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

FB-02
Mathematics Paper-II (New Course)

Seat No. :


Time : three Hours] [Total Marks: 105


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


1. (a) Define adjoint of a matrix. If A = [aij]n is a square matrix of order n, prove that
A (adj A) = (adj A) A = |A| In.
OR
(a) Define Transpose of a matrix.
If A is m ? n matrix and B is n ? p matrix then prove that
(AB)T = BT AT.
(b) Define rank of a matrix.
? 3 2 0 -1 ?

obtain the rank of a matrix

? 1 -1 2 2 ? .



(c) Attempt any 1 :

? 0 1 -3 -1 ?

(1) Define symmetric and skew-symmetric matrices.



Express the matrix A =


symmetric matrices.

? 2 3 1 ?
? 4 3 1 ?
? ?
? 11 2 4 ?



as a sum of symmetric and skew-



(2) For the matrix A =

? 2 1 4 ?
? 4 3 1 ? , find A–1.
? ?
? 1 2 4 ?



2. (a) Define eigen value and eigen vector of a square matrix. obtain the eigen values
and corresponding eigen vectors of the matrix

?
A = ?
?

1 0 0 ?
2 1 0 ?
?
3 0 0 ?




?
(b) Obtain the characteristic formula of the matrix A = ?
?

2 1 1 ?
0 1 0 ? , and also obtain
?
1 1 2 ?

the matrix represented by the expression
A8 – 5A7 + 7A6 – 3A5 + A4 – 5A3 + 8A2 – 2A + I.
(c) Using Crammer’s rule, find the solution of the system of linear equations provided by
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 11z = 5

OR
2. (a) State and prove Caley-Hamilton theorem.
(b) Discuss the consistency of the subsequent system of linear equations, obtain its solution using row-reduction method.
x + y + z = 3
x + 2y + 3z = 4
x + 4y + 9z = 6


?
(c) Using Caley-Hamilton theorem obtain the inverse of the matrix ?
?
?


7 –1 3 ?
6 1 4 ? .
?
2 4 8 ?


3. (a) Explain Ferrari’s method to solve a Bi-Quadratic formula
ax4 + 4bx3 + 6cx2 + 4dx + e = 0.
a, b, c, d, e ? R, a ? 0
OR
(a) Explain Cardon’s method to solve cubic formula ax3 + 3bx2 + 3cx + d = 0.
a, b, c, d ? R, a ? 0


(b) Attempt any 2 :
(1) Solve the equation x4 – 10x3 + 44x2 – 104x + 96 = 0 using Ferrari’s method.
(2) Solve the formula x3 + 6x2 – 12x + 32 = 0 using Cardon’s method.
(3) If ?? ?? ? are the roots of the formula x3 – 12x + 16 = 0 then find the equations whose roots are
(i) (? – ?)2, (? – ?)2, (? – ?)2 and
(ii) ? (? + ?), ? (? + ?), ? (? + ?).



l
4. (a) In usual notations find the polar formula r = one + e cos ? of a conic in R2.

OR

(a) Obtain the polar formula of circle having centre at (?, ?) and radius ‘a’. If circle passes through pole, then what is its formula ?

(b) Attempt any 2 :

(1) If P'SP and Q'SQ are mutually perpendicular focal chords of conic, then


prove that

1 SP·SP' +

1
SQ·SQ' = constant.


(2) Prove that the formula r = a cos ? + b sin ? (a, b non-zero constants)
represents a circle. Also obtain its centre and radius.

(3) Find Cartesian and spherical co-ordinates of a point whose cylindrical co-
?
ordinates are (4, four , 4).


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

–r2 – 2–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 and radius r in R3.


x – ?
l =


y – ?
m =


z – ?
n



(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 semi-vertical 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 their 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 = 19.


x2
(2) If the tangent plane to the ellipsoid 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 – ? = 1



touches the conicoid


x2
a2 +


y2
b2 –


z2
c2 = 1.

Earning:   Approval pending.