B.Sc-B.Sc Mathematics 1st Sem Algebra and Geometry(University of Pune, Pune-2013)

                                         F.Y. B.Sc.

                                                                  MATHEMATICS

                                                   Algebra and Geometry

SEAT No. :

[Total No. of Pages : 3

(2008 Pattern) (Paper - I)

Time : 3 Hours]                                                                                                [Max. Marks :80

Instructions to the candidates:

1) All questions are compulsory.

2) Figures to the right indicate full marks.

Q1) Attempt all the subquestions:

a) Define power set of a set. Let A = {a, b, c}. Find power set of A.

f(10) -f(3)

b) Find the value of   f(4) wheref is Euler’s phi-function.

Find the quotient and remainder when 3x4 + 6x3 +8x2- 2x -3 is divided

by 2x3 + x2-9 .

Find the centre of the conic 3x2 + 2xy + 3y2- 4x + 2 y +1= 0 .

Obtain the equation of line joining the points (1, 2, 3) and (–2, 1, –2).

Find the equation of the sphere on AB as a diameter where A (2, –3, 1)

and B (–1, –2, 4).

A=⎡2 4⎤

h) Reduce the matrix

⎢3 2⎥ to row echelon form. Hence find its rank.

⎣

Q2) Attempt any four of the following:

1. Let f :R ®R be a function defined by f = 5x - 2 . Show that the
2. Let R be defined on the set of integers Z by xRy⇒5x + 6 y is divisible

function f is bijective. Also find a formula for f

by 11, for x, yÎZ . Show that R is an equivalence relation.

c)       If P is prime and a,b are integers. Show that P ab then P a or P b .

d)       In Z12 , Calculate

1. (+1)-1.
2. - 5( 4 + 5).
3. Find the modulus and argument of z =(-1+ i)3.

f)        Find the values of a and b if 2 and –3 are the roots of the equation

2 x 4 + 3 x 3- 1 2 x 2 + ax + b = 0 .

Q3) Attempt any two of the following:                                                                   [16]

If a and b are any two integers with a ¹ 0 then prove that there exist

unique integers q and r such that b = aq + r, where 0£ r < a .

State and prove De Moivre’s theorem.

Find greatest common divisor of 3587 and 1819 and express it in the

form 3587 m + 1819 n.

i) For any two complex numbers z1 and z2, show that z1 + z2 2 + z1 - z2 2 = 2 z1 2 + z2 2 .

1. 1. Solve the equation 2x3- 7x2 + 7x - 2 = 0; given that the two roots

are reciprocals of each other.

Q4) Attempt any four of the following:                                                                  [16]

a) If under rotation of axes, without shifting the origin, the expression

ax2 + 2hxy + by 2 is transformed to a′x′2 + 2h′x′y′ + b′ y′2 then show that

a + b = a′ + b′ .

• cos2a + cos2 b + cos2 g =1.

Find the point where the line passing through the point (0, –1, 2) and

having direction ratios (2, –1, 3) meets the plane x- y - 2z = 0 .

2

d)       Find the equation of the sphere passing through the circle

x2 + y2 + z2 + 2x- 2 y - 2z -1= 0; 2x - 2 y + z -1= 0 and passing through

the point (3, –1, 1).

e)  Find the points at which the line

                  x-7= y-6= z +5

x 2 + y 2 + z 2- 2 x + 3 y - 5 z - 31= 0 .

⎡1 3 4 3⎤

Reduce the matrix A =⎢3 9 12 9⎥ to row echelon form. Hence find

                                               ⎢

                                              ⎥⎢1 3 4 1⎥     its rank.

Q5) Attempt any two of the following:                                                                   [16]

1. Reduce the equation 5x2 + 6xy + 5y2-10x - 6 y - 3= 0 to the standard

form and name the conic.

b)       i) Derive equation of the plane in the normal form.

x-1 y - 8 z - 2                 x +1 y- 2 z + 4

c) Show that the two lines-1 = 7 = 2 and 1 = -1 = 1 are coplanar and find the equation of plane containing them.

Show that for every real numberl the equation S+ lU = 0  represents a sphere containing the circle of intersection of the sphere Sº x2 + y2 + z2 +2ux+2vy+2wz + d =0 and U º ax + by + cz + d′= 0 . Show that the plane 2x – 2y + z + 12 = 0 touches the sphere x2 + y2 + z2- 2x - 4 y + 2z =3 . Also find the point of contact.

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