(e) Reduce the equation
2 10 5 5 6 25 0
2 2
x - xy + y + x - y - =
to the standard form.
(f) Find the transformed equation of the
curve
(2x + 4y +1) (4x - 2y + 3) = 5
when two perpendicular lines
2x + 4y +1 = 0 and 4x - 2y + 3 = 0
are taken as coordinate axes.
22ZP/145 ( Turn Over )
( 4 )
3. Answer any five questions : 10×5=50
(a) (i) Obtain the equation of the plane
through (–3, 0, –7), (–1, 3, 6) and
parallel to the z-axis. 5
(ii) If
ax hxy by gx fy c
2 2
+ 2 + + 2 + 2 + = 0
represents two straight lines
equidistant from the origin,
show that
f g c bf ag
4 4 2 2
- = ( - ) 5
(b) (i) Transform the equation
14 4 11 44 58 71 0
2 2
x - xy + y - x - y + =
to rectangular axes through the
point (2, 3) and inclined at an
angle tan ( )
-1 2. 5
(ii) Find the equation to the sphere
that passes through the circle
x y z x y z
2 2 2
+ + = 1, 3 + 2 + 7 = 5
and touches the plane
y + 4z = 11. 5
22ZP/145 ( Continued )
( 5 )
(c) (i) Prove that the plane
ax + by + cz = 0 cuts the cone
xy + yz + zx = 0 in perpendicular
lines if
1 1 1
0
a b c
+ + =
5
(ii) Find the equation of the right
circular cylinder whose axis is
the line
x y z
-
= =
2 2 -3
and radius equal to 12. 5
(d) (i) A plane passes through a fixed
point (a,b, g) and cuts the
coordinate axes in A, B, C. Prove
that the locus of the centre of
the sphere OABC is given by
a b g
x y z
+ + = 2
5
(ii) Find the pole of the line
lx + my + n = 0 with respect to
the parabola y ax
2
= 4 . 5
22ZP/145 ( Turn Over )
( 6 )
(e) (i) Find the condition that the lines
y = mx and y = m¢x may be
conjugate diameters of the conic
ax hxy by gx fy c
2 2
+ 2 + + 2 + 2 + = 0
Hence find the diameter of the
conic 15 20 16 1
2 2
x - hxy + y =
conjugate to the diameter
y + 2x = 0. 3+2=5
(ii) Let
r
r = (-1, 2, 3) denotes a
position vector with length r = r
r
and
r
a is a constant vector.
Determine
div {r (a r )}
4
r r
́ and curl {r (a r )}
4
r r
́ 5
(f) (i) If
r
f = x yi - zxj + yzk
2 2 2
$ $ $
, find
div
r
f and grad
r
f . 5
(ii) A particle moves along the curve
x = 3t - 2t + 7t y = 9 2t
4 2
, tan( )
and z e
t
=
- 5
2
sin( ), where t is the
time. Determine the velocity and
acceleration at any time t and
their magnitude at t = 3. 5
22ZP/145 ( Continued )
( 7 )
(g) (i) Find the directional derivatives
of
u yz x e zx y xy z e
x z
= (3 - 2 ) + (2 - 5 ) (7 + 3 )
5
at (1, –1, 0) in the direction
2 3 4
$ $ $
i - j + k. 5
(ii) Prove that r r
(n + 2)
r
is irrotational.
Find n when it is solenoidal. 5
bnbvrf 1 qaX1QA ~~~~~~~!Q`1!1!aaaaaaaaaaaaaaaaaA QZaaaaaaaaaaaaaaaaaaaaaaaaaaaax hxy by
2 2
+ 2 +
changes to
a¢x ¢ + h ¢x ¢y¢ + b¢y¢
2 2 2
then prove that
a + b = a¢ + b¢ and ab - h = a¢b¢ - h ¢
2 2
22ZP/145 ( Continued )
( 3 )
(b) Find the length and the equations of
the shortest distance between the
lines
x - y z
=
-
=
1 +
2
2
3
2
4
and x + y z
=
-
=
1 -
1
1
2
2
3
(c) Find the polar equation of the conic
section in the general form
l
r
= 1 + e cosq
(d) Find the equation of the right
circular cylinder whose axis is the
line
x - y z
=
-
=
2 -
5
3
6
4
7
and radius equal to 10.
(e) Reduce the equation
2 10 5 5 6 25 0
2 2
x - xy + y + x - y - =
to the standard form.
(f) Find the transformed equation of the
curve
(2x + 4y +1) (4x - 2y + 3) = 5
when two perpendicular lines
2x + 4y +1 = 0 and 4x - 2y + 3 = 0
are taken as coordinate axes.
22ZP/145 ( Turn Over )
( 4 )
3. Answer any five questions : 10×5=50
(a) (i) Obtain the equation of the plane
through (–3, 0, –7), (–1, 3, 6) and
parallel to the z-axis. 5
(ii) If
ax hxy by gx fy c
2 2
+ 2 + + 2 + 2 + = 0
represents two straight lines
equidistant from the origin,
show that
f g c bf ag
4 4 2 2
- = ( - ) 5
(b) (i) Transform the equation
14 4 11 44 58 71 0
2 2
x - xy + y - x - y + =
to rectangular axes through the
point (2, 3) and inclined at an
angle tan ( )
-1 2. 5
(ii) Find the equation to the sphere
that passes through the circle
x y z x y z
2 2 2
+ + = 1, 3 + 2 + 7 = 5
and touches the plane
y + 4z = 11. 5
22ZP/145 ( Continued )
( 5 )
(c) (i) Prove that the plane
ax + by + cz = 0 cuts the cone
xy + yz + zx = 0 in perpendicular
lines if
1 1 1
0
a b c
+ + =
5
(ii) Find the equation of the right
circular cylinder whose axis is
the line
x y z
-
= =
2 2 -3
and radius equal to 12. 5
(d) (i) A plane passes through a fixed
point (a,b, g) and cuts the
coordinate axes in A, B, C. Prove
that the locus of the centre of
the sphere OABC is given by
a b g
x y z
+ + = 2
5
(ii) Find the pole of the line
lx + my + n = 0 with respect to
the parabola y ax
2
= 4 . 5
22ZP/145 ( Turn Over )
( 6 )
(e) (i) Find the condition that the lines
y = mx and y = m¢x may be
conjugate diameters of the conic
ax hxy by gx fy c
2 2
+ 2 + + 2 + 2 + = 0
Hence find the diameter of the
conic 15 20 16 1
2 2
x - hxy + y =
conjugate to the diameter
y + 2x = 0. 3+2=5
(ii) Let
r
r = (-1, 2, 3) denotes a
position vector with length r = r
r
and
r
a is a constant vector.
Determine
div {r (a r )}
4
r r
́ and curl {r (a r )}
4
r r
́ 5
(f) (i) If
r
f = x yi - zxj + yzk
2 2 2
$ $ $
, find
div
r
f and grad
r
f . 5
(ii) A particle moves along the curve
x = 3t - 2t + 7t y = 9 2t
4 2
, tan( )
and z e
t
=
- 5
2
sin( ), where t is the
time. Determine the velocity and
acceleration at any time t and
their magnitude at t = 3. 5
22ZP/145 ( Continued )
( 7 )
(g) (i) Find the directional derivatives
of
u yz x e zx y xy z e
x z
= (3 - 2 ) + (2 - 5 ) (7 + 3 )
5
at (1, –1, 0) in the direction
2 3 4
$ $ $
i - j + k. 5
(ii) Prove that r r
(n + 2)
r
is irrotational.
Find n when it is solenoidal. 5
