Gujarat University 2007 B.Sc Mathematics FB-02 -II (Old ) - Question Paper

FB-02
Mathematics Paper-II
(Old Course)

Seat No. :




Time : three Hours] [Total Marks : 105


Instructions : (1) There are 7 ques. in this ques. paper.
(2) Attempt all ques..
(3) All ques. carry equal marks.



1. (a) Define vector space.


If –x = (x , x ),


–y = (y , y ) ? R2 and –x


+ –y


= (x


+ x , y


+ y ) and

1 2 1 2

1 2 1 2


? –x = (?x , 0) ? ? R are described in R2, is R2 a vector space ? Justify your
answer.

OR

(a) Define subspace of the vector space. If A and B are 2 subspaces of vector space V then prove that A + B is also subspace of V.

(b) Attempt any 2 :

(1) Prove that A = {(x, y, z) | x + y + z = 0} is a subspace of R3.

(2) Determine whether the vectors (2, 1, 1) and (1, 1, 2) belongs to the subspace SP {(1, 2, 1), (–1, 3, 2), (4, 5, –3)} of R3 or not.

(3) If A is a non-empty subset of a vector space V then prove that

(i) [A] = A ? A is a subspace of V (ii) [[A]] = [A]


2. (a) Define linear independence and linear dependence of vector of Rn.

Prove that every subset of a linearly independent set is linearly independent.

OR



(a) Every linearly independent subset of a finite dimensional vector space V can be
extended up to a basis of V.

(b) Attempt any 2 :

(1) Obtain co-ordinates of the vector (25, 25) of the vector space R2 with respect to the basis {(3, 4), (4, 3)}.


(2) If


–x, –y, –z are 3 linearly independent vectors of R3, prove that –x


+ –y,

–y +

–z,

–z +

–x are also linearly independent vectors.


(3) Extend the set {(1, 0, 1)} to 2 various bases of vector space R3.



3. (a) Define linear transformation. Prove that if T : U ? V is linear transformation

– – – –
? T(? x + ? y) = ? T(x) + ?T(y).

? x, y ? U, ?? ? ? R.

OR

(a) T : U ? V is linear transformation. Prove that T is one-one if and only if
N(T) = {?U}.


(b) Attempt any 2 :

(1) Prove that T : R3 ? R2, T (x, y, z) = (x + y, y + z) is linear transformation.
Also obtain N(T) and R(T).

(2) Prove that the linear transformation
T : R3 ? R3, T (x, y, z) = (x + y + z, y + z, z) is non-singular. Also obtain T –1.

(3) For linear transformation T : R2 ? R3, T(1, 1) = (2, 0, 1) and
T(2, –1) = (1, –1, 1), then obtain T(x, y). Also obtain T(2, 3).



4. (a) Define matrix associated with a linear transformation.

If T : R2 ? R2, T(x, y) = (x cos ? – y sin ?, x sin ? + y cos ?) ? ? R is linear transformation and B1 = B2 = {e1, e2} are bases in R2 then obtain [T : B1, B2].



OR

(a) In usual notations find the polar formula of a conic in R2 as l




= one + e cos ?.



(b) Attempt any 2 :

(1) Obtain the linear transformation T : R2 ? R3 so that A = [T : B1, B2]

?
where A = ?
?

1 2
0 1
–1 3

?
? and B1 = {(1, 2), (–2, 1)} and
?

B2 = {(1, –1, –1), (1, 2, 3), (–1, 0, 2)} are ordered basis of R2 and R3
respectively.

(2) Which conic is represented by the formula 15 – 3r = r cos ? ? find length of latusrectum & its Cartesian formula.

(3) Find the polar formula of the straight line passing through (2, ?/6) and
( 3, ?/3). obtain the length of perpendicular drawn from the pole upon it.



5. (a) Obtain the formula of the tangent plane to the sphere x2 + y2 + z2 = a2 at the point P(?? ?? ?) in R3.
OR
(a) Prove that the intersection of sphere and plane is a circle. (b) Attempt any 2 :
(1) For what value of ‘a’ the plane x + y + z = a touches the sphere

x2 + y2 + z2 – 2x – 2y – 2z – 13 = 0 ? find the point of contact.

(2) Prove that the spheres x2 + y2 + z2 + 4x + 4y + 4z – 13 = 0 and

x2 + y2 + z2 – 20x – 36y – 14z + 73 = 0 touch every other externally.

(3) Find centre and radius of the circle
–r two – two –r · (1, –2, 3) + three = 0, –r · (1, 5, –7) = 45.




6. (a) Obtain the formula of an enveloping cone having generator touching a sphere

x2 + y2 + z2 = a2 and passing through a point (?? ?? ?) in R3.

OR

(a) Obtain the formula of right circular cylinder having axis


x – ?
l =

y – ?
m =

z – ?
n and radius r in R3.



(b) Attempt any 2 :

(1) Obtain the formula of a cone having vertex at (?? ?? ?) and guiding curves y2 = 4ax, z = 0.

(2) Prove that the formula xy + yz + zx = 0 represents a right circular cone and also obtain its axis, the vertex and the semivertical angle.

(3) If the axis of the right circular cylinder passing through (4, –5, 3) is parallel to z-axis and passes through point (5, –2, 6) then obtain the formula of right circular cylinder.



7. (a) Obtain the condition that the plane lx + my + nz = p touches the central conicoid ax2 + by2 + cz2 = one and also obtain point of contact.

OR

(a) Obtain the formula of the tangent plane at point (?? ?? ?) to the paraboloid ax2 + by2 = 2z.

(b) Attempt any 2 :

(1) Find the formula of the tangent plane and the point of contact to the hyperboloid 5x2 - 4y2 + 7z2 = 139 of 1 sheet parallel to the plane
20x + 4y – 21z = 139.



(2) If the tangent plane to the ellipsoid


x2
a2 +


y2
b2 +


z2
c2 = one meets the co-

ordinate axes in A, B, C then prove that the locus of centroid of ? ABC is

a2
x2 +

b2
y2 +

c2
z2 = 9.


(3) Prove that for all values of ?, the plane


2x y 2z


?x 2y z ?

a + b +

c + ? ?a –

b – c –

? = one touches the conicoid


x2 y2 z2

a2 +

b2 – c2 = 1.

Earning:   Approval pending.