Bangalore University 2006 B.Sc Mathematics - Question Paper

I Sem B.SC Examination, Mar/Apr 2006
(Y2K7 scheme)(2007-08 and onwards)
MATHEMATICS
Time:3 hrs
Max marks:80
Instruction:Answer all the parts
SECTION-A
ans any ten questions(10*2=20)
1)Define symmetric and skew symmetric matrix and provide an example for every
2) obtain the chararcteristic roots of the matrix
1 4
3 2
3)State the necessary and sufficient condition for a non empty subset of group G
To be a subgroup of G
4)Find the cross product of vector a=2i+3j-2k and vector b=12i-4j+8k are
Orthogonal
5)Find the cross product of vector a=2i+3j+4k and vector b=i-k
6)The centroid of a triangle is(4,-1,1).If 2 of the vctorices are(4,-2,7) and
(5,-4,1) obtain the 3rd vertex.
7)Find the direction cosine of a line equally inclined with the co-ordinate axes
8)Find the centre and the radius of the sphere 4x2+4y2+4z2-16x-24y+16=0
9)Find the 2nd derivative of log(sin x) w.r.t. x
10)Find the nth derivative of y=sin3x
11 if z=xy obtain delta z/delt x,delta z/delta y
12)Evaluate
Integration cos tita/root of 4-sin2 tita dtita
With lower limit 0 and higher limit pie/2
13)Evaluate
Integration of 1/2x2+x-3 dx
12)Solve dy/dx=ex+y
15)Solve dy/dx +(cot x)y=1

SECTION-B

II ans any 2 questions(2*5=10)
1)solve the subsequent system of formula by matrix method:
2x+3y=8,2y-z=1,2x+3z=11
2)Verify Cayley Hemilton theorem for the
Matrix a= [3 two an hence obtain A-1
-1 4]
3)Find a matrix P such that P-1AP is a diagonal matrix where A=[3 4
-2 -3]
III Answe any 2 questions(2*5=10)
1)Write the multiplication modulo six table for the set G={1,2,3,4,5} is(G,X6) a
Group?Give cause
2)Show that the set of all 4th root of unity form a group under usual
Multiplication
3)Find the area of the parallelogram whose diagonals are 3i-2j-k and -4i+2j+k
4)Prove that[vectora*b;vectorb*c;vectorc*a]=[vectora,vectorb,vectorc]2
IV ans any 2 questions(2*5=10)
1)Find the angle ranging from the diagonals of a cube
2)Find the formula of the plane passing through the point (-2,4,0)(-3,2,-9)
And parallel to the line whose direction ratio are 3,-2,3
3)Derive the formula of a sphere wi9th a centre (x1,y1,z1) and radius R
4)Find the formula of the right circular cone whose vertex is(2,3,4) axis
Makes equal angles with the coordinatues axes and semivertical angle is 300

V ans any four (4*5=20)
1)Find the nth drivative of cos4x
2) If y=sin-1x ,show that
(1-x2)yn+2-(2n+1)xyn+1-n2yn=0
3) If u=log(x4+y4/x-y), show that x *delta u/delta x +y*delta u/delta y=3
4)Evaluate the integration of(x-1)/(x-2)(x-3)dx
5) Evaluate the integration of 1/(3-cos x) dx
6)Evaluate the integration of x *sin3x*cos x dx
7) Evaluate the integration sec4tita d tita with upper limits pie/4 and lower limits 0

VI Answe any 2 (2*5=10
1)Solve y dy/dx=ex+y
2)Solve dy/dx=x2+y2/xy
3)Verify the exactness and solve the formula
(5x4+3x2y2-2xy3)dx+(2x3y-3x2y2-5y4)dy=0
4)Solve cos2x dy/dx +y=tan x

Earning:   Approval pending.